BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Ken Ono (University of Virginia)
DTSTART:20200907T140000Z
DTEND:20200907T150000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/1/">Variations of Lehmer’s Conjecture on the Nonvanishing of the Raman
 ujan tau-function</a>\nby Ken Ono (University of Virginia) as part of EIMI
  Number Theory Seminar\n\n\nAbstract\nIn the spirit of Lehmer's unresolved
  speculation on the nonvanishing of Ramanujan's tau-function\, it is natur
 al to ask whether a fixed integer is a value of $\\tau(n)$\, or is a Fouri
 er coefficient of any given newform.  In joint work with J. Balakrishnan\,
  W. Craig\, and W.-L. Tsai\, the speaker has obtained some results that wi
 ll be described here. For example\, infinitely many spaces are presented f
 or which the primes $\\ell\\leqslant 37$ are not absolute values of coeffi
 cients of any new forms with integer coefficients. For Ramanujan’s tau-f
 unction\, such results imply\, for $n>1$\, that $\\tau(n)\\notin \\{\\pm \
 \ell\\\,:\\\, \\ell<100\n\\\,\\text{is odd prime}\\}$.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruce Berndt (University of Illinois)
DTSTART:20200916T160000Z
DTEND:20200916T170000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/2/">The Circle Problem of Gauss\, the Divisor Problem of Dirichlet\, and
  Ramanujan's Interest in Them</a>\nby Bruce Berndt (University of Illinois
 ) as part of EIMI Number Theory Seminar\n\n\nAbstract\nLet $r_2(n)$ denote
  the number of representations of the positive integer $n$ as a sum of two
  squares\, and let $d(n)$ denote the number of positive divisors of $n$.  
 Gauss and Dirichlet were evidently the first mathematicians to derive asym
 ptotic formulas for $\\sum_{n\\leq x}r_2(n)$ and $\\sum_{n\\leq x}d(n)$\, 
 respectively\, as $x$ tends to infinity.  The magnitudes of the error term
 s for the two asymptotic expansions are unknown.  Determining the exact or
 ders of the error terms are the Gauss Circle Problem and Dirichlet's Divis
 or Problem\, respectively\, and they represent two of the most famous and 
 difficult unsolved problems in number theory.\n\nBeginning with his first 
 letter to Hardy\, it is evident that Ramanujan had a keen interest in the 
 Divisor Problem\, and from a paper written by Hardy and published  in 1915
 \, shortly after Ramanujan arrived in England\, we learn that Ramanujan  w
 as also greatly interested in the Circle Problem. In a fragment published 
 with his Lost Notebook\, Ramanujan stated two doubly infinite series ident
 ities involving Bessel functions that we think Ramanujan derived to attack
  these two famous unsolved problems. The identities are difficult to prove
 .  Unfortunately\, we cannot figure out how Ramanujan might have intended 
 to use them. We survey what is known about these two unsolved problems\, w
 ith a concentration on Ramanujan's two marvelous and mysterious identities
 .  Joint work with Sun Kim\, Junxian Li\, and Alexandru Zaharescu is discu
 ssed.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthias Beck (San Francisco State University and Freie Universita
 et Berlin)
DTSTART:20200924T150000Z
DTEND:20200924T160000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/3/">Partitions with fixed differences between largest and smallest parts
 </a>\nby Matthias Beck (San Francisco State University and Freie Universit
 aet Berlin) as part of EIMI Number Theory Seminar\n\n\nAbstract\nEnumerati
 on results on integer partitions form a classic body of mathematics going 
 back to at least Euler\, including numerous applications throughout mathem
 atics and some areas of physics. We study the number $p(n\,t)$ of partitio
 ns of $n$ with difference $t$ between largest and smallest parts. For exam
 ple\, $p(n\,0)$ equals the number of divisors of $n$\, the function $p(n\,
 1)$ counts the nondivisors of $n$\, and $p(n\,2) = \\binom{ \\left\\lfloor
  \\frac n 2 \\right\\rfloor }{ 2 }$. Beyond these three cases\, the existi
 ng literature contains few results about $p(n\,t)$\, even though concrete 
 evaluations of this partition function are featured in several entries of 
 Sloane's Online Encyclopedia of Integer Sequences. \n\nOur main result is 
 an explicit formula for the generating function $P_t(q) := \\sum_{ n \\ge 
 1 } p(n\,t) \\\, q^n$. Somewhat surprisingly\, $P_t(q)$ is a rational func
 tion for $t>1$\; equivalently\, $p(n\,t)$ is a quasipolynomial in $n$ for 
 fixed $t>1$ (e.g.\, the above formula for $p(n\,2)$ is an example of a qua
 sipolynomial with period 2). Our result generalizes to partitions with an 
 arbitrary number of specified distances.\n\nThis is joint work with George
  Andrews (Penn State) and Neville Robbins (SF State).\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Garvan (University of Florida)
DTSTART:20201001T150000Z
DTEND:20201001T170000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/4/">A new approach to Dyson's rank conjectures</a>\nby Frank Garvan (Uni
 versity of Florida) as part of EIMI Number Theory Seminar\n\n\nAbstract\nI
 n 1944 Dyson defined the rank of a partition as the largest part minus the
  number of parts\, and conjectured that the residue of the rank mod 5 divi
 des the partitions of 5n+4 into five equal classes. This gave a combinator
 ial explanation of Ramanujan's famous partition\ncongruence mod 5. He made
  an analogous conjecture for the rank mod 7 and the partitions of 7n+5. In
  1954 Atkin and Swinnerton-Dyer proved Dyson's rank conjectures by constru
 cting several Lambert-series identities basically using the theory of elli
 ptic functions. In 2016 the author gave another proof using the theory of 
 weak harmonic Maass forms. In this talk we\ndescribe a new and more elemen
 tary approach using Hecke-Rogers series.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Goettingen University)
DTSTART:20201005T140000Z
DTEND:20201005T160000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/5
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/5/">On the distribution of Campana points on toric varieties</a>\nby Dam
 aris Schindler (Goettingen University) as part of EIMI Number Theory Semin
 ar\n\n\nAbstract\nIn this talk we discuss joint work with Marta Pieropan o
 n the distribution of Campana points on toric varieties. We discuss how th
 is problem leads us to studying a generalised version of the hyperbola met
 hod\, which had first been developed by Blomer and Bruedern. We show how d
 uality in linear programming is used to interpret the counting result in t
 he context of a general conjecture of Pieropan-Smeets-Tanimoto-Varilly-Alv
 arado.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fernando Rodriguez-Villegas (The Abdus Salam International Centre 
 for Theoretical Physics)
DTSTART:20201015T140000Z
DTEND:20201015T160000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/6/">Mixed Hodge numbers and factorial ratios</a>\nby Fernando Rodriguez-
 Villegas (The Abdus Salam International Centre for Theoretical Physics) as
  part of EIMI Number Theory Seminar\n\n\nAbstract\nThe factorial ratios of
  the title are numbers such as $\\frac{(30n)!n!}{(6n)!(10n)!(15n)!}$ consi
 dered by Chebyshev in his work on the distribution of prime numbers\, whic
 h are integral for all n in a non-obvious way. I will discuss how integral
 ity is related to the lack of interior points of the first few dilations o
 f an associated polytope and the vanishing of certain Hodge numbers of ass
 ociated varieties. This work is an offshoot of an ongoing project on hyper
 geometric motives joint with D. Roberts and M. Watkins.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Raum (Chalmers Technical University\, Gothenburg\, Sweden)
DTSTART:20201019T160000Z
DTEND:20201019T170000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/7/">Relations among Ramanujan-type Congruences</a>\nby Martin Raum (Chal
 mers Technical University\, Gothenburg\, Sweden) as part of EIMI Number Th
 eory Seminar\n\n\nAbstract\nWe present a new framework to access relations
  among Ramanujan-type congruences of a weakly holomorphic modular form. Th
 e framework is strong enough to apply to all Shimura varieties\, and cover
 s half-integral weights if unary theta series are available. We demonstrat
 e effectiveness in the case of elliptic modular forms of integral weight\,
  where we obtain a characterization of Ramanujan-type congruences in terms
  of Hecke congruences. Finally\, we showcase concrete computer calculation
 s\, exploring the information encoded by our framework in the case of elli
 ptic modular forms of half-integral weight. This leads to an unexpected di
 chotomy between Ramanujan-type congruences found by Atkin and by Ono\, Ahl
 gren-Ono.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Lovejoy (CNRS)
DTSTART:20201028T170000Z
DTEND:20201028T180000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/8/">Quantum q-series identities</a>\nby Jeremy Lovejoy (CNRS) as part of
  EIMI Number Theory Seminar\n\n\nAbstract\nAs analytic identities\, classi
 cal $q$-series identities like the Rogers-Ramanujan identities are equalit
 ies between functions for $|q|<1$. In this talk we discuss another type of
  $q$-series identity\, called a quantum q-series identity\, which is valid
  only at roots of unity. We note some examples from work of Cohen\, Bryson
 -Ono-Pitman-Rhoades\, and  Folsom-Ki-Vu-Yang\, and then show how these and
  many more quantum identities follow from classical q-hypergeometric trans
 formations.  In the second part of the talk we discuss examples of quantum
  q-series identities arising from knot theory.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lola Thompson (Utrecht University)
DTSTART:20201112T170000Z
DTEND:20201112T180000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/9
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/9/">Counting quaternion algebras\, with applications to spectral geometr
 y</a>\nby Lola Thompson (Utrecht University) as part of EIMI Number Theory
  Seminar\n\n\nAbstract\nWe will introduce some classical techniques from a
 nalytic number theory and show how they can be used to count quaternion al
 gebras over number fields subject to various constraints. Because of the c
 orrespondence between maximal subfields of quaternion algebras and geodesi
 cs on arithmetic hyperbolic manifolds\, these counts can be used to produc
 e quantitative results in spectral geometry. This talk is based on joint w
 ork with B. Linowitz\, D. B. McReynolds\, and P. Pollack.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Andrews (Pennsylvania State University)
DTSTART:20201119T150000Z
DTEND:20201119T170000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/10
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/10/">How Ramanujan May Have Discovered the Mock Theta Functions</a>\nby 
 George Andrews (Pennsylvania State University) as part of EIMI Number Theo
 ry Seminar\n\n\nAbstract\nThe mock theta functions made their first appear
 ance in Ramanujan's last letter to Hardy.  Ramanujan explains that he is t
 rying to find functions apart from theta functions that behave like theta 
 functions near the unit circle.  Where did he ever get the idea that such 
 functions might exist?  Why in the world would he consider the special q-s
 eries that he lists in his last letter?  The object of this talk is to pro
 vide a plausible explanation for the discovery of mock theta functions.\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amanda Folsom (Amherst College)
DTSTART:20201201T150000Z
DTEND:20201201T160000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/11
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/11/">Almost harmonic Maass forms and Kac-Wakimoto characters</a>\nby Ama
 nda Folsom (Amherst College) as part of EIMI Number Theory Seminar\n\n\nAb
 stract\nWe explain the modular properties of certain characters due to Kac
  and Wakimoto pertaining to $sl(m|n)^{}$\, where n is a positive integer. 
 We prove that these characters are essentially holomorphic parts of new au
 tomorphic objects we call "almost harmonic Maass forms\," which generalize
  both harmonic Maass forms and almost holomorphic modular forms. By using 
 new methods involving meromorphic Jacobi forms\, this generalizes prior wo
 rks of Bringmann-Ono and Bringmann-Folsom\, which treat the case n=1.  Thi
 s is joint work with Kathrin Bringmann (University of Cologne).\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Osburn (University College Dublin)
DTSTART:20201208T150000Z
DTEND:20201208T170000Z
DTSTAMP:20260404T111008Z
UID:EIMINT/12
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/EIMIN
 T/12/">Generalized Fishburn numbers\, torus knots and quantum modularity</
 a>\nby Robert Osburn (University College Dublin) as part of EIMI Number Th
 eory Seminar\n\n\nAbstract\nThe Fishburn numbers are a sequence of positiv
 e integers with numerous combinatorial interpretations and interesting asy
 mptotic properties. In 2016\, Andrews and Sellers initiated the study of a
 rithmetic properties of these numbers. In this talk\, we discuss a general
 ization of this sequence using knot theory and the quantum modularity of t
 he associated Kontsevich-Zagier series.\n\nThe first part is joint work wi
 th Colin Bijaoui (McMaster)\, Hans Boden (McMaster)\, Beckham Myers (Harva
 rd)\, Will Rushworth (McMaster)\, Aaron Tronsgard (Toronto) and Shaoyang Z
 hou (Vanderbilt) while the second part is joint work with Ankush Goswami (
 RISC).\n
LOCATION:https://stable.researchseminars.org/talk/EIMINT/12/
END:VEVENT
END:VCALENDAR