Speakers

Title: Congruences for the partition function 

Abstract: I will discuss recent work with a number of co-authors about congruences for the ordinary partition function and for the generalized frobenius partition functions (which are a natural higher level generalization).  The methods largely involve the theory of half-integral weight modular forms. 

Title: Finite trigonometric sums: identity of Ramanujan, evaluations, relations

Abstract: Trigonometric sums arise in the famous, unsolved Gauss Circle Problem. In his lost notebook, Ramanujan recorded an elegant identity connected with the Circle Problem. This leads to sums of products of trig functions and several open problems connected with them. We evaluate analogues of Ramanujan sums, one of which leads to a theorem similar to the Franel–Landau criterion for the Riemann Hypothesis. Certain finite trigonometric sums are evaluated in closed form. Reciprocity and/or three sum relations are established for others. Much of this research was conducted with Sun Kim and Alexandru Zaharescu.

Title: On the q-Chu-Vandermonde identity and its consequences 

Abstract: In this talk, I will first introduce the q-Chu-Vandermonde identity and then connect it to the study of Rogers-Ramanujan identities and Ramanujan-Gollnitz Gordon identities. The talk is based on my recent works with Song Heng Chan and Zhi-Guo Liu. 

Title: On a modular differential equation and the Rogers-Ramanujan functions 

Abstract: I will first review my old work on a second-order modular differential equation, and then revisit the Rogers-Ramanujan functions in relation to the Bianchi normal form of elliptic curves. 

Title: Matching Coefficients, Sign patterns, and congruences of certain infinite products related to an elegant identity of Ramanujan for the Rogers-Ramanujan continued fraction 

Abstract: The Rogers-Ramanujan continued fraction, $R(q)$, defined by

$$R(q):=\dfrac{q^{1/5}}{1}_{+}\dfrac{q}{1}_{+}\dfrac{q^2}{1}_{+}\dfrac{q^3}{1}_{+~\cdots,}\quad|q|<1,$$

has the $q$-product representation

$$R(q)=\dfrac{(q;q^5)_\infty(q^4;q^5)_\infty}{(q^2;q^5)_\infty(q^3;q^5)_\infty}.$$

In his first letter to Hardy, written on January 16, 1913, \textbf{Ramanujan} sent \textbf{Hardy} an elegant identity expressing 

$R^5(q)$ in terms of $R(q^5)$; namely,

$$R^5(q)=R(q^5)~\dfrac{1-2R(q^5)+4R^2(q^5)-3R^3(q^5)+R^4(q^5)}{1+3R(q^5)+4R^2(q^5)+2R^3(q^5)+R^4(q^5)}.$$

In this talk, we will discuss some results on matching coefficients, sign patterns, and congruences of certain infinite products related to this identity. 

This is a joint work with \textbf{Abhishek Sarma} and \textbf{Pranjal Talukdar}. 

Title: Elliptic functions and Basic hypergeometric series 

Abstract: Motivated by techniques from the theory of elliptic functions, we present new, simple proofs of several fundamental results in the theory of basic hypergeometric series. These include the q-binomial theorem, Ramanujan’s 1ψ1 summation formula, Rogers’ 6ϕ5 summation formula, Bailey’s 6ψ6 summation formula. This talk is based on joint work with Heng Huat Chan and Zhi-Guo Liu.

Title: The Rogers-Ramanujan dissection of a theta function

Abstract: Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which, as shown by Andrews, not only gives a famous modular relation between the Rogers-Ramanujan functions G(q) and H(q) as a corollary but also a relation between two fifth order mock theta functions and G(q) and H(q). In this talk, we will discuss a generalization of Ramanujan's relation which gives an infinite family of such identities. Our result shows that a theta function can always be ``dissected'' as a finite sum of products of generalized Rogers-Ramanujan functions. 

Several new and well-known results are shown to be consequences of our theorem, for example, a generalization of the Jacobi triple product identity and Andrews' relation between two of his generalized third order mock theta functions. As will be shown, the identities resulting from our main theorem for s>2 transcend the modular world and hence look difficult to be written in the form of a modular relation. Using asymptotic analysis, we also offer compelling evidence that explains how Ramanujan may have arrived at his generalized modular relation. This is joint work with Gaurav Kumar.

Title: On the Ramanujan conjecture and triple product L-functions 

Abstract: The original Ramanujan conjecture concerns the absolute value of the coefficients of the $\Delta$-function.  It was proved by Deligne using his proof of the Weil conjectures.  However, this is not the end of the story.  Ramanujan's conjecture and the Selberg eigenvalue conjecture admit a common generalization, namely that all cuspidal automorphic representations of GL_n are tempered.  This conjecture is unknown even for cuspidal automorphic representations of GL_2 over the rational numbers.  As one of the first indications of the power of Langlands functoriality, Langlands proved that the Ramanujan conjecture would follow from special cases of his functoriality conjecture.  Following his argument, I will explain how my recent work with P. Gu, C-H. Hsu, and S. Leslie provides a small step towards proving enough of Langlands functoriality to deduce the Ramanujan conjecture. 

Title: Detection and Combinatorics of Partitions 

Abstract: An irreducible subgroup of GL(n) that stabilizes a line in a given representation of GL(n) is said to be detected by that representation. The notion of detection was originally motivated by Langlands’ beyond endoscopy proposal, but one can study it using concrete methods in representation theory, algebraic combinatorics, and in particular partition theory. I will survey my work in this subject and propose some open problems.   

Title: Ramanujan type congruences for Rogers-Ramanujan and Rogers-Selberg Quotients 

Abstract: We present a general approach to determine congruences for quotients of Rogers-Ramanujan functions and Rogers-Selberg functions. The goal is to characterize a broad class of quotients whose Fourier coefficients satisfy Ramanujan-type congruences for sufficiently large primes. Exponent sets corresponding to such quotients are described by approaching the problem geometrically.  Ramanujan's integral formulation of the Rogers-Ramanujan continued fraction leads to an inductive proof that certain tetrahedral regions contain lattice points corresponding to quotients that satisfy Ramanujan-type congruences. We present an algorithm for finding and proving all congruence classes in the exponents modulo each prime for which Ramanujan-type congruences occur. The theory of theta cycles can be used to preclude congruences for certain classes of exponents. 

Title: Asymptotic distribution of partition Statistics 

Abstract: Various partition statistics have been introduced to understand the arithmetic and/or combinatorial properties of integer partitions. In this talk, we will focus on two of them: the reciprocal sums of parts and the number of $t$-hooks of a partition.

Let ${\rm srp}(\lambda)$ denote the sum of reciprocal parts of a partition $\lambda$, and $n_t (\lambda)$ be the number of $t$-hooks in $\lambda$. We investigate the distribution of these statistics by introducing random variables that take values ${\rm srp}(\lambda)$ or $n_t(\lambda)$, assuming $\lambda$ is chosen uniform randomly from certain sets of partitions. Utilizing generating functions for moments, we will present their asymptotic means and variances. Additionally, we will discuss modularity properties of the generating functions for moments of the reciprocal sums.

This talk is based on joint works with Kathrin Bringmann and Eunmi Kim, and with Hyunsoo Cho, Eunmi Kim, and Ae Ja Yee. 

Title: Recursion formuals for modular traces 

Abstract: (joint work with Soyoung Choi, Hee Doo Kang and Kyeong Seok Min)

The traces of singular moduli were originally introduced by Zagier and their modularity was established by Bruinier and Funke. In this talk I will explain how to derive recursion formulas satisfied by modular traces of weakly weakly holomorphic modular functions and more generally modular traces of certain weak Maass forms of weight zero. The two main tools for deriving such recursions are Jacobi forms and modular equations. I will give concrete examples in the case of Hauptmodul for genus zero function field. 

Title: Distribution of zeros of derivatives of L-functions 

Abstract: Motivated by the close connection of the zeros of the derivative of the Riemann zeta function, we study the zeros of higher order derivatives of certain L-functions. This is joint work with Rahul Kumar. 

Title: Hauptmoduln and even-order mock theta functions modulo 2 

Abstract: The Fourier coefficients c(n) of the elliptic modular j-function are always even for n \not\equiv 7 (mod 8). In contrast, for n \equiv 7 (mod 8), it is conjectured that ``half" of the coefficients take odd values. In this talk, we first observe in detail when c(8n − 1) is odd and show that the coefficients share the same parity as the coefficients of the 2nd order mock theta function μ_2(q). Furthermore, we prove that this phenomenon also holds among several hauptmoduln and between hauptmoduln and even-order mock theta functions. This is a joint work with Soon-Yi Kang, Seonkyung Kim, and Jaeyeong Yoo (Kangwon National University) 

Title: Summation formulas and Hurwitz L-functions 

Abstract: In this talk, I will briefly explore the rich history of summation formulas, beginning with a brief overview of their classical applications. Building on this, I will introduce a new summation formula that applies to a broad class of arithmetical functions, offering a unifying perspective.

As one of the key applications, I will define a novel analogue of the Hurwitz zeta function for a class of arithmetical functions and delve into its analytic properties, such as its analytic continuation, explicit location of poles, and their special values.

This work is in collaboration with Professor Madeline Dawsey. 

Title: The largest size of an $(s,s+2)$-core with the same parity parts 

Abstract: Olsson and Stanton initiated the computation of the largest size of a simultaneous core partition. Since then, there have been various works on the largest size of a core partition. In particular, Nam and Yu computed the largest size of an $(s,s+1)$-core partition, where all the parts are of the same parity. In this paper, we evaluate the largest size of an $(s,s+2)$-core partition, where all the parts are of the same parity. 

Title: RANK, TWO-COLOR PARTITIONS AND MOCK THETA FUNCTIONS 

Abstract: In this talk, we establish that the number of partitions of a natural number with

positive odd rank is equal to the number of two-color partitions (red and blue), where the smallest part is even (say 2n) and all red parts are even and lie within the interval (2n; 4n]. This led us to derive a new representation for the third order mock theta function f3(q) and an analogue of the fundamental identity for the smallest part partition function Spt(n), both of which are of significant interest in their own right. We also consider the odd smallest part version of the above two-color partition, whose generating function involves another third order mock theta function φ3(q). This is joint work with Professor George Andrews 

Title: Closed formulas for the representation numbers of certain quaternary quadratic forms 

Abstract: The famous Lagrange four-square theorem states that every positive integer can be expressed as a sum of four squares. A quantitative version of this theorem was discovered by Jacobi around 1830. Since then, this result has been generalized in various directions. For example, for some diagonal quaternary forms with coefficients of the form $2^{k}$, closed formulas for the representation numbers have been studied by several authors. In this talk. we further extend these results by deriving closed formulas for the representation numbers of all Bell-type quaternary quadratic forms whose class number is at most two. This is joint work with Chang Heon Kim, Kyoungmin Kim, and Soonhak Kwon. 

Title: GBG-rank generating functions for ordinary, self-conjugate, and doubled distinct partitions 

Abstract: In 2008, Berkovich and Garvan introduced the BG-rank to establish generalizations and refinements of Ramanujan's modulo $5$ partition congruence. Since then, the BG-rank has been used to prove various partition congruences and refinements, and has recently played an important role in the enumeration of Kleshchev bipartitions.

In this talk, we introduce the BG-rank, its generalization, the GBG-rank, and the GBG-rank generating functions for ordinary, self-conjugate, and doubled distinct partitions. We discuss how these combinatorial statistics are applied in the theory of partitions and beyond. 

Title: Newman's conjecture for the partition function 

Abstract: Let $p(n)$ denote the partition function, which is the number of distinct ways of representing $n$ as a sum of positive integers. In 1960, Newman conjectured that for each positive integer $M$ and each non-negative integer $r<M$, there are infinitely many positive integers n such that $p(n)\equiv r \pmod{M}$. In this talk, I will present our recent progress on Newman’s conjecture in the case where $M$ has more than one distinct prime divisor.

If time permits, I will also introduce analogous results for other partition functions such as the t-core partition function. This is a joint work with Dohoon Choi. 

Title: ON A PRODUCT OF THREE THETA FUNCTIONS AND THE NUMBER OF REPRESENTATIONS OF INTEGERS AS MIXED TERNARY SUMS INVOLVING SQUARES, TRIANGULAR, PENTAGONAL AND OCTAGONAL NUMBERS 

Abstract:  In this talk, we derive a general formula to express the product of three theta functions as a linear combination of other products of three theta functions. Moreover, we use the main formula to deduce a general formula for the product of two theta functions. Furthermore, as applications, we extract several theorems in the theory of representation of integers as mixed ternary sums involving squares, triangular numbers, generalized pentagonal numbers and generalized octagonal numbers. This is joint work with N A S Bulkhali and G K Keerthana. 

Title: Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions

Abstract: We establish Kronecker limit type formula for the generalized Mordell-Tornheim zeta function $\Theta(r,r,t,x)$ as a function of the third argument around $t=1-r$. We then show that the above Kronecker limit type formula is equivalent to the two-term functional equation of the higher Herglotz function obtained by Vlasenko and Zagier. We also show the equivalence between a previously known Kronecker limit type formula for $\Theta(1,1,t,x)$ around $t=0$ and the two-term functional equation of the Herglotz-Zagier function obtained by Zagier. Using the theory of the Mordell-Tornheim zeta function, we obtain results of Herglotz, Ramanujan, Guinand, Zagier and Vlasenko-Zagier as corollaries.

We further generalize a special case of $\Theta(r,s,t,x)$ by incorporating two Gauss sums, each associated to a Dirichlet character, and decompose it in terms of an interesting integral which involves the Fekete polynomial as well as the character polylogarithm. This result gives infinite families of functional equations of Herglotz-type integrals out of which only two, due to Choie and Kumar, were known so far. The talk is based on my recent works with Atul Dixit and Sumukha Sathyanarayana.