Ali Uncu originally discovered the following (Crank-Mex Theorem): The number of partitions of n with non-negative crank is equal to the number of partitions of n with odd mex, where the mex of a partition is the least positive integer that is not a part of the partition. It should be noted that Hopkins and Sellers (as well as others) subsequently independently discovered this theorem. This talk will be devoted to a discussion of how one goes about discovering a proof for the following refinement: Theorem. The number of partitions of n with non-negative crank and k parts that aren't ones equal; the number of partitions on n with odd mex and k parts that aren't ones. A greatly desired, combinatorial proof has eluded us.

About five years ago, Dan Romik considered the Taylor coefficients of Jacobi's theta function $\theta_3(\tau)$ at $\tau=i$ and showed that, under a certain normalisation, these coefficients become integers. He conjectured that, when taken modulo any fixed prime power, these coefficients are eventually periodic, and if $p$ is congruent to 3 modulo 4 then, more precisely, these coefficients eventually vanish modulo any fixed power of $p$. I shall first explain that, in a sense, periodicity was already known (in a more general context), but very hidden (and nobody noticed). The corresponding argument however only provides astronomic bounds on the period length, and it can not address whether the sequence eventually vanishes modulo a prime power. I shall then present joint work with Thomas W. M\"uller in which we prove all of Romik's (not-always-)conjectures together with reasonably good bounds on period lengths respectively on start of vanishing. Furthermore, we show that similar results hold for Jacobi's theta function $\theta_2(\tau)$, all Eisenstein series and even weight modular forms. I shall close with some conjectures on "actual" period lengths respectively on start of vanishing.

In this talk, I will discuss on hyperbolicity of Jensen polynomials through the lens of Griffin, Ono, Rolen, and Zagier’s framework. In particular, will restrict to Jensen polynomials associated with sequences related to partition statistics. This is an ongoing joint work with Kathrin Bringmann and Larry Rolen.

We derive asymptotic expansions for weighted partition numbers satisfying certain conditions. As applications we partially settle some conjectures by Berkovic and Garvan, and by Seo and Yee, on the nonnegativity of the coefficients of certain infinite products, and a conjecture by Chan and Yesilyurt on the periodicity of the signs of the coefficients of a non-theta product. This is joint work with Nian Hong Zhou.

In the recent years, the study of cylindric partitions (from many different angles) has been fruitful. Corteel-Welsh’s recurrence relation and Borodin’s product formula used together proved many beautiful sum-product identities. We will add another angle to the studies and look at cylindric partitions and study these objects with an imposed bound using MacMahon’s partition analysis. We will present a companion to Foda-Quano’s finite Andrews-Gordon identities and infinite hierarchies over these results. This is joint work with Runqiao Li.

We study a level 5 representation in the principal picture of the affine algebra in the hope of generalizing a level 3 identity and a more recent level 4 identity by D. Nandi.

1992, Stefano Capparelli conjectured two partition theorems by a study of vertex operators in Lie Algebras. A few months later, in 1992, Andrews proved the Capparelli conjectures by the use of generating function polynomials. Subsequently in 1992, Alladi-Andrews-Gordon generalized and refined the Capparelli partition theorems by the method of weighted words and gave both combinatorial and q-hypergeometric proofs. In 1994, I noticed that the Capparelli partition theorems are part of an infinite hierarchy of theorems. I this talk I shall present recent partition and hypergeometric results on this infinite hierarchy established jointly with my PhD student Yazan Alamoudi, including polynomial (finite) versions of the hierarchy.

The function $p(n,m,N)$ enumerates the partitions of $n$ into at most m parts with no part larger than $N$. For $m,N\ge 0$, the standard generating function for $p(n,m,N)$ is given by the Gaussian polynomial, also known as the $q$-binomial coefficient: $\binom{N+m}{m}_q$. $$ \binom{N+m}{m}_q = \sum_{n=0}^{mN} p(n,m,N)q^n = \frac{(q;q)_{N+m}}{(q;q)_m (q;q)_N} $$ In this talk we will establish a completely new set of generating functions for $p(n,m,N)$ for $m\le 6$. For example: Proposition 1: $$ \sum_{N=0}^{\infty}p(2N-A,4,N)z^n = \begin{cases} \sum_{N=0}^{\infty} p(2N-2a,4,N)z^N = \frac{z^a(1+z^2-z^{a+1})}{(1-z)^2 (1-z^2) (1-z^3)} & \sum_{N=0}^{\infty} p(2N-(2a+1),4,N)z^N = \frac{z^a(1+z^2-z^{a+2})}{(1-z)^2 (1-z^2) (1-z^3)}. & \end{cases} $$ As $A$ spans the integers, we obtain each and every coefficient of each and every Gaussian polynomial of the form $\binom{N+4}{4}_q$. With these new generating functions established, we are able to quickly prove a host of identities such as the following surprising example: Example: Let $N$ be any nonnegative integer. Then $$ p(2N-2,4,N) = p(2N-1,4,N) = p(2N+1,4,N) = p(2N+2,4,N). $$ Very recently this example was proved independently by Lie algebra researchers D. Burde and F. Wagemann. Our results allow us to prove the unimodality of Gaussian polynomials for $m\le 6$.

In this talk, we introduce new combinatorial objects called $d$--fold partition diamonds, which generalize both the classical partition function (for unrestricted integer partitions) and the partition diamonds of Andrews, Paule, and Riese. We consider two counting functions related to these combinatorial objects, the second of which we call "Schmidt type'' $d$--fold partition diamonds, which has counting function $s_d(n)$. After finding the generating function for $s_d(n)$, we identify a surprising connection to a well--known family of polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan--like congruences satisfied by $s_d(n)$ for various values of $d$, including the following family: for all $d\geq 1$ and all $n\geq 0,$ $s_d(2n+1) \equiv 0 \pmod{2^d}.$ This is joint work with Dalen Dockery, Marie Jameson, and Samuel Wilson (all of the University of Tennessee).

q-series appear in various contexts of mathematics, e.g. in the theory of VOAs, knots, and partitions. Some of these q-series are known to be modular functions, but a complete classification of modular q-series seems out of reach. For a family of q-series called Nahm sums, the modularity is known to be related to the vanishing of elements in the so-called Bloch group. However, a precise conjecture is unknown. In this talk, I present recent results concerning this relationship.

In 1988, Koblitz conjectured the infinitude of primes p for which the group order $|E(F_p)|$ is prime for elliptic curves over Q, drawing an analogy with the twin prime conjecture. He also posed a related question about the quantity $|E(F_{p^l})| / |E(F_p)|$, viewed as an analogue of $(p^l - 1)/(p - 1)$. Motivated by earlier work on $|E(F_p)|$, we study, for primes $l \ge 2$, lower bounds for the number of primes $p \le x$ for which $|E(F_{p^l})| / |E(F_p)|$ has a bounded number of prime factors. The CM case is unconditional, relying on Huxley’s large sieve in number fields. For non-CM curves, we prove analogous results using the Chebotarev density theorem under GRH. In the CM case, we also use a vector sieve to show the infinitude of primes $p$ such that $|E(F_{p^2})|$ is an almost prime.

In this talk, we introduce a new class of mock Eisenstein series that resemble classical Eisenstein series in many ways, but with a key distinction: only their non-holomorphic completions transform like (quasi)modular forms. We show how the partition rank generating function can be expressed in terms of partition traces of these functions. A key feature of our construction is that the completions satisfy a holomorphic anomaly equation--a phenomenon typically seen in the context of quantum field theory and string theory. We also show that the Fourier coefficients of the mock Eisenstein series are integral.