tag:blogger.com,1999:blog-70669578776947322012024-03-16T17:56:25.520+05:30Seminar Series: Topics in Special Functions and Number TheoryOrganizers: Gaurav Bhatnagar (Ashoka University) , Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU). Contact: sfandnt@gmail.comGaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.comBlogger87125tag:blogger.com,1999:blog-7066957877694732201.post-54935231919290259532024-03-16T17:55:00.005+05:302024-03-16T17:55:40.389+05:30Gaurav Bhatnagar (Ashoka University) -- Thursday, Mar 21, 2024 - 4:00 PM (IST)<p><span style="font-family: inherit; font-size: medium;"> Dear all,</span></p><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;">The next talk is by Gaurav Bhatnagar of Ashoka University. The announcement is as follows. <br /></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><div><span style="font-family: inherit; font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title:</b> Elliptic enumeration and identities<br /></span></div><div><span style="font-family: inherit; font-size: medium;"><b>Speaker:</b> Gaurav Bhatnagar <span class="gmail_sendername" dir="auto" style="color: #222222;">(Ashoka University, )</span><br /><b>When:</b> Mar 21, 2024, 4:00 PM- 5:00 PM IST <b><br /></b></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span><div><span style="font-family: inherit; font-size: medium;"><b>Where</b>: Zoom: Please write to the organisers for the link.</span></div><div><span style="font-family: inherit; font-size: medium;">Live LInk: https://youtube.com/live/cpqHK-R2oXg?feature=share</span></div><div><b><span style="font-family: inherit; font-size: medium;"><br /></span></b></div><div><b><span style="font-family: inherit; font-size: medium;">Abstract</span></b></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;">Many of the ideas of $q$-counting and $q$-hypergeometrics are now being extended to the elliptic case. The approach is not very far from the $q$-case. In this talk, we show several examples to illustrate this idea. First we extend some Fibonacci identities using combinatorial methods. Many such identities can be found by telescoping, so we next use telescoping to find <span style="background-color: white;">elliptic extensions of elementary identities such as the sum of the first<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white;"><span class="gmail-Apple-converted-space"> </span>odd or even numbers, the geometric sum and the sum of the first<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white;"><span class="gmail-Apple-converted-space"> </span>cubes. </span><span style="background-color: white;">In the course of our study, we obtained an identity with many parameters, which appears to be new even in the<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white;">$q$-case. Finally, we introduce elliptic hypergeometric series and give an extension of some important identities of Liu. As applications, we find 5 double summations and 4 new elliptic transformation formulas. Again, these are new in the $q$-hypergeometric case, where the nome $p$ is 0. </span></span></div><div><span style="background-color: white;"><span style="font-family: inherit; font-size: medium;"><br /></span></span></div><div><span style="background-color: white;"><span style="font-family: inherit; font-size: medium;">This is a report of joint work with Archna Kumari and Michael Schlosser. </span><br /></span></div><div><span style="background-color: white; font-family: sans-serif; font-size: 14px;"></span></div></div></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-52750128823128044142024-03-02T19:59:00.006+05:302024-03-16T17:53:22.487+05:30Shivani Goel (IIIT, Delhi) - Thursday, Mar 7, 2024 - 4:00 PM (IST)<p> Dear all,</p><div><br /></div><div>The next talk is by Shivani Goel, of the Indraprastha Institute of Information Technology (IIIT), Delhi. The announcement is as follows. <br /></div><div><br /></div><div><div><b>Talk Announcement: </b><br /><br /><b>Title:</b> <b style="color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: 12.8px;"></b><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">Distribution and applications of Ramanujan sums</span><b></b></div><div><b>Speaker:</b> Shivani Goel<span class="gmail_sendername" dir="auto" face="Arial, Helvetica, sans-serif" style="color: #222222;"> (IIIT, Delhi)</span><br /><b>When:</b> Mar 7, 2024, 4:00 PM- 5:00 PM IST <b><br /></b></div><div><br /><div><b>Where</b>: Zoom: Ask the organisers for the link.</div><div>Live LInk: https://youtube.com/live/mQ9EiVeqimI?feature=share</div><div><b><br /></b></div><div><b>Abstract</b></div><div><div style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;">While studying the trigonometric series expansion of certain arithmetic functions, Ramanujan, in 1918, defined a sum of the $n^{th}$ power of the primitive $q^{th}$ roots of unity and denoted it as $c_q(n)$. These sums are now known as Ramanujan sums.<br /><br />Our focus lies in the distribution of Ramanujan sums. One way to study distribution is via moments of averages. Chan and Kumchev initially considered this problem. They estimated the first and second moments of Ramanujan sums. Building upon their work, we extend the estimation of the moments of Ramanujan sums for cases where $k\ge 3$. Apart from this, We derive a limit formula for higher convolutions of Ramanujan sums to give a heuristic derivation of the Hardy-Littlewood formula for the number of prime $k$-tuplets less than $x$. </div><br class="gmail-Apple-interchange-newline" /><br /></div></div></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/mQ9EiVeqimI?si=5y8RD3-cK3eFo_hu" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-90478426604800924172024-02-19T17:00:00.007+05:302024-03-02T20:02:32.337+05:30Pedro Ribeiro (Porto, Portugal) - Thursday, Feb 22, 2024 - 4:00 PM (IST)<p><span style="font-size: medium;"> Dear all,</span></p><div class="Ar Au Ao" id=":2pl" jslog="171097"><div aria-controls=":2rv" aria-expanded="false" aria-label="Message Body" aria-multiline="true" aria-owns=":2rv" class="Am aiL Al editable LW-avf tS-tW tS-tY" g_editable="true" hidefocus="true" id=":2ph" itacorner="6,7:1,1,0,0" role="textbox" spellcheck="false" style="direction: ltr; min-height: 214px;" tabindex="1"><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">This week's talk is by <span class="gmail_sendername" dir="auto" face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;">Pedro Ribeiro</span> of the University of Porto, Portugal. We apologize for the late notification. The talk announcement follows. <br /></span></div><div><span style="font-size: medium;"><br /></span></div><div><div><span style="font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title:</b> <b style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"></b><span face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;">Generalizations (in the spirit of Koshliakov) of some formulas from Ramanujan's Lost Notebook</span><b></b></span></div><div><span style="font-size: medium;"><b>Speaker:</b> <span class="gmail_sendername" dir="auto" face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;">Pedro Ribeiro (University of Porto, Portugal)</span> <br /><b>When:</b> Feb 22, 2024, 4:00 PM- 5:00 PM IST (10:30 am Western European Time (WET)) <b><br /></b></span></div><div><span style="font-size: medium;"><br /></span><div><span style="font-size: medium;"><b>Where</b>: Zoom:</span></div><div><span style="font-size: medium;">Live Link: <a href="https://youtube.com/live/yq4SVSlTD2o?feature=share">https://youtube.com/live/yq4SVSlTD2o?feature=share</a></span></div><div><span style="font-size: medium;"><br /></span></div><div><b><span style="font-size: medium;">Abstract</span></b></div><div><span style="font-size: medium;"><span face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;">In his lost notebook, Ramanujan recorded beautiful identities. These include earlier versions of Guinand's formula for the divisor function and the transformation formula for the logarithm of Dedekind's </span><img alt="\eta" class="gmail-CToWUd" height="11" src="https://blogger.googleusercontent.com/img/proxy/AVvXsEhxh5VEtWVUURn2i4SFJfWgKiDXhbwYhSICJ03NH7z7XExeQALdKk4jjQfhzDWAjgseJwU0rkrsLyDwlxmEeENpbuQIxbeUQibiFOt8Ej24N2lkAgqVtFJFFWV3cwClHTWBx5G1MsSO8TEOSPFltYjxXSGujGaHlXGg0UxZ6QejkLDkkhwPDHM=s0-d-e1-ft&bg=ffffff&fg=000000&s=0&latex=%5Ceta" style="caret-color: rgb(34, 34, 34); color: #222222; display: inline; font-family: Arial, Helvetica, sans-serif; vertical-align: -4px;" width="7" /><span face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;">-function. </span><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><span face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;">In our presentation we will describe some generalizations of these formulas using a beautiful theory due to the forgotten mathematician N. S. Koshliakov. Our work will be presented under the point of view initiated by A.<span class="gmail-Apple-converted-space"> </span><span class="gmail-il">Dixit</span><span class="gmail-Apple-converted-space"> </span>and R. Gupta, the first mathematicians of our century who have extended Koshliakov's theory in several directions. </span><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /><br style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;" /></span><span face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="font-size: medium;">This talk is based on joint work with Semyon Yakubovich. </span><br /></span></div></div></div><div><span face="Arial, Helvetica, sans-serif" style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="font-size: medium;"><br /></span></span></div></div></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/yq4SVSlTD2o?si=eXAtyeVtu7yrBMT8" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-66878806428067341652024-02-06T13:59:00.004+05:302024-02-19T17:02:21.863+05:30Arvind Ayyer (IISc.) - Thursday, Feb 8, 2024 - 4:30 PM (IST)<p><span style="font-family: inherit; font-size: medium;"> Dear all,</span></p><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;">The next talk is by Arvind Ayyer of the Indian Institute of Science, Bangalore, India. The talk announcement follows. Please note that the talk is half an hour later than our usual meeting time. <br /></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><div><span style="font-family: inherit; font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title:</b> <span style="background-color: white; color: #222222;">A new combinatorial formula for the modified Macdonald polynomials</span><b><br /></b></span></div><div><span style="font-family: inherit; font-size: medium;"><b>Speaker:</b> Arvind Ayyer (IISc, Bangalore, India)<br /><b>When:</b> Feb 8, 2024, 4:30 PM- 5:30 PM IST (<b>Note special time) </b></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span><div><span style="font-family: inherit; font-size: medium;"><b>Where</b>: Zoom: Please write to the organisers for the link</span></div><div><span style="font-family: inherit; font-size: medium;"><b>Live Link:</b> <a data-saferedirecturl="https://www.google.com/url?q=https://youtube.com/live/udOYGrz0zaI?feature%3Dshare&source=gmail&ust=1707294255905000&usg=AOvVaw1YRe6ZHZ45R-sHeHliinnM" href="https://youtube.com/live/udOYGrz0zaI?feature=share" target="_blank">https://youtube.com/live/<wbr></wbr>udOYGrz0zaI?feature=share</a></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><b><span style="font-family: inherit; font-size: medium;">Abstract</span></b></div><div><span style="font-family: inherit; font-size: medium;"><span style="background-color: white; color: #222222;">Macdonald polynomials are a remarkable family of symmetric</span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">functions that are known to have connections to combinatorics, algebraic</span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">geometry and representation theory. The modified Macdonald polynomials </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">are obtained from the Macdonald polynomials using an operation called </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">plethysm. A combinatorial formula for the latter was given by Haglund, </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">Haiman and Loehr in a celebrated work (JAMS, 2004). We will give a new </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">combinatorial formula (ALCO 2023).</span><br style="color: #222222;" /><br style="color: #222222;" /><span style="background-color: white; color: #222222;">Recently, a formula for the symmetric Macdonald polynomials was given by </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">Corteel, Mandelshtam and Williams in terms of objects called multiline </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">queues, which also compute probabilities of a statistical mechanics </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">model called the multispecies ASEP on a ring. It is natural to ask </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">whether the modified Macdonald polynomials can be obtained using a </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">combinatorial gadget for some other statistical mechanics model. We </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">answer this question in the affirmative via a multispecies totally </span><br style="color: #222222;" /><span style="background-color: white; color: #222222;">asymmetric zero-range process (TAZRP) in (arXiv:2209.09859).</span><br style="color: #222222;" /><br style="color: #222222;" /><span style="background-color: white; color: #222222;">These are joint works with J. Martin and O. Mandelshtam.</span></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span><br /></div></div></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/udOYGrz0zaI?si=rGH-5ouFgUAoRSU7" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-2485219632761765352024-01-19T07:01:00.003+05:302024-01-25T20:51:34.112+05:30Ramanujan Special: Frank Garvan (Florida) - Thursday, Jan 25, 2024 - 7:30 PM (IST)<p><span style="font-family: inherit; font-size: medium;"> Happy new year. </span></p><div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;">The first talk of the year (on January 25, 2023) is a ``Ramanujan Special". This year's speaker is Frank Garvan. Please note that the talk will be later than usual. A report on the activities of this seminar in 2023 appears in the SIAM newsletter OPSFNET. We hope this year is equally exciting for our group. Please consider the seminar to present your latest preprint. <br /></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div></div><div><span style="font-family: inherit; font-size: medium;"><b>Talk Announcement: The 2024 Ramanujan Special</b><br /><br /><b>Title:</b> Identities for Ramanujan's Mock Theta Functions and Dyson's Rank</span><p class="MsoNormal" style="color: #222222; margin: 0px;"><span style="font-family: inherit; font-size: medium;">Function</span></p><span style="font-family: inherit; font-size: medium;"><b>Speaker:</b> Frank Garvan (University of Florida, USA)<br /><b>When:</b> Jan 25, 2024, 7:30 PM- 8:30 PM IST (9 AM EST) (<b>Note special time) </b></span></div><div><span style="font-family: inherit; font-size: medium;"><b>(</b>EST= IST - 10:30)<br /></span><div><span style="font-family: inherit; font-size: medium;"><b>Where</b>: Zoom: Please write to the organisers for the link.</span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;">Live LInk: <a data-saferedirecturl="https://www.google.com/url?q=https://youtube.com/live/KWUjDWfjFwg?feature%3Dshare&source=gmail&ust=1705714186623000&usg=AOvVaw2EM4sB4N1lJkuKocyv1Pzz" href="https://youtube.com/live/KWUjDWfjFwg?feature=share" target="_blank">https://youtube.com/live/<wbr></wbr>KWUjDWfjFwg?feature=share</a></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><b><span style="font-family: inherit; font-size: medium;">Abstract<br /></span></b></div></div><span style="font-size: medium;"><span style="font-family: inherit;">In Ramanujan's Lost Notebook there are identities connecting </span><span style="color: #222222; font-family: inherit;">Ramanujan's fifth order mock theta functions and Dyson's rank mod 5. </span><span style="color: #222222; font-family: inherit;">We extend these connections to Zagier's higher order mock theta functions. </span><span style="color: #222222; font-family: inherit;">We consider Dyson's problem of giving a group-theoretic structure to</span></span><div><p class="MsoNormal" style="color: #222222; margin: 0px;"><span style="font-size: medium;"><span style="font-family: inherit;">the mock theta functions analogous to Hecke's theory of modular forms.<u> </u></span><span style="font-family: inherit;">From this much surprising symmetry and q-series identities arise in joint </span><span style="font-family: inherit;">work with Rishabh Sarma and Connor Morrow.</span></span></p><p class="MsoNormal" style="color: #222222; margin: 0px;"><span style="font-family: inherit; font-size: large;"><br /></span></p><p class="MsoNormal" style="color: #222222; margin: 0px;"><span style="font-family: inherit; font-size: large;"><br /></span></p></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/KWUjDWfjFwg?si=h43oJ_lXwzf85oyQ" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-26784453624601760962023-12-04T13:30:00.007+05:302023-12-07T21:31:18.061+05:30David Bradley (Maine) - Thursday Dec 7, 2023 - 6:30 PM (IST) (NOTE. Special Time)<p> <span face="arial, sans-serif" style="font-size: large;">Dear all,</span></p><div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;">The talk this week is </span><span face="arial, sans-serif" style="font-size: medium;">by David Bradley of the University of Maine. Please note the special time. Since Professor Bradley is located in the US, we are starting later than usual. <br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;">This will be the final talk of the year. We will come back next year with a Ramanujan special, and hope that we get an opportunity to meet in person in the upcoming conference and holiday season. We wish you happy holidays and a great new year. <br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;">The announcement is as follows. <br /></span></div></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Talk Announcement: </b></span><span face="arial, sans-serif" style="font-size: medium;"><br /></span><div><span face="arial, sans-serif" style="font-size: medium;"><b><br /></b></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Title:</b> </span><span face="arial, sans-serif" style="font-size: medium;">On Fractal Subsets of Pascal's "Pyramid" and the Number of </span><span face="arial, sans-serif" style="font-size: medium;">Multinomial Coefficients Congruent to a Given Residue Modulo a Prime</span><span style="font-size: medium;"><span face="arial, sans-serif"><br class="gmail-Apple-interchange-newline" /><b><br /></b></span></span></div><div><span style="font-size: medium;"><span face="arial, sans-serif"><b>Speaker:</b> David Bradley (University of Maine, USA)<br /><b><br /></b></span></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>When:</b> Thursday, Thursday Dec 7 23, 2023 - 6:30 PM (IST) (8AM EST/ 2PM (CET))<br /><br /></span><div><span face="arial, sans-serif" style="font-size: medium;"><b>Where</b>: Zoom: Ask the organisers for a link</span></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Live Link: </b></span><span style="font-size: medium;"><span face="arial, sans-serif"><a href="https://youtube.com/live/jXpB3U41ELs?feature=share">https://youtube.com/live/jXpB3U41ELs?feature=share</a> <br /></span></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b><br /></b></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Abstract. </b></span><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><div style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="font-size: medium;"><span face="arial, sans-serif" style="color: black;">We obtain an explicit formula and an asymptotic formula for the number of multinomial coefficients which are congruent to a given residue modulo a prime, and which arise in the expansion of a multinomial raised to any power less than a given power of that prime. Each such multinomial coefficient can be associated with a certain Cartesian product of intervals contained in the unit cube. For a fixed prime, the union of these products forms a set which depends on both the residue and the power of the prime. In the limit as the power of the prime increases to infinity, the sequence of unions converges in the Hausdorff metric to a non-empty compact set which is independent of the residue. We calculate the fractal dimension of this limiting set, and consider its monotonicity properties as a function of the prime. </span><span face="arial, sans-serif" style="color: black;">To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first </span><span face="arial, sans-serif" style="color: black;"><i>p</i> </span><span face="arial, sans-serif" style="color: black;">rows of Pascal’s "pyramid" which are congruent to a given nonzero residue<i> </i></span><span face="arial, sans-serif" style="color: black;"><i>r</i> </span><span face="arial, sans-serif" style="color: black;">modulo the prime<i> </i></span><span face="arial, sans-serif" style="color: black;"><i>p</i></span><span face="arial, sans-serif" style="color: black;">. Unfortunately, numerical computations with large prime moduli suggests that if there is such a formula, it is extremely complicated. Nevertheless, the evidence indicates that for sufficiently large primes </span><span face="arial, sans-serif" style="color: black;"><i>p</i></span><span face="arial, sans-serif" style="color: black;">, the number of<i><span class="gmail-Apple-converted-space"> </span><b>binomial</b></i><span class="gmail-Apple-converted-space"> </span>coefficients in Pascal's<span class="gmail-Apple-converted-space"> </span><i><b>triangle</b></i> which are congruent to </span><span face="arial, sans-serif" style="color: black;"><i>r</i> </span><span face="arial, sans-serif" style="color: black;">mod </span><span face="arial, sans-serif" style="color: black;"><i>p</i> </span><span face="arial, sans-serif" style="color: black;">for </span><span face="arial, sans-serif" style="color: black;"><i>r</i> </span><span face="arial, sans-serif" style="color: black;">= 1, 1 </span><span face="arial, sans-serif" style="color: black;"><<span class="gmail-Apple-converted-space"> </span><i>r</i><span class="gmail-Apple-converted-space"> </span><<span class="gmail-Apple-converted-space"> </span><i>p</i></span><span face="arial, sans-serif" style="color: black;">−</span><span face="arial, sans-serif" style="color: black;">1, and<i> </i></span><span face="arial, sans-serif" style="color: black;"><i>r</i> </span><span face="arial, sans-serif" style="color: black;">= </span><span face="arial, sans-serif" style="color: black;"><i>p</i></span><span face="arial, sans-serif" style="color: black;">−</span><span face="arial, sans-serif" style="color: black;">1 is well approximated by the respective linear functions of<span class="gmail-Apple-converted-space"> </span><i>p</i><span class="gmail-Apple-converted-space"> </span>given </span></span></div><div style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="font-size: medium;"><span face="arial, sans-serif" style="color: black;">by 3</span><span face="arial, sans-serif" style="color: black;"><i>p</i></span><span face="arial, sans-serif" style="color: black;">, </span><span face="arial, sans-serif" style="color: black;"><i>p</i>/</span><span face="arial, sans-serif" style="color: black;">2, and<i> </i></span><span face="arial, sans-serif" style="color: black;"><i>p</i></span><span face="arial, sans-serif" style="color: black;">. </span><span face="arial, sans-serif" style="color: black;">In particular, for large primes </span><span face="arial, sans-serif" style="color: black;"><i>p</i> </span><span face="arial, sans-serif" style="color: black;">there are approximately six times as many occurrences of the residue 1 in the first </span><span face="arial, sans-serif" style="color: black;"><i>p</i> </span><span face="arial, sans-serif" style="color: black;">rows of Pascal’s triangle reduced modulo </span><span face="arial, sans-serif" style="color: black;"><i>p</i> </span><span face="arial, sans-serif" style="color: black;">than there are of any other residue </span><span face="arial, sans-serif" style="color: black;"><i>r </i></span><span face="arial, sans-serif" style="color: black;">in the range 1 </span><span face="arial, sans-serif" style="color: black;"><<i><span class="gmail-Apple-converted-space"> </span>r<span class="gmail-Apple-converted-space"> </span></i><<span class="gmail-Apple-converted-space"> </span><i>p</i> </span><span face="arial, sans-serif" style="color: black;">− </span><span face="arial, sans-serif" style="color: black;">1, and three times as many as<i> </i></span><span face="arial, sans-serif" style="color: black;"><i>r</i> </span><span face="arial, sans-serif" style="color: black;">= </span><span face="arial, sans-serif" style="color: black;"><i>p</i> </span><span face="arial, sans-serif" style="color: black;">− </span><span face="arial, sans-serif" style="color: black;">1. On the other hand, if we let the nonnegative integer </span><span face="arial, sans-serif" style="color: black;"><i>k</i> </span><span face="arial, sans-serif" style="color: black;">vary while keeping the prime </span><span face="arial, sans-serif" style="color: black;"><i>p</i> </span><span face="arial, sans-serif" style="color: black;">fixed, and look at the relative frequency of various residue classes that occur in the first </span><span face="arial, sans-serif" style="color: black;"><i>p</i></span><span face="arial, sans-serif" style="color: black; vertical-align: 4pt;"><i>k</i> </span><span face="arial, sans-serif" style="color: black;">rows, the seemingly substantial differences in frequency between </span><span face="arial, sans-serif" style="color: black;"><i>r</i> </span><span face="arial, sans-serif" style="color: black;">= 1, 1 </span><span face="arial, sans-serif" style="color: black;"><<i><span class="gmail-Apple-converted-space"> </span>r<span class="gmail-Apple-converted-space"> </span></i><<span class="gmail-Apple-converted-space"> </span><i>p</i></span><span face="arial, sans-serif" style="color: black;">−</span><span face="arial, sans-serif" style="color: black;">1, and </span><span face="arial, sans-serif"><i><span style="color: black;">r </span><span style="color: black;">= </span><span style="color: black;">p</span></i><span style="color: black;">−</span><span style="color: black;">1 when </span><span style="color: black;"><i>k</i> </span><span style="color: black;">= 1 are increasingly dissipated as </span><span style="color: black;"><i>k</i> </span><span style="color: black;">grows without bound. We show that in the limit as </span><span style="color: black;"><i>k</i> </span><span style="color: black;">tends to infinity, </span><span style="color: black;">all </span><span style="color: black;">nonzero residues are equally represented with asymptotic proportion 1</span><span style="color: black;">/</span><span style="color: black;">(</span><span style="color: black;"><i>p</i> </span><span style="color: black;">− </span><span style="color: black;">1).</span></span></span></div></div><div style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="font-size: medium;"><span face="arial, sans-serif"><span style="color: black;"><br /></span></span></span></div><div style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="font-size: medium;"><span face="arial, sans-serif"><span style="color: black;"><br /></span></span></span></div><div><span face="arial, sans-serif" style="font-size: medium;"></span></div></div></div> <div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/jXpB3U41ELs?si=-SfakqQKTrctbWbc" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-7705753163989261032023-11-19T09:39:00.004+05:302023-12-04T13:34:00.108+05:30Sonika Dhillon (ISI, Delhi) - Thursday, Thursday Nov 23, 2023 - 4:00 PM (IST) <p> <span face="arial, sans-serif" style="font-size: large;">Dear all,</span></p><div><div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;">The talk this week is by Sonika Dhillon, </span><span face="arial, sans-serif" style="font-size: medium;">ISI, Delhi. The announcement is as follows. <br /></span></div></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Talk Announcement: </b></span><span face="arial, sans-serif" style="font-size: medium;"><br /></span><div><span face="arial, sans-serif" style="font-size: medium;"><b>Title:</b> </span><span face="arial, sans-serif" style="font-size: medium;">Linear independence of numbers<br /><b>Speaker:</b> Sonika Dhillon (ISI, Delhi)<br /><b>When:</b> Thursday, Thursday Nov 23, 2023 - 4:00 PM (IST) <br /><br /></span><div><span face="arial, sans-serif" style="font-size: medium;"><b>Where</b>: Zoom: Write to the organisers for the link</span></div><div><div><span face="arial, sans-serif" style="font-size: medium;"><b><br /></b></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Live Link:</b> </span><span face="arial, sans-serif" style="font-size: medium;"><a data-saferedirecturl="https://www.google.com/url?q=https://youtube.com/live/JSqZ8FLhKdU?feature%3Dshare&source=gmail&ust=1700453258272000&usg=AOvVaw0AOqQudHgzHjJ3NFGJ-Ezt" href="https://youtube.com/live/JSqZ8FLhKdU?feature=share" target="_blank">https://youtube.com/live/<wbr></wbr>JSqZ8FLhKdU?feature=share</a></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b><br /></b></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Abstract. </b></span></div><span face="arial, sans-serif" style="font-size: medium;">Let $\psi(x)$ denote the digamma function that is the logarithmic derivative of $\Gamma$ function.<br />In 2007, Murty and Saradha studied the linear independence of special values of digamma function $\psi(a/q)+\gamma$ over some specific numbers fields which also imply the non-vanishing of $L(1,f)$ for any rational-valued Dirichlet type function $f$. In 2009, Gun, Murty and Rath studied the non-vanishing of $L'(0,f)$ for even Dirichlet-type periodic $f$ in terms of $L(1,\hat{f})$ and established that this is related to the linear independence of logarithm of gamma values. In this direction, they made a conjecture which they call it as a variant of Rohrlich conjecture concerning the linear independence of logarithm of gamma values. In this talk, first we will discuss the linear independence of digamma values over the field of algebraic numbers. Later, we provide counterexamples<br />to this variant of Rohrlich conjecture.</span></div></div></div></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/JSqZ8FLhKdU?si=A3bsBZ_dx2oH3SfA" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-52801974978767127472023-11-05T09:14:00.005+05:302023-11-19T09:41:53.203+05:30Sagar Shrivastava (TIFR, India) - Thursday, Thursday Nov 9, 2023 - 4:00 PM (IST) <p><span style="font-size: medium;"> <span face="arial, sans-serif">Dear all,</span></span></p><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;">The talk in the coming week is by Sagar Shrivastava, School of Mathematics, Tata Institute of Fundamental Research (TIFR).<br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Talk Announcement: </b></span><span face="arial, sans-serif" style="font-size: medium;"><br /></span><div><span face="arial, sans-serif" style="font-size: medium;"><b>Title:</b> </span><span face="arial, sans-serif" style="font-size: medium;"><span style="background-color: white; color: #222222; font-variant-ligatures: normal;">Representations, Determinants and Branching rules</span><br /><b>Speaker:</b> Sagar Shrivastava (TIFR, India)<br /><b>When:</b> Thursday, Thursday Nov 9, 2023 - 4:00 PM (IST) <br /><br /></span><div><span face="arial, sans-serif" style="font-size: medium;"><b>Where</b>: Zoom: Please write to the organisers for the link</span></div><div><span face="arial, sans-serif" style="font-size: medium;"><br /></span></div><div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Live Link:</b> </span><span face="arial, sans-serif" style="font-size: medium;"><a href="https://youtube.com/live/S4DQRvK46no?feature=share" target="_blank">https://youtube.com/live/S4DQRvK46no?feature=share</a><br /></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b><br /></b></span></div><div><span face="arial, sans-serif" style="font-size: medium;"><b>Abstract. </b></span></div><span face="arial, sans-serif" style="font-size: medium;"><span style="background-color: white; color: #222222; font-variant-ligatures: normal;">Branching rules/laws (restriction of representations) also known as symmetry breaking in physics has been an active area of research since the onset of the topic by Herman Weyl in 1950. In this talk, I would give a brief description of Highest weight theory and the determinantal form of the Weyl character formula. I would proceed to talk about branching from $GL_n$ to $GL_{n-1}$ and give an idea about the other classical groups.</span><span style="background-color: white; color: #222222; font-variant-ligatures: normal;"> </span><br /><br /></span></div></div></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/1RE4GMBfyPo?si=iNSC-uJ4dCCpOp5f" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-18179058934414773262023-10-22T17:58:00.007+05:302023-11-05T09:15:35.497+05:30Seamus Albion (Vienna, Austria) - Thursday Oct 26, 2023 - 4:00 PM (IST)<p><span style="font-family: inherit; font-size: medium;"> Dear all,</span></p><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;">The next talk is by Seamus Albion of the University of Vienna. The announcement is as follows.</span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;"><span><b>Talk Announcement: </b></span><span><br /></span></span><div><span style="font-family: inherit; font-size: medium;"><span><b>Title:</b> </span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">An elliptic $A_n$ Selberg integral</span><br /><b>Speaker:</b> Seamus Albion (Vienna, Austria)<br /><b>When:</b> Thursday, Thursday Oct 26, 2023 - 4:00 PM (IST) (12:30 PM CEST)</span><div><span style="font-family: inherit; font-size: medium;"><b>Where</b>: Zoom: Write to the organisers to get the link</span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;"><span><b>Live Link:</b> </span><a href="https://youtube.com/live/S4DQRvK46no?feature=share" target="_blank">https://youtube.com/live/S4DQRvK46no?feature=share</a></span></div><div><span style="font-family: inherit; font-size: medium;"><b>Abstract. <br /></b></span></div><div><span style="font-family: inherit; font-size: medium;"><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">Selberg's multivariate extension of the beta integral appears<span class="gmail-Apple-converted-space"> </span></span><br style="caret-color: rgb(34, 34, 34); color: #222222;" /><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">all over mathematics: in random matrix theory, analytic number theory,<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">multivariate orthogonal polynomials and conformal field theory. The goal<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">of my talk will be to explain a recent unification of two important<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">generalisations of the Selberg integral, namely the Selberg integral<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">associated with the root system of type A_n due to Warnaar and the<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">elliptic Selberg integral conjectured by van Diejen and Spiridonov and<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">proved by Rains. The key tool in our approach is the elliptic<span class="gmail-Apple-converted-space"></span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">interpolation kernel, also due to Rains. This is based on joint work with<span class="gmail-Apple-converted-space"> </span></span><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">Eric Rains and Ole Warnaar.</span></span></div></div></div><div><span style="font-family: inherit; font-size: medium;"><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;"><br /></span></span></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/S4DQRvK46no?si=FKhlfg_6_GLPgnkR" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-31628352094942511012023-10-07T08:19:00.003+05:302023-10-07T08:19:37.102+05:30David Wahiche (Univeriste de Tours, France) - Thursday, October 12, 2023 - 4:00 PM (IST)<p> <span style="font-size: large;">Dear all,</span></p><div class="Ar Au Ao" id=":bsjj"><div aria-controls=":bsm1" aria-label="Message Body" aria-multiline="true" aria-owns=":bsm1" class="Am Al editable LW-avf tS-tW tS-tY" g_editable="true" hidefocus="true" id=":bsjf" itacorner="6,7:1,1,0,0" role="textbox" spellcheck="false" style="direction: ltr; min-height: 286px;" tabindex="1"><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">The next talk is by David Wahiche of the University of Tours, France. The title and abstract is below. <br /></span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Talk Announcement: </b></span><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Title:</b> </span><span style="font-size: medium;"><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;">From Macdonald identities to Nekrasov--Okounkov type formulas</span><br /><span style="font-family: garamond, "times new roman", serif;"><b>Speaker:</b> David Wahiche (Universite' de Tours, France)<br /><b>When:</b> Thursday, Oct 12, 2023, 4:00 PM- 5:00 PM IST <br /></span></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Where</b>: Zoom: Ask the organisers for a link</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Live Link:</b><a href=" https://youtube.co/live/9BNeHB2umCM?feature=share" target="_blank"> </a></span><span style="font-size: medium;"><a href=" https://youtube.co/live/9BNeHB2umCM?feature=share" target="_blank">https://youtube.co/live/9BNeHB2umCM?feature=share</a></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Abstract.</b></span></div><div><p style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><span style="font-size: medium;">Between 2006 and 2008, using various methods coming from representation theory (Westbury), gauge theory (Nekrasov--Okounkov) and combinatorics (Han), several authors proved the so-called Nekrasov–Okounkov formula which involves hook lengths of integer partitions.</span></p><p style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><span style="font-size: medium;">This formula does not only cover the generating series for P, but more generally gives a connection between powers of the Dedekind η function and integer partitions. Among the generalizations of the Nekrasov--Okounkov formula, a (q, t)-extension was proved by Rains and Warnaar, by using refined skew Cauchy-type identities for Macdonald polynomials. The same result was also obtained independently by Carlsson–Rodriguez-Villegas by means of vertex operators and the plethystic exponential. As mentioned in both of these papers, the special case q=t of their formula correspond to a q version of the Nekrasov--Okounkov formula, which was already obtained by Dehaye and Han (2011) and Iqbal et al. (2012).</span></p><p style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><span style="font-size: medium;">Motivated by the work of Han et al. around the generalizations of the Nekrasov--Okounkov formula, one way of deriving Nekrasov--Okounkov formula is by using the Macdonald identities for infinite affine root systems (Macdonald 1972), which can be thought as extension of the classical Weyl denominator formula.</span></p><p style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><span style="font-size: medium;">In this talk, I will try to explain how some reformulations of the Macdonald identities (Macdonald 1972, Stanton 1989, Rosengren and Schlosser 2006) can be decomposed in the basis of characters for each infinite of the 7 infinite affine root systems by the Littlewood decomposition. This echoes a representation theoretic interpretation of the Macdonald identities (see the book of Carter for instance) and an ongoing project with Cédric Lecouvey, I will mention some partial results we get.</span></p><p style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><span style="font-size: medium;">At last, I will briefly explain how to go from these reformulations of Macdonald identities to q Nekrasov--Okounkov type formulas.</span></p></div></div></div></div></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-54588389040252898002023-09-25T14:22:00.004+05:302023-10-06T11:33:41.827+05:30Seema Kushwaha (IIIT, Allahabad) - Thursday Sept 28, 2023 - 4:00 PM (IST)<p><span style="font-family: garamond, "times new roman", serif; font-size: large;">Dear all, sorry for the late announcement. The next talk is by Seema Kushwaha of IIIT, Allahabad. We are back to our usual time now. Hope to see you later this week.</span><span style="font-family: garamond, "times new roman", serif; font-size: large;"> </span></p><div class="Ar Au Ao" id=":uh"><div aria-controls=":158" aria-label="Message Body" aria-multiline="true" aria-owns=":158" class="Am Al editable LW-avf tS-tW tS-tY" g_editable="true" hidefocus="true" id=":ud" itacorner="6,7:1,1,0,0" role="textbox" spellcheck="false" style="direction: ltr; min-height: 242px;" tabindex="1"><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Talk Announcement: </b></span><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Title:</b> </span><span style="font-size: medium;"><span style="font-family: garamond, "times new roman", serif;">Farey-subgraphs and Continued Fractions<br /><b>Speaker:</b> Seema Kushwaha (IIIT, Allahabad)<br /><b>When:</b> Thursday, Sept 28, 2023, 4:00 PM- 5:00 PM IST <br /></span></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Where</b>: Zoom: Please send email to the organisers for a link.</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Live Link:</b> </span><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><a data-saferedirecturl="https://www.google.com/url?q=https://youtube.com/live/UzJEHaj123Y?feature%3Dshare&source=gmail&ust=1695641600317000&usg=AOvVaw2B4iIZz_VgVh2MCqFIJl9y" href="https://youtube.com/live/UzJEHaj123Y?feature=share" target="_blank">https://youtube.com/live/<wbr></wbr>UzJEHaj123Y?feature=share</a></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Abstract. <br /></b></span></div><div><span style="font-size: medium;"><span style="font-family: garamond, "times new roman", serif;">Let $p$ be a prime and $l\in\mathbb{N}$. Let \begin{equation*}\label{X_n}<br />\mathcal{X}_{p^l}=\left\{\frac{x}{y}:~x,y\in\mathbb{Z},~ y>0,~\mathrm{gcd}(x,y)=1~\textnormal{and}~{p^l}|y\right\}\cup\{\infty\}.<br />\end{equation*} <br />The set $\mX_{p^l}$ is the vertex set of a connected graph where vertices $x/y$ and $u/v$ are adjacent if and only if $ xv-uy=\pm p^l.$ These graphs give rise to a family of continued fraction, namely, $\f_{p^l}$-continued fractions \cite{seema_fareysubgraphs}.<br /><br /> Let $\mathcal{X}$ be a subset of the extended set of rational numbers. A {\it best $\mathcal{X}$-approximation} of a real number is a notion which is analogous to best rational approximation. <br /></span></span><span style="font-size: medium;"><span style="font-family: garamond, "times new roman", serif;"><br />An element $u/v$ of $\mX$ is called a \textit{best $\mX$-approximation} of $x\in\R$, if for every $u'/v'\in\mX$ different from $u/v$ with $0< v' \le v$, we have $|vx-u|<|v'x-u'|$. <br /> <br />In this talk, we will discuss the existence and uniqueness of $\f_{p^l}$-continued fractions and their approximation properties. </span></span></div><div><br /></div></div><div><br /></div></div></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/UzJEHaj123Y?si=zK-wZjG9c8HbtEkx" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-76927985817966701152023-09-14T18:00:00.000+05:302023-09-19T09:44:07.146+05:30Shashank Kanade (University of Denver) - Thursday Sept 14, 2023 - 6:00 PM (IST)<p><span style="font-family: trebuchet; font-size: medium;"> Dear all,</span></p><div><span style="font-family: trebuchet; font-size: medium;"><br /></span></div><div><span style="font-family: trebuchet; font-size: medium;">We are back after an extended summer break. I hope many of us had an opportunity to meet each other and further our research goals. </span></div><div><span style="font-family: trebuchet; font-size: medium;"><br /></span></div><div><span style="font-family: trebuchet; font-size: medium;">The next talk is by Shashank Kanade, University of Denver. It is a little later in the evening from our usual time.<br /></span></div><div><span style="font-family: trebuchet; font-size: medium;"><br /></span></div><span style="font-family: trebuchet; font-size: medium;"><span><b>Talk Announcement: </b></span><br /></span><div><span style="font-family: trebuchet; font-size: medium;"><span><br /><b>Title:</b> </span><span style="background-color: white;"><span class="gmail-Apple-converted-space"></span>On the $A_2$ Andrews--Schilling--Warnaar identities</span><br /><span><br /><b>Speaker:</b> Shashank Kanade (University of Denver)</span></span></div><div><span style="font-family: trebuchet; font-size: medium;"><span><br /><b>When:</b> Thursday, Sept 14, 2023, 6:00 PM- 7:00 PM IST (6:30 AM MDT)<br /></span></span><div><span style="font-family: trebuchet; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: trebuchet; font-size: medium;"><b>Where</b>: Zoom: please write to the organisers for the link</span></div><div><span style="font-family: trebuchet; font-size: medium;"><span><br /></span></span></div><div><span style="font-family: trebuchet; font-size: medium;"><span><b>Live Link</b>: </span><a href="https://youtube.com/live/jnsJGm69sjM?feature=share">https://youtube.com/live/jnsJGm69sjM?feature=share</a></span></div><div><span style="font-family: trebuchet; font-size: medium;"><b><br /></b></span></div><div><span style="font-family: trebuchet; font-size: medium;"><b>Abstract <br /></b></span></div><div><div><span style="font-family: trebuchet; font-size: medium;">I will give a description of my work with Matthew C. Russell on the $A_2$</span></div><div><span style="font-family: trebuchet; font-size: medium;">Andrews--Schilling--Warnaar identities. Majority of our single variable</span></div><div><span style="font-family: trebuchet; font-size: medium;">sum=product conjectures have been proven by S. O. Warnaar; I will also</span></div><div><span style="font-family: trebuchet; font-size: medium;">explain what remains. Bi-variate versions of our conjectures are largely open. <br /></span></div><div style="color: #222222;"><span style="font-size: medium;"><span style="background-color: white;"><span style="font-family: trebuchet;"><br /></span><br /></span></span></div></div></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/jnsJGm69sjM?si=X60u-CyVyHhvueTa" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-3272918824821600482023-05-21T19:48:00.005+05:302023-05-26T06:24:11.723+05:30Michael Schlosser (Vienna, Austria) - Thursday May 25, 2023 - 4:00 PM (IST)<p><span style="font-size: medium;"> <span style="font-family: garamond, "times new roman", serif;">Dear all,</span></span></p><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">The next talk is by Michael Schlosser of the University of Vienna, Austria. <br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">After this talk we will be taking a break for the summer. Hopefully, we will get an opportunity to meet in person during this time. </span><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white; color: #222222;"></span></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white; color: #222222;"></span></span></div><div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title:</b> </span><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">Bilateral identities of the Rogers-Ramanujan type</span><br /><br /><b>Speaker:</b> Michael Schlosser (University of Vienna, Austria)<br /><b>When:</b> May 25, 2023, 4:00 PM- 5:00 PM IST (12:30 PM CEST)<br /></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Where</b>: Zoom. Please write to the organisers for a link</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">Live Link:</span><span style="font-family: garamond, "times new roman", serif; font-size: medium;"> <a href="https://youtube.com/live/VO3hTqh8TSw?feature=share">https://youtube.com/live/VO3hTqh8TSw?feature=share</a></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Abstract <br /></b></span></div><div><div style="color: #222222;"><span style="font-size: medium;"><span style="font-family: garamond, "times new roman", serif;"><span style="background-color: white;">The first and second Rogers-Ramanujan (RR) identities have a </span><span style="background-color: white;">prominent history. They were originally discovered and proved </span><span style="background-color: white;">in 1894 by Leonard J. Rogers, and then independently rediscovered </span><span style="background-color: white;">by the legendary self-taught Indian mathematician Srinivasa Ramanujan </span><span style="background-color: white;">some time before 1913. They were also independently discovered and </span><span style="background-color: white;">proved in 1917 by Issai Schur. About the RR identities Hardy remarked</span><br /><br /><span style="background-color: white;">`It would be difficult to find more beautiful formulae than the</span><br /><span style="background-color: white;">``Rogers-Ramanujan'' identities, ...'</span><br /><br /><span style="background-color: white;">Apart from their intrinsic beauty, the RR identities have served </span><span style="background-color: white;">as a stimulus for tremendous research around the world. </span><span style="background-color: white;">The RR and related identities have found interpretations in</span><br /><span style="background-color: white;">various areas including combinatorics, number theory, probability </span><span style="background-color: white;">theory, statistical mechanics, representations of Lie algebras, </span><span style="background-color: white;">vertex algebras, knot theory and conformal field theory.</span><br /><br /><span style="background-color: white;">In this talk, a number of bilateral identities of the RR type will be </span><span style="background-color: white;">presented. We explain how these identities can be derived by </span><span style="background-color: white;">analytic means using identities for bilateral basic hypergeometric </span><span style="background-color: white;">series. Our results include bilateral extensions of the RR and</span><br /><span style="background-color: white;">of the Göllnitz-Gordon identities, and of related identities </span></span><span style="background-color: white; font-family: garamond, "times new roman", serif;">by Ramanujan, Jackson, and Slater.</span></span></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;"><br /></span></span></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;">Corresponding results for multiseries are given as well, </span><span style="background-color: white;">including multilateral extensions of the Andrews-Gordon </span><span style="background-color: white;">identities, of Bressoud's even modulus identities, and others.</span></span></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;"><br /></span></span></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;">This talk is based on the speaker's preprint <a href="https://arxiv.org/abs/1806.01153" target="_blank">arXiv:1806.011153v2</a> </span><span style="background-color: white;">(which has been accepted for publication in Trans. Amer. Math. Soc.).</span></span></div></div></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;"><br /></span></span></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;"><br /></span></span></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/VO3hTqh8TSw" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-73887341622042376142023-05-07T09:00:00.007+05:302023-05-20T17:19:48.696+05:30Bishal Deb (University College, London) - Thursday May 11, 2023 - 4:00 PM (IST)<p> <span style="font-family: garamond, "times new roman", serif; font-size: large;">Dear all,</span></p><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">The next talk is by <span style="background-color: white; color: #222222;">Bishal Deb of University College, London. Here is the announcement. <br /></span></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title:</b> </span><span style="font-size: medium;"><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-family: garamond, "times new roman", serif;">The "quadratic family" of continued fractions and combinatorial sequences</span><span style="font-family: garamond, "times new roman", serif;"><br /><br /><b>Speaker:</b> Bishal Deb (University College, London)<br /><b>When:</b> May 11, 2023, 4:00 PM- 5:00 PM IST (11:30 AM BST)<br /></span></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Where</b>: Zoom:</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">Live Link: </span><span style="font-size: medium;"><a href="https://youtube.com/live/sl0ef-T5c3o?feature=share">https://youtube.com/live/sl0ef-T5c3o?feature=share</a></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Abstract <br /></b></span></div><div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;">We will begin this talk by introducing some combinatorial sequences whose Stieltjes-type continued fraction coefficients increase linearly. We briefly mention the work of Sokal and Zeng where they systematically studied multivariate generalisations of these continued fractions for factorials, Bell numbers and double factorials.<span class="gmail-Apple-converted-space"> </span></span><br /><br /><span style="background-color: white;">Next, we will define the Genocchi and median Genocchi numbers and introduce D-permutations, a class of permutations which enumerate these numbers. We mention some multivariate continued fractions counting various statistics on D-permutations.</span><br /><br /><span style="background-color: white;">Finally, we move to the secant numbers and introduce cycle-alternating permutations; these are another class of permutations which enumerate the secant numbers. We mention some multivariate continued fractions counting various statistics on cycle-alternating permutations. We then describe the entries in the Jacobi-Rogers matrix of our continued fraction using alternating Laguerre digraphs, which are a class of directed graphs. If time permits, we will briefly state some remarks on the Jacobian elliptic functions.</span><br /><br /><span style="background-color: white;">This talk will be based on joint work with Alan Sokal.</span></span><span style="font-family: garamond, "times new roman", serif;"><br /></span></div></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;"><br /></span></span></div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white;"><br /></span></span></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/sl0ef-T5c3o" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-33363745598342226302023-04-25T06:49:00.006+05:302023-04-28T06:17:28.679+05:30Rahul Kumar (Penn State University) - Thursday, April 27 - 4:00 PM<p><span style="font-size: medium;"><span style="background-color: white; color: #222222; font-family: garamond, "times new roman", serif;">The next talk is by Rahul Kumar,</span><span style="background-color: white; color: #222222; font-family: garamond, "times new roman", serif;"> </span><span style="background-color: white; color: #222222; font-family: garamond, "times new roman", serif;">Fulbright-Nehru Postdoctoral Fellow, Penn State University. </span><span style="background-color: white; color: #222222; font-family: garamond, "times new roman", serif; font-variant-ligatures: normal;"><span style="font-variant-ligatures: normal;"> The announcement is as follows. </span></span></span></p><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title: Arithmetic properties of the Herglotz-Zagier-Novikov function</b><br /><br /><b>Speaker:</b> <span style="background-color: white; color: #222222; font-variant-ligatures: normal;"><span style="font-variant-caps: inherit;">Rahul Kumar</span></span> <span style="background-color: white; color: #222222;">(<span style="font-variant-ligatures: normal;"><span style="font-variant-ligatures: normal;">Penn State University</span></span></span>)<br /><b>When:</b> Apr 27, 2023, 4:00 PM- 5:00 PM IST <br /></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Where</b>: Zoom: Write to sfandnt@gmail.com for a link</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">Live Link: https://youtube.com/live/5iJRbNZOksM?feature=share</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Abstract</b></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">The Kronecker limit formulas are concerned with the constant term in the Laurent series expansion of certain Dirichlet series at $s=1$. Various special functions appear in Kronecker limit formulas; one of them is \emph{Herglotz function}. Recently, Radchenko and Zagier extensively studied the properties of the Herglotz function, such as its special values, connection to Stark's conjecture, etc. This function appeared in the work of Herglotz, and Zagier. After providing an overview of the history of this research area, we will discuss the arithmetic properties of a Herglotz-type function that appears in a Kronecker limit formula derived by Novikov. For example, we will present the two- and three-term functional equations satisfied by it along with its special values. This is joint work with Professor YoungJu Choie.</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/5iJRbNZOksM?start=22" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-77419263428344279822023-04-07T16:15:00.006+05:302023-04-25T06:53:41.048+05:30A. Sankaranarayanan (Hyderabad) - April 13, 2023 - 4:00 PM<p> <span style="font-family: garamond, "times new roman", serif; font-size: medium;">Dear all,</span></p><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><span style="background-color: white; color: #222222;">The next talk is by A. Sankaranarayanan of the <span style="font-variant-ligatures: normal;">School of Mathematics and Statistics, <span style="font-variant-ligatures: normal;">University of Hyderabad. The announcement is as follows. <br /></span></span></span></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title: </b><span style="background-color: white; color: #222222; font-variant-ligatures: normal;">On the Rankin-Selberg L-function related to the Godement-Jacquet L-function</span><br /><br /><b>Speaker:</b> <span style="background-color: white; color: #222222; font-variant-ligatures: normal;">A. Sankaranarayanan</span> <span style="background-color: white; color: #222222;">(<span style="font-variant-ligatures: normal;"><span style="font-variant-ligatures: normal;">University of Hyderabad</span></span></span>)<br /><b>When:</b> Apr 13, 2023, 4:00 PM- 5:00 PM IST <br /></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Where</b>: Zoom: Write to organisers for the link</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">Live Link: <a href="https://youtube.com/live/ijtihr4Bi14?feature=share">https://youtube.com/live/ijtihr4Bi14?feature=share</a></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Abstract <br /></b></span></div><div><div style="color: #222222;"><div dir="auto" style="background-color: white; font-variant-ligatures: normal;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;">We discuss the Riesz mean of the Coefficients of the Rankin-Selberg L-function related to the Godement-Jacquet L-function.</span></div><div dir="auto" style="background-color: white; font-variant-ligatures: normal;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div dir="auto" style="background-color: white; font-variant-ligatures: normal;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;">This is a joint work with Amrinder Kaur and recently appeared in Acta Mathematica Hungarica.</span></div><div dir="auto" style="background-color: white; font-variant-ligatures: normal;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"></span></div></div>
<div style="text-align: center;"><span style="font-size: medium;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/ijtihr4Bi14" title="YouTube video player" width="560"></iframe></span></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-82295574076201389652023-03-29T05:53:00.005+05:302023-03-30T17:54:47.296+05:30Christophe Vignat (Tulane) - Thursday, Mar 30 - 4:00 PM<p><span style="font-size: medium;"> <span style="font-family: garamond, "times new roman", serif;">The next talk is by</span><span style="font-family: garamond, "times new roman", serif;"> </span><span style="background-color: white; color: #222222; font-family: garamond, "times new roman", serif;">Christophe Vignat. It will be at our usual time of 4 PM IST. </span></span></p><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title:</b> <span style="background-color: white;">Dirichlet Series Under Standard Convolutions: Variations on Ramanujan’s Identity for Odd Zeta Values </span><br /><br /><b>Speaker:</b> Christophe Vignat (<span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">Université Paris-Saclay, CentraleSupelec, Orsay, France and </span>Tulane University)<br /><b>When:</b> Mar 30, 2023, 4:00 PM- 5:00 PM IST (1:30 PM EEST)<br /></span><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Where</b>: Zoom. Write to the organisers for a link.</span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;">Live Link: <a data-saferedirecturl="https://www.google.com/url?q=https://youtube.com/live/nx60NHwUUjg?feature%3Dshare&source=gmail&ust=1680135707810000&usg=AOvVaw08uMpw7WPKQnEG-xrbTfML" href="https://youtube.com/live/nx60NHwUUjg?feature=share" target="_blank">https://youtube.com/live/<wbr></wbr>nx60NHwUUjg?feature=share</a></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><b>Abstract <br /></b></span></div><div><div style="color: #222222;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span><div style="background-color: white; color: black;"><span style="font-size: medium;"><span style="font-family: garamond, "times new roman", serif;">I will show a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; this formula is inspired by a famous identity of Ramanujan. </span><span style="font-family: garamond, "times new roman", serif;">Some specializations of this convolution formula produce new identities and allow to recover several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and his collaborators, or the convolution of Bernoulli numbers by Y. Komori and his collaborators. </span></span></div><div style="background-color: white; color: black;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div style="background-color: white; color: black;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;">If time permits, I will also exhibit some matrix product representations for the Riemann zeta function evaluated at even and odd integers.</span></div><div style="background-color: white; color: black;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div><div style="background-color: white; color: black;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;">This is joint work with P. Chavan, S. Chavan and T. Wakhare.</span></div><div style="background-color: white; color: black;"><span style="font-family: garamond, "times new roman", serif; font-size: medium;"><br /></span></div></div></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/nx60NHwUUjg" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-21633084428514583312023-03-11T08:21:00.006+05:302023-03-19T14:33:42.490+05:30Bruce Berndt (UIUC) - Thurs Mar 16 - 6:00 PM (Note Special Time)<p><span style="font-size: medium;"> <span>Dear all,</span></span></p><div class="Ar Au" id=":g18b"><div aria-controls=":g1b1" aria-label="Message Body" aria-multiline="true" aria-owns=":g1b1" class="Am Al editable LW-avf tS-tW tS-tY" g_editable="true" hidefocus="true" id=":g187" itacorner="6,7:1,1,0,0" role="textbox" spellcheck="false" style="direction: ltr; min-height: 258px;" tabindex="1"><div class="gmail-Ar gmail-Au gmail-Ao" id="gmail-:fx8j"><div aria-controls=":fxbb" aria-label="Message Body" aria-multiline="true" class="gmail-Am gmail-Al editable gmail-LW-avf gmail-tS-tW gmail-tS-tY" id="gmail-:fx8f" role="textbox" style="direction: ltr; min-height: 258px;" tabindex="1"><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">We are happy to announce our next talk is by Professor Bruce Berndt, the world's biggest authority on Ramanujan's mathematics and related areas. </span><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">Please note the special time. It is two hours later than usual. Please circulate this announcement in your department. </span><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;">Please see a further announcement below.<br /></span></div><div><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><b>Talk Announcement: </b><br /><br /><b>Title:</b> <span face="Tahoma, Geneva, sans-serif" style="background-color: white;">Finite Trigonometric Sums: Evaluations, Estimates, Reciprocity Theorems</span><br /><br /><b>Speaker:</b> <span face="Arial, Helvetica, sans-serif" style="background-color: white; color: #222222;">Bruce Berndt (University of Illinois at Urbana Champaign)</span><br /><b>When:</b> Mar 16, 2022, 6:00 PM- 7:00 PM IST (7:30 AM - 8:30 AM (CDT))<br /></span><div><span style="font-size: medium;"><b>Where</b>: Zoom. Please write to the organisers for the link.</span></div><div><span style="font-size: medium;"><b>Live Video Link:</b> <a href="https://youtube.com/live/rCDRKyv_880?feature=share">https://youtube.com/live/rCDRKyv_880?feature=share</a></span></div><div><span style="font-size: medium;"><b><br /></b></span></div><div><span style="font-size: medium;"><b>Abstract <br /></b></span></div><div><div style="background-color: white; font-family: Tahoma, Geneva, sans-serif;"><span style="font-size: medium;">First, motivated by a theorem in Ramanujan's lost notebook, Martino Fassina, Sun Kim, Alexandru Zaharescu, and the speaker developed representations for finite sums of products of trig functions for which we provided theorems and several conjectures. <span class="gmail-Apple-converted-space"> </span><br /></span></div><div style="background-color: white; font-family: Tahoma, Geneva, sans-serif;"><span style="font-size: medium;"><br /></span></div><div style="background-color: white; font-family: Tahoma, Geneva, sans-serif;"><span style="font-size: medium;">Second, a paper of Richard McIntosh served as motivation. First, he made a very interesting conjecture, which was recently proved by Likun Xie, Zaharescu, and the speaker. Second, he examined a particular trigonometric sum, which inspired Sun Kim, Zaharescu, and the speaker to evaluate in closed form several classes of trigonometric sums, and find reciprocity theorems for others. <span class="gmail-Apple-converted-space"> <br /></span></span></div><div class="gmail-adL"><span style="font-size: medium;"><br /></span></div><div class="gmail-adL" style="text-align: center;"><span style="font-size: medium;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/rCDRKyv_880" title="YouTube video player" width="560"></iframe></span></div><div class="gmail-adL"><span style="font-size: medium;"><br /></span></div><div class="gmail-adL"><span style="font-size: medium;"><br /></span></div><div class="gmail-adL"><span style="font-size: medium;"><b>New conference announcement<br /></b></span></div><div class="gmail-adL"><span style="font-size: medium;">A new conference on algebraic combinatorics has been announced by Arvind Ayyer. It is called <span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;"><br /></span></span></div><div class="gmail-adL"><span style="font-size: medium;"><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">Meru Annual Combinatorics<span class="gmail-Apple-converted-space"> </span></span><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">Conference</span></span></div><div class="gmail-adL"><span style="font-size: medium;"><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;"><span face="Inconsolata, sans-serif" style="box-sizing: inherit; caret-color: rgb(0, 0, 0); color: black;">Dates:</span><span face="Inconsolata, sans-serif" style="background-color: oldlace; caret-color: rgb(0, 0, 0); color: black;"><span class="gmail-Apple-converted-space"> </span>29th to 31st May, 2023</span></span></span></div><div class="gmail-adL"><span style="font-size: medium;"><span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">Conference website: <a href="https://www.imsc.res.in/~amri/meru/" rel="noreferrer" style="color: #1155cc;" target="_blank">https://www.imsc.res.in/~amri/<wbr></wbr>meru/</a></span></span></div><div class="gmail-adL"><span style="font-size: medium;"><br /></span></div></div></div><span style="font-size: medium;"><br /><br /></span></div></div></div></div>
<br />Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-39475532134179906792023-02-26T13:31:00.003+05:302023-03-02T17:02:46.178+05:30B. Ramakrishnan, ISI, Tezpur - Thursday, Mar 2, 2023 - 4:00 PM<p>The next talk is by <span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">B. Ramakrishnan (popularly known as </span>Ramki), formerly of HRI, Allahabad, and now in ISI, Tezpur. </p><div><br /></div><b>Talk Announcement: </b><br /><br /><b>Title:</b> <span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">An extension of Ramanujan-Serre derivative map and some applications.</span><br /><br /><b>Speaker:</b> <span face="Arial, Helvetica, sans-serif" style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222; font-size: small;">B. Ramakrishnan</span> (Indian Statistical Institute North-East Center, Tezpur)<br /><b>When:</b> Mar 2, 2022, 4:00 PM- 5:00 PM IST <br /><div><b>Where</b>: Zoom: Please write to organisers for the link.</div><div><b>Live</b>: <a href="https://youtube.com/live/dbmTBbKLhRc?feature=share" rel="nofollow" target="_blank">Youtube link</a></div><div><br /></div><div><b>Abstract <br /></b></div><div><div><br /></div><div style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;">In this talk, we present a simple extension of the Ramanujan-Serre derivative map and </div><div style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;">describe how it can be used to derive a general method for explicit evaluation of convolution sums of the divisor functions. We provide explicit examples for four types of convolution sums.</div><div style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><br /></div><div style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;">This is a joint work with Brundaban Sahu and Anup Kumar Singh. </div></div><div style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><br /></div><div style="caret-color: rgb(34, 34, 34); color: #222222; font-family: Arial, Helvetica, sans-serif;"><br /></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/dbmTBbKLhRc" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-91701267095507917082023-02-11T12:53:00.001+05:302024-01-09T19:57:20.634+05:30Galina Filipuk, University of Warsaw - Thursday, Feb 16, 2023 - 4:00 PM<div><br /></div><div>We are back to our usual time with a talk by Galina Filipuk all the way from Warsaw, Poland. Please note that we will be open to changing the time, since speakers from the US find this time to be very inconvenient, and we surely would like speakers from the US. The discussions in the previous talk went quite late into the night (for New Zealand) and we thank Shaun Cooper for a very nice talk. <br /></div><div><br /></div><b>Talk Announcement: </b><br /><br /><b>Title:</b> (Quasi)-Painleve equations and Painleve equivalence problem<br /><br /><b>Speaker:</b> Galina Filipuk (University of Warsaw, Poland)<br /><b>When:</b> Feb 16, 2023, 4:00 PM- 5:00 PM IST (11:30 CET in Warsaw)<br /><div><b>Where</b>: Zoom: Please write to sfandnt@gmail.com for the link.</div><div><br /></div><div><b>Abstract <br /></b></div><div><div><br /></div><div>Painleve equations are second order nonlinear differential equations solutions of which have no movable critical points (algebraic singularities). They appear in many applications (e.g., in the theory of orthogonal polynomials) but in disguise. How to find a transformation to the canonical form? This is known as the Painleve equivalence problem.<br />The so-called geometric approach may help in many cases.</div><div><br /></div><div>In this talk I shall present some recent results on the geometric approach for the Painleve and quasi-Painleve equations.<br /></div><div><br /><br /></div></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-81248934939834326832023-01-28T07:17:00.009+05:302023-02-05T14:11:29.124+05:30Ramanujan Special: Shaun Cooper (Massey University) - Thursday, Feb 2, 2023 - 2:30 PM<p> Happy new year. </p><div><br /></div><div>The first talk of the year (on Feb 2, 2023) is a "Ramanujan Special". This year's speaker is Shaun Cooper. Please note that the talk will be earlier than usual. </div><div><br /></div><div>The last year was quite exciting for our group with many talks as well as a mini course. We hope this year is equally exciting. Please consider the seminar to present your latest preprint. <br /></div><br /><b>Talk Announcement: Ramanujan Special</b><br /><br /><b>Title:</b> Apéry-like sequences defined by four-term recurrence relations: theorems and conjectures<br /><b><br /></b><div><b>Speaker:</b> Shaun Cooper (Massey University, Auckland, New Zealand)<br /><b><br /></b><div><b>When:</b> Feb 2, 2022, 2:30 PM- 3:30 PM IST (<b>Note special time) (</b>IST= GMT - 5:30)<br /><div><b><br /></b></div><div><b>Where</b>: Zoom. Write to sfandnt@gmail.com for a link.</div><div><br /></div><div><b>Abstract <br /></b></div><div><br /></div><div>The Apéry numbers are famous for having been introduced and used by R. Apéry to prove that~$\zeta(3)$ is irrational. They may be defined by the recurrence relation<br />$$<br />(n+1)^3A(n+1)=(2n+1)(17n^2+<wbr></wbr>17n+5)A(n)-n^3A(n-1),<br />$$<br />with the single initial condition $A(0)=1$ being enough to start the recurrence. The Apéry numbers are all integers, a fact not obvious from the recurrence relation, and they satisfy interesting congruence properties. The generating function<br />$$<br />y=\sum_{n=0}^\infty A(n)w^n<br />$$<br />has a splendid parameterisation given by<br />$$<br />y = \prod_{j=1}^\infty \frac{(1-q^{2j})^7(1-q^{3j})^<wbr></wbr>7}{(1-q^{j})^5(1-q^{6j})^5}<br />\quad<br />\mbox{and}<br />\quad<br />w=q\,\prod_{j=1}^\infty \frac{(1-q^{j})^{12}(1-q^{6j})<wbr></wbr>^{12}}{(1-q^{2j})^{12}(1-q^{<wbr></wbr>3j})^{12}}.<br />$$<br />In this talk I will briefly survey other sequences defined by three-term recurrence relations that have properties similar to those satisfied by the Apéry numbers described above. I will also introduce some sequences defined by four-term recurrence relations and describe some of their properties.</div><div><br /></div><div>Several conjectures will be presented.<br /></div><div><br /></div><div>Here are the <a href="https://drive.google.com/file/d/16u3-BB5gq4CzOk6bjRd4Z_rvZZM_8FlG/view?usp=sharing" target="_blank">slides</a> of the talk.</div><div><br /></div><div><br /></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/HHBNuvkxUew" title="YouTube video player" width="560"></iframe></div></div></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-78407039052363400102022-11-12T07:26:00.004+05:302022-12-03T15:14:33.218+05:30Nishu Kumari (IISc, Bangalore) - Thursday Nov 17, 2022 - 4:00 PM to 5:00 PM<p><span style="font-family: inherit; font-size: medium;"> Dear all, </span></p><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;">The next talk is by Nishu Kumari, a graduate student in IISc, Bangalore. The announcement is below.</span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;"><b>Talk Announcement</b><b><br /></b></span></div><div><span style="font-family: inherit; font-size: medium;"><br /></span></div><div><span style="font-family: inherit; font-size: medium;"><span style="background-color: white; font-variant-ligatures: normal;"><span style="color: #212121;"><b>Title:</b> Factorization of Classical Characters twisted by Roots of Unity</span><span style="color: #212121;"></span><div dir="ltr" style="color: #212121;"><br /></div><div style="color: #212121;"><b>Speaker:</b> Nishu Kumari (IISc, Bangalore)<br /></div><div dir="ltr" style="color: #212121;"><br /></div><div dir="ltr"><div style="color: #212121;"><b>When:</b> Thursday, November 17, 2022 - 4:00 PM - 5:00 PM (IST) <br /></div><div style="color: #212121;"><br /></div><div><span face="arial, sans-serif"><span><b style="color: #212121;">Where:</b><span style="color: #212121;"> </span><span face="arial,sans-serif"><span style="color: #212121;">Zoom</span><b><span style="color: #212121;">. Please write to the <span style="caret-color: rgb(33, 33, 33);">organisers for the link.</span></span></b></span></span></span><div style="color: #212121;"><br /></div></div><div style="color: #212121;"><span face="arial, sans-serif"><span face="arial, sans-serif"><b>Live Link:</b></span></span> <a data-saferedirecturl="https://www.google.com/url?q=https://youtu.be/nFarazOy7eI&source=gmail&ust=1668304292239000&usg=AOvVaw3ZP66CKWA6KdNve0o9r_vn" href="https://youtu.be/nFarazOy7eI" target="_blank">https://youtu.be/nFarazOy7eI</a></div><div style="color: #212121;"><span face="arial, sans-serif"></span></div><span style="color: #212121;"></span><p style="color: #212121;"><b>Tea or Coffee</b>: Please bring your own.<br /><br /><b>Abstract:</b></p></div></span>Schur polynomials are the characters of irreducible representations of classical groups of type A parametrized by partitions. For a fixed integer $t \geq 2$ and a primitive $t$'th root of unity \omega, Schur polynomials evaluated at elements $\omega^{k} x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, were considered by D. J. Littlewood (AMS press, 1950)<span style="color: #2e2e2e; line-height: 1.5;"> </span>and independently by D. Prasad (Israel J. Math., 2016). They characterized partitions for which the specialized Schur polynomials are nonzero and showed that if the Schur polynomial is nonzero, it factorizes into characters of smaller classical groups of type A. </span><div dir="ltr" style="background-color: white; color: #212121; font-variant-ligatures: normal;"><span style="line-height: 1.5;"><span style="font-family: inherit; font-size: medium;"><br /></span></span></div><div style="background-color: white; color: #212121; font-variant-ligatures: normal;"><span style="font-family: inherit; font-size: medium;"><span style="line-height: 1.5;">In this talk, I will present a generalization of the factorization result to the characters of classical groups of type B, C and D. We give a uniform approach for all cases. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. This is joint work with A. Ayyer and is available </span><a data-saferedirecturl="https://www.google.com/url?q=https://doi.org/10.1016/j.jalgebra.2022.06.015&source=gmail&ust=1668304292239000&usg=AOvVaw2VZSK943uP1SGxKjB35x29" href="https://doi.org/10.1016/j.jalgebra.2022.06.015" style="color: #0078d4; line-height: 1.5;" target="_blank">here</a>. (Preprint: <a data-saferedirecturl="https://www.google.com/url?q=https://arxiv.org/abs/2109.11310&source=gmail&ust=1668304292239000&usg=AOvVaw1bVDYPdp0Jlu4U7OkkMxHO" href="https://arxiv.org/abs/2109.11310" target="_blank">https://arxiv.org/abs/2109.<wbr></wbr>11310</a>)</span></div></div><div style="background-color: white; color: #212121; font-variant-ligatures: normal;"><span style="font-family: inherit; font-size: medium;"><br /></span></div><div style="background-color: white; color: #212121; font-variant-ligatures: normal;"><span style="font-family: inherit; font-size: medium;"><br /></span></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/nFarazOy7eI" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-40084696857521869612022-11-01T14:29:00.002+05:302022-11-03T18:11:30.055+05:30Nicholas Smoot (RISC, Johannes Kepler University, Linz, Austria) - Thursday Nov 3, 2022 - 4:00 PM to 5:00 PM <p><span style="font-size: large;">The talk this week is by Nicholas Smoot from the Research Institute of Symbolic Computation (RISC) at Johannes Kepler University (JKU), Linz, Austria. The announcement is below.</span></p><div><br /></div><div><p style="color: #222222;"><span style="font-size: medium;"><span style="background-color: white;"><b>Talk Announcement:</b></span></span></p><div><div style="color: #222222;"><span style="font-family: inherit; font-size: medium;"><span style="background-color: white;"><span face="arial, sans-serif"><b>Title: </b></span></span><span style="background-color: white; white-space: pre-wrap;"> <span style="color: black;">Partitions, Kernels, and Localization</span></span><span style="background-color: white;"></span></span></div><div style="color: #222222;"><span style="font-size: medium;"><br /></span></div><div style="color: #222222;"><span style="font-size: medium;"><span id="m_4753550418791714068m_6817113731689661263gmail-m_-5156083776740988679gmail-m_5608636544501622636gmail-m_-9026948515018795222gmail-m_5996750849471202718gmail-m_-2245253888636014354gmail-m_643489453249097870gmail-m_4385757875682979210gmail-m_-6116591908518297465gmail-m_8363861980176606183gmail-m_8164958050465424504gmail-m_-4432346833018561358gmail-m_-7527141843211794468m_7455997257251026206gmail-m_7737114434378993800m_3744415322078576468m_8012140437112174855gmail-m_-401428299463079870gmail-m_-3841287407571171818gmail-MathJax-Element-2-Frame"></span></span></div><div style="color: #222222;"><span style="font-family: inherit; font-size: medium;"><span style="background-color: white;"><b>Speaker</b>: Nicholas A. Smoot (RISC at JKU, Austria)</span></span></div><div style="color: #222222;"><span style="font-size: medium;"><span style="background-color: white;"><span face="arial, sans-serif"><b> </b></span></span></span></div><div style="color: #222222;"><span style="font-size: medium;"><b>When:</b> Thursday, November 3, 2022 - 4:00 PM - 5:00 PM (IST) <br /></span></div><div style="color: #222222;"><span style="font-size: medium;"><br /></span></div><div><span style="font-size: medium;"><span style="background-color: white;"><span face="arial, sans-serif"><span><b style="color: #222222; font-family: inherit;">Where:</b><span style="color: #222222; font-family: inherit;"> Zoom. Please write to the organisers if you require the </span><span style="color: #222222;"><span style="caret-color: rgb(34, 34, 34);">link</span></span><span style="color: #222222; font-family: inherit;"> (sfandnt@gmail.com)</span></span></span></span></span></div><div style="color: #222222;"><span style="font-size: medium;"><span style="background-color: white;"><span face="arial, sans-serif"><span style="font-family: inherit;"><br /></span></span></span></span></div><div style="color: #222222;"><span style="font-family: inherit; font-size: medium;"><span style="background-color: white;"><span face="arial, sans-serif"><span face="arial, sans-serif"><b>Live Link: </b></span></span></span></span><span style="font-size: medium;"><a data-saferedirecturl="https://www.google.com/url?q=https://www.blogger.com/blog/post/edit/7066957877694732201/2757915598200132892%23&source=gmail&ust=1667379428927000&usg=AOvVaw1FjaLKx4oVVxBw9TfYoUJK" href="https://www.blogger.com/blog/post/edit/7066957877694732201/2757915598200132892#" target="_blank">https://youtu.be/bXp1LMWTcqY</a></span></div><div style="color: #222222;"><span style="font-size: medium;"><span face="arial, sans-serif"></span></span></div><span style="color: #222222; font-family: inherit; font-size: medium;"><span style="background-color: white;"></span></span><p style="color: #222222;"><span style="font-size: medium;"><b>Tea or Coffee</b>: Please bring your own.<br /><br /><b>Abstract:</b> </span></p></div><p><span style="font-size: medium;"><span style="background-color: white; color: #222222;"><span style="color: black;">Since Ramanujan's groundbreaking work, a large variety of infinite congruence families for partition functions modulo prime powers have been discovered. These families vary enormously with respect to the difficulty of proving them. We will discuss the application of the localization method to proving congruence families by walking through the proof of one recently discovered congruence family for the counting function for 5-elongated plane partitions. In particular, we will discuss a critical aspect of such proofs, in which the associated generating functions of a given congruence family are members of the kernel of a certain linear mapping to a vector space over a finite field. We believe that this approach holds the key to properly classifying congruence families.</span></span></span></p><p><br /></p></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/bXp1LMWTcqY" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-27579155982001328922022-10-16T07:18:00.003+05:302022-12-03T15:15:38.829+05:30Sunil L. Naik (IMSc, Chennai) - Thursday, October 20, 2022 - 4:00 PM - 5:00 PM (IST) <p><span style="font-family: inherit;">Our next speaker is Sunil Naik, a grad student in IMSc, Chennai. </span></p><p style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="background-color: white;"><b><span style="font-family: inherit;">Talk Announcement:</span></b></span></p><div style="caret-color: rgb(34, 34, 34); color: #222222;"><div><span style="font-family: inherit;"><span style="background-color: white;"><span face="arial, sans-serif"><b>Title: </b></span></span><span style="background-color: white; white-space: pre-wrap;"> </span><span style="background-color: white;">Prime factors of non-zero Hecke eigen values</span></span></div><div><span style="background-color: white;"><span style="font-family: inherit;"><br /></span></span></div><div><span style="background-color: white;"><span face="arial, sans-serif"><span style="font-family: inherit;"><span id="gmail-m_-5156083776740988679gmail-m_5608636544501622636gmail-m_-9026948515018795222gmail-m_5996750849471202718gmail-m_-2245253888636014354gmail-m_643489453249097870gmail-m_4385757875682979210gmail-m_-6116591908518297465gmail-m_8363861980176606183gmail-m_8164958050465424504gmail-m_-4432346833018561358gmail-m_-7527141843211794468m_7455997257251026206gmail-m_7737114434378993800m_3744415322078576468m_8012140437112174855gmail-m_-401428299463079870gmail-m_-3841287407571171818gmail-MathJax-Element-2-Frame"></span></span></span></span></div><div><span style="font-family: inherit;"><span style="background-color: white;"><b>Speaker</b>: Sunil L. Naik (IMSc, Chennai)</span></span></div><div><span style="background-color: white;"><span face="arial, sans-serif"><b><span style="font-family: inherit;"> </span></b></span></span></div><div><span style="background-color: white;"><span style="font-family: inherit;"><b>When:</b> Thursday, October 20, 2022 - 4:00 PM - 5:00 PM (IST) <br /></span></span></div><div><span style="background-color: white;"><span style="font-family: inherit;"><br /></span></span></div><div><span style="background-color: white;"><span face="arial, sans-serif"><span style="font-family: inherit;"><b>Where:</b> <span face="arial,sans-serif">Zoom<b>. Please ask the organisers for a link<br /></b></span></span></span></span><div></div><div><span style="background-color: white;"><span style="font-family: inherit;"><br /></span></span></div></div><div><span style="font-family: inherit;"><span style="background-color: white;"><span face="arial, sans-serif"><span face="arial, sans-serif"><b>Live Link: </b></span></span></span></span><a href="https://youtu.be/Y6-A5Ab74Ec">https://youtu.be/Y6-A5Ab74Ec</a></div><div><span style="background-color: white;"><span style="font-family: inherit;"><span face="arial, sans-serif"></span></span></span></div><span style="font-family: inherit;"><span style="background-color: white;"></span></span><p><span style="background-color: white;"><span style="font-family: inherit;"><b>Tea or Coffee</b>: Please bring your own.<br /><br /><b>Abstract:</b> </span></span></p></div><p><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">The non-vanishing as well as primitivity of the values of the Fourier </span><br style="caret-color: rgb(34, 34, 34); color: #222222;" /><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">coefficients of non-CM Hecke eigen forms, in particular the Ramanujan </span><br style="caret-color: rgb(34, 34, 34); color: #222222;" /><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">$\tau$ function is a deep and mysterious theme in Number theory.</span><br style="caret-color: rgb(34, 34, 34); color: #222222;" /><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">In this talk, we will report on our recent work on the number of </span><br style="caret-color: rgb(34, 34, 34); color: #222222;" /><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">distinct prime factors of the values of the Fourier coefficients of </span><br style="caret-color: rgb(34, 34, 34); color: #222222;" /><span style="background-color: white; caret-color: rgb(34, 34, 34); color: #222222;">non-CM Hecke eigen forms, in particular the Ramanujan $\tau$ function.</span></p><p><br /></p>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/Y6-A5Ab74Ec" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0tag:blogger.com,1999:blog-7066957877694732201.post-27320784410390145762022-09-17T12:28:00.004+05:302023-02-05T14:11:49.552+05:30Kaneenika Sinha (IISER, Pune) - Mini Course - Thursday, Sept 22 and Oct 6 - 4-5:00 PM (IST)<p><span style="font-size: large;"><span style="font-family: inherit;">We are happy that Kaneenika Sinha (IISER, Pune) has consented to give a mini-course on </span><b style="font-family: inherit;">Central limit theorems in number theory</b><span style="font-family: inherit;">. The course will comprise two lectures. The announcement is below. Graduate students who are interested in number theory are especially welcome to hear Professor Sinha. </span></span></p><p style="caret-color: rgb(34, 34, 34); color: #222222;"><span style="background-color: white;"><b><span style="font-family: inherit; font-size: large;">Mini-course announcement</span></b></span></p><div style="caret-color: rgb(34, 34, 34); color: #222222;"><div><span style="font-family: inherit; font-size: large;"><span style="background-color: white;"><span face="arial, sans-serif"><b>Title: </b></span></span><span style="background-color: white; white-space: pre-wrap;"> </span><span face="Arial, Helvetica, sans-serif" style="background-color: white;">Central Limit theorems in Number Theory</span></span></div><div><span style="background-color: white;"><span style="font-family: inherit; font-size: large;"><br /></span></span></div><div><span style="background-color: white;"><span face="arial, sans-serif"><span style="font-family: inherit;"><span id="gmail-m_-5156083776740988679gmail-m_5608636544501622636gmail-m_-9026948515018795222gmail-m_5996750849471202718gmail-m_-2245253888636014354gmail-m_643489453249097870gmail-m_4385757875682979210gmail-m_-6116591908518297465gmail-m_8363861980176606183gmail-m_8164958050465424504gmail-m_-4432346833018561358gmail-m_-7527141843211794468m_7455997257251026206gmail-m_7737114434378993800m_3744415322078576468m_8012140437112174855gmail-m_-401428299463079870gmail-m_-3841287407571171818gmail-MathJax-Element-2-Frame" style="font-size: large;"></span></span></span></span></div><div><span style="background-color: white;"><span style="font-family: inherit; font-size: large;"><b>Speaker</b>: Kaneenika Sinha (IISER, Pune)</span></span></div><div><span style="background-color: white;"><span style="font-family: inherit; font-size: large;"><br /></span></span></div><div><span style="background-color: white;"><b><span style="font-family: inherit; font-size: large;">Abstract:</span></b></span></div><div><span style="caret-color: rgb(0, 0, 0); color: black;"><span style="font-family: inherit; font-size: large;">The goal of these lectures is to review a theme that binds the study of different types of arithmetic functions, namely central limit theorems. After reviewing the "prototype" theorem in this theme, namely the classical Erdos-Kac theorem about the prime-omega function, we will survey different types of central limit theorems in the context of zeroes of zeta functions, eigenvalues of Hecke operators acting on spaces of cusp forms and eigenvalues of regular graphs.</span></span></div><div><span style="font-family: inherit; font-size: large;"><br /></span></div><div><span style="font-size: large;"><b>Where:</b><span style="background-color: white;"> </span><span face="arial,sans-serif">Zoom<b>. Please ask the organisers for a link</b></span></span></div><div><span style="font-family: inherit; font-size: large;"><br /></span></div><div><span style="background-color: white;"><span style="font-family: inherit; font-size: large;"><b>Talk 1: </b> Thursday, September 22, 2022 - 4:00 PM - 5:00 PM (IST) <br /></span></span></div><div><span style="font-family: inherit; font-size: large;"><br /></span></div><div><span style="font-size: large;"><span style="font-family: inherit;"><span style="background-color: white;"><span face="arial, sans-serif"><span face="arial, sans-serif"><b>Live Link: </b></span></span></span></span><a href="https://youtu.be/YBencJpl4bI">https://youtu.be/YBencJpl4bI</a></span></div><div><span style="font-family: inherit; font-size: large;"><br /></span></div><div><span style="font-family: inherit; font-size: large;"><b>Talk 2: </b><span style="background-color: white;"> Thursday, October 6, 2022 - 4:00 PM - 5:00 PM (IST) </span></span></div><div><span style="font-family: inherit;"><span style="background-color: white; font-size: large;"><br /></span></span></div><div><span style="font-size: large;"><span style="background-color: white;"><span face="arial, sans-serif"><span face="arial, sans-serif"><b>Live Link: </b></span></span></span><a href="https://youtu.be/2wsTmPoVvhI">https://youtu.be/2wsTmPoVvhI</a></span></div><div><span style="font-size: large;"><br /></span></div><div><span style="background-color: white;"><span style="font-family: inherit;"><span face="arial, sans-serif" style="font-size: large;"></span></span></span></div><span style="font-family: inherit;"><span style="background-color: white; font-size: large;"></span></span><p><span style="background-color: white;"><span style="font-family: inherit; font-size: large;"><b>Tea or Coffee</b>: Please bring your own.</span></span></p><p><span style="background-color: white;"><b><span style="font-size: x-large;"><br /></span></b></span></p><p><span style="background-color: white;"><b><span style="font-size: large;">Talk 1</span></b></span></p></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/YBencJpl4bI" title="YouTube video player" width="560"></iframe></div><div style="text-align: center;"><br /></div><div style="text-align: left;"><b><span style="font-size: large;">Talk 2</span></b></div><div style="text-align: center;"><br /></div>
<div style="text-align: center;"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/2wsTmPoVvhI" title="YouTube video player" width="560"></iframe></div>Gaurav Bhatnagarhttp://www.blogger.com/profile/17526386852258537327noreply@blogger.com0