Related papers: Dwork-type supercongruences through a creative $q$…

Let $f(q)$ denote Ramanujan's mock theta function \[f(q) = \sum_{n=0}^{\infty} a(n) q^{n} := 1+\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{(1+q)^{2}(1+q^{2})^{2}\cdots(1+q^{n})^{2}}.\] It is known that there are many linear congruences for the…

Tverberg-type theory aims to establish sufficient conditions for a simplicial complex $\Sigma$ such that every continuous map $f\colon \Sigma \to \mathbb{R}^d$ maps $q$ points from pairwise disjoint faces to the same point in…

"Divergent" Ramanujan-type series for $1/\pi$ and $1/\pi^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by…

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…

We introduce an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In…

A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to…

In this paper we attack the Erdos-Straus conjecture by means of the structure of its solutions, extending and improving the results of a previous paper. Using previous results and supported by the works of Elsholtz and Tao and Monks and…

In the genus 1 case, mirror symmetry reduces to the statement that a certain family of generating functions, relating to an elliptic curve, are quasimodular. In their proof of this fact, Kaneko and Zagier used a related family of generating…

Traces of singular moduli were introduced and studied by Zagier in 1998. Being simultaneously the (traces of) values of a modular function ($j$-invariant) and Fourier coefficients of modular forms - which constitutes Zagier's duality -…

In this paper, we use the Banach fixed point theorem to examine the existence of meromorphic solutions to the following first-order $q$-difference equation \begin{align}\tag{{\dag}}\label{dagger}…

Using the $3D$ mirror symmetry we construct a system of polynomials $T_s(z)$ with integral coefficients which solve the quantum differential equitation of $X=T^{*} Gr(k,n)$ modulo $p^s$, where $p$ is a prime number. We show that the…

We present a new conformal algebra. It is Z2 x Z2 graded and generated by three N=1 superconformal algebras coupled to each other by nontrivial relations of parafermionic type. The representation theory and unitary models of the algebra are…

Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+\beta) \equiv 0 \pmod{\ell}.$…

Two families of q-Schur algebras associated to Hecke algebras of type D are introduced, and related to a family used by Geck, Gruber and Hiss [10], [11]. We prove that the algebras in one family, called the q-Schur^{1.5} algebras, are…

Let $p$ be prime, and let $p_{[1,p]}(n)$ denote the function whose generating function is $\prod (1-q^n)^{-1}(1 - q^{pn})^{-1}$. This function and its generalizations $p_{[c^{\ell}, d^m]}(n)$ are the subject of study in several recent…

We provide a full and rigorous proof of a theorem attributed to \.Zuk, stating that random groups in the Gromov density model for d > 1/3 have property (T) with high probability. The original paper had numerous gaps, in particular, crucial…

This is an expository paper which gives a quick introduction to Dwork's conjecture about p-adic meromorphic continuation of his unit root zeta function arising from algebraic geometry. Special emphasis is given to the case of elliptic…

We study the confluence property of abstract rewriting systems internal to cubical categories. We introduce cubical contractions, a higher-dimensional generalisation of reductions to normal forms, and employ them to construct cubical…

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite…

We generalize the patterns of supercongruences of Ramanujan-type observed by L. Van Hamme and W. Zudilin to series involving simple square roots anywhere and not only in the result of the sum. To support our observations we give some…