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BEGIN:VEVENT
SUMMARY:Ken Ono (University of Virginia)
DTSTART:20200907T140000Z
DTEND:20200907T150000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/1/">V
 ariations of Lehmer’s Conjecture on the Nonvanishing of the Ramanujan ta
 u-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 specul
 ation on the nonvanishing of Ramanujan's tau-function\, it is natural to a
 sk whether a fixed integer is a value of $\\tau(n)$\, or is a Fourier coef
 ficient of any given newform.  In joint work with J. Balakrishnan\, W. Cra
 ig\, and W.-L. Tsai\, the speaker has obtained some results that will be d
 escribed here. For example\, infinitely many spaces are presented for whic
 h the primes $\\ell\\leqslant 37$ are not absolute values of coefficients 
 of any new forms with integer coefficients. For Ramanujan’s tau-function
 \, such results imply\, for $n>1$\, that $\\tau(n)\\notin \\{\\pm \\ell\\\
 ,:\\\, \\ell<100\n\\\,\\text{is odd prime}\\}$.\n
LOCATION:https://researchseminars.org/talk/EIMINT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruce Berndt (University of Illinois)
DTSTART:20200916T160000Z
DTEND:20200916T170000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/2/">T
 he Circle Problem of Gauss\, the Divisor Problem of Dirichlet\, and Ramanu
 jan's Interest in Them</a>\nby Bruce Berndt (University of Illinois) as pa
 rt of EIMI Number Theory Seminar\n\n\nAbstract\nLet $r_2(n)$ denote the nu
 mber of representations of the positive integer $n$ as a sum of two square
 s\, and let $d(n)$ denote the number of positive divisors of $n$.  Gauss a
 nd Dirichlet were evidently the first mathematicians to derive asymptotic 
 formulas for $\\sum_{n\\leq x}r_2(n)$ and $\\sum_{n\\leq x}d(n)$\, respect
 ively\, as $x$ tends to infinity.  The magnitudes of the error terms for t
 he two asymptotic expansions are unknown.  Determining the exact orders of
  the error terms are the Gauss Circle Problem and Dirichlet's Divisor Prob
 lem\, respectively\, and they represent two of the most famous and difficu
 lt 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\, shor
 tly after Ramanujan arrived in England\, we learn that Ramanujan  was also
  greatly interested in the Circle Problem. In a fragment published with hi
 s Lost Notebook\, Ramanujan stated two doubly infinite series identities i
 nvolving Bessel functions that we think Ramanujan derived to attack these 
 two famous unsolved problems. The identities are difficult to prove.  Unfo
 rtunately\, we cannot figure out how Ramanujan might have intended to use 
 them. We survey what is known about these two unsolved problems\, with a c
 oncentration on Ramanujan's two marvelous and mysterious identities.  Join
 t work with Sun Kim\, Junxian Li\, and Alexandru Zaharescu is discussed.\n
LOCATION:https://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:20260422T225755Z
UID:EIMINT/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/3/">P
 artitions with fixed differences between largest and smallest parts</a>\nb
 y Matthias Beck (San Francisco State University and Freie Universitaet Ber
 lin) as part of EIMI Number Theory Seminar\n\n\nAbstract\nEnumeration resu
 lts on integer partitions form a classic body of mathematics going back to
  at least Euler\, including numerous applications throughout mathematics a
 nd some areas of physics. We study the number $p(n\,t)$ of partitions of $
 n$ with difference $t$ between largest and smallest parts. For example\, $
 p(n\,0)$ equals the number of divisors of $n$\, the function $p(n\,1)$ cou
 nts the nondivisors of $n$\, and $p(n\,2) = \\binom{ \\left\\lfloor \\frac
  n 2 \\right\\rfloor }{ 2 }$. Beyond these three cases\, the existing lite
 rature contains few results about $p(n\,t)$\, even though concrete evaluat
 ions of this partition function are featured in several entries of Sloane'
 s Online Encyclopedia of Integer Sequences. \n\nOur main result is an expl
 icit 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 function fo
 r $t>1$\; 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 quasipolyn
 omial with period 2). Our result generalizes to partitions with an arbitra
 ry number of specified distances.\n\nThis is joint work with George Andrew
 s (Penn State) and Neville Robbins (SF State).\n
LOCATION:https://researchseminars.org/talk/EIMINT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Garvan (University of Florida)
DTSTART:20201001T150000Z
DTEND:20201001T170000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/4/">A
  new approach to Dyson's rank conjectures</a>\nby Frank Garvan (University
  of Florida) as part of EIMI Number Theory Seminar\n\n\nAbstract\nIn 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 divides the
  partitions of 5n+4 into five equal classes. This gave a combinatorial exp
 lanation of Ramanujan's famous partition\ncongruence mod 5. He made an ana
 logous conjecture for the rank mod 7 and the partitions of 7n+5. In 1954 A
 tkin and Swinnerton-Dyer proved Dyson's rank conjectures by constructing s
 everal Lambert-series identities basically using the theory of elliptic fu
 nctions. In 2016 the author gave another proof using the theory of weak ha
 rmonic Maass forms. In this talk we\ndescribe a new and more elementary ap
 proach using Hecke-Rogers series.\n
LOCATION:https://researchseminars.org/talk/EIMINT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damaris Schindler (Goettingen University)
DTSTART:20201005T140000Z
DTEND:20201005T160000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/5/">O
 n the distribution of Campana points on toric varieties</a>\nby Damaris Sc
 hindler (Goettingen University) as part of EIMI Number Theory Seminar\n\n\
 nAbstract\nIn this talk we discuss joint work with Marta Pieropan on the d
 istribution of Campana points on toric varieties. We discuss how this prob
 lem leads us to studying a generalised version of the hyperbola method\, w
 hich had first been developed by Blomer and Bruedern. We show how duality 
 in linear programming is used to interpret the counting result in the cont
 ext of a general conjecture of Pieropan-Smeets-Tanimoto-Varilly-Alvarado.\
 n
LOCATION:https://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:20260422T225755Z
UID:EIMINT/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/6/">M
 ixed Hodge numbers and factorial ratios</a>\nby Fernando Rodriguez-Villega
 s (The Abdus Salam International Centre for Theoretical Physics) as part o
 f EIMI Number Theory Seminar\n\n\nAbstract\nThe factorial ratios of the ti
 tle are numbers such as $\\frac{(30n)!n!}{(6n)!(10n)!(15n)!}$ considered b
 y Chebyshev in his work on the distribution of prime numbers\, which are i
 ntegral for all n in a non-obvious way. I will discuss how integrality is 
 related to the lack of interior points of the first few dilations of an as
 sociated polytope and the vanishing of certain Hodge numbers of associated
  varieties. This work is an offshoot of an ongoing project on hypergeometr
 ic motives joint with D. Roberts and M. Watkins.\n
LOCATION:https://researchseminars.org/talk/EIMINT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Raum (Chalmers Technical University\, Gothenburg\, Sweden)
DTSTART:20201019T160000Z
DTEND:20201019T170000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/7/">R
 elations among Ramanujan-type Congruences</a>\nby Martin Raum (Chalmers Te
 chnical University\, Gothenburg\, Sweden) as part of EIMI Number Theory Se
 minar\n\n\nAbstract\nWe present a new framework to access relations among 
 Ramanujan-type congruences of a weakly holomorphic modular form. The frame
 work is strong enough to apply to all Shimura varieties\, and covers half-
 integral weights if unary theta series are available. We demonstrate effec
 tiveness in the case of elliptic modular forms of integral weight\, where 
 we obtain a characterization of Ramanujan-type congruences in terms of Hec
 ke congruences. Finally\, we showcase concrete computer calculations\, exp
 loring the information encoded by our framework in the case of elliptic mo
 dular forms of half-integral weight. This leads to an unexpected dichotomy
  between Ramanujan-type congruences found by Atkin and by Ono\, Ahlgren-On
 o.\n
LOCATION:https://researchseminars.org/talk/EIMINT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Lovejoy (CNRS)
DTSTART:20201028T170000Z
DTEND:20201028T180000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/8
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/8/">Q
 uantum q-series identities</a>\nby Jeremy Lovejoy (CNRS) as part of EIMI N
 umber Theory Seminar\n\n\nAbstract\nAs analytic identities\, classical $q$
 -series identities like the Rogers-Ramanujan identities are equalities bet
 ween functions for $|q|<1$. In this talk we discuss another type of $q$-se
 ries identity\, called a quantum q-series identity\, which is valid only a
 t roots of unity. We note some examples from work of Cohen\, Bryson-Ono-Pi
 tman-Rhoades\, and  Folsom-Ki-Vu-Yang\, and then show how these and many m
 ore quantum identities follow from classical q-hypergeometric transformati
 ons.  In the second part of the talk we discuss examples of quantum q-seri
 es identities arising from knot theory.\n
LOCATION:https://researchseminars.org/talk/EIMINT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lola Thompson (Utrecht University)
DTSTART:20201112T170000Z
DTEND:20201112T180000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/9/">C
 ounting quaternion algebras\, with applications to spectral geometry</a>\n
 by Lola Thompson (Utrecht University) as part of EIMI Number Theory Semina
 r\n\n\nAbstract\nWe will introduce some classical techniques from analytic
  number theory and show how they can be used to count quaternion algebras 
 over number fields subject to various constraints. Because of the correspo
 ndence between maximal subfields of quaternion algebras and geodesics on a
 rithmetic hyperbolic manifolds\, these counts can be used to produce quant
 itative results in spectral geometry. This talk is based on joint work wit
 h B. Linowitz\, D. B. McReynolds\, and P. Pollack.\n
LOCATION:https://researchseminars.org/talk/EIMINT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Andrews (Pennsylvania State University)
DTSTART:20201119T150000Z
DTEND:20201119T170000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/10/">
 How Ramanujan May Have Discovered the Mock Theta Functions</a>\nby George 
 Andrews (Pennsylvania State University) as part of EIMI Number Theory Semi
 nar\n\n\nAbstract\nThe mock theta functions made their first appearance in
  Ramanujan's last letter to Hardy.  Ramanujan explains that he is trying t
 o find functions apart from theta functions that behave like theta functio
 ns near the unit circle.  Where did he ever get the idea that such functio
 ns might exist?  Why in the world would he consider the special q-series t
 hat he lists in his last letter?  The object of this talk is to provide a 
 plausible explanation for the discovery of mock theta functions.\n
LOCATION:https://researchseminars.org/talk/EIMINT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amanda Folsom (Amherst College)
DTSTART:20201201T150000Z
DTEND:20201201T160000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/11/">
 Almost harmonic Maass forms and Kac-Wakimoto characters</a>\nby Amanda Fol
 som (Amherst College) as part of EIMI Number Theory Seminar\n\n\nAbstract\
 nWe explain the modular properties of certain characters due to Kac and Wa
 kimoto pertaining to $sl(m|n)^{}$\, where n is a positive integer. We prov
 e that these characters are essentially holomorphic parts of new automorph
 ic objects we call "almost harmonic Maass forms\," which generalize both h
 armonic Maass forms and almost holomorphic modular forms. By using new met
 hods involving meromorphic Jacobi forms\, this generalizes prior works of 
 Bringmann-Ono and Bringmann-Folsom\, which treat the case n=1.  This is jo
 int work with Kathrin Bringmann (University of Cologne).\n
LOCATION:https://researchseminars.org/talk/EIMINT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Osburn (University College Dublin)
DTSTART:20201208T150000Z
DTEND:20201208T170000Z
DTSTAMP:20260422T225755Z
UID:EIMINT/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EIMINT/12/">
 Generalized Fishburn numbers\, torus knots and quantum modularity</a>\nby 
 Robert Osburn (University College Dublin) as part of EIMI Number Theory Se
 minar\n\n\nAbstract\nThe Fishburn numbers are a sequence of positive integ
 ers with numerous combinatorial interpretations and interesting asymptotic
  properties. In 2016\, Andrews and Sellers initiated the study of arithmet
 ic properties of these numbers. In this talk\, we discuss a generalization
  of this sequence using knot theory and the quantum modularity of the asso
 ciated Kontsevich-Zagier series.\n\nThe first part is joint work with Coli
 n Bijaoui (McMaster)\, Hans Boden (McMaster)\, Beckham Myers (Harvard)\, W
 ill Rushworth (McMaster)\, Aaron Tronsgard (Toronto) and Shaoyang Zhou (Va
 nderbilt) while the second part is joint work with Ankush Goswami (RISC).\
 n
LOCATION:https://researchseminars.org/talk/EIMINT/12/
END:VEVENT
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