# Publications

In this paper we construct the modular Cauchy kernel $\Xi_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=\gamma z_1, ~\gamma \in \Gamma_0(N) \right\}.$$

The function $\Xi_N(z_1, z_2)$ is used in two cases and for two different purposes. Firstly, we prove generalization of the Zagier theorem (\cite{La}, \cite{Za3}) for the Hecke subgroups $\Gamma_0(N)$ of genus $g>0$. Namely, we obtain a kind of ``kernel function'' for the Hecke operator $T_N(m)$ on the space of the weight 2 cusp forms for $\Gamma_0(N)$, which is the analogue of the Zagier series $\omega_{m, N}(z_1,\bar{z_2}, 2)$. Secondly, we consider an elementary proof of the formula for the infinite Borcherds product of the difference of two normalized Hauptmoduls, ~$J_{\Gamma_0(N)}(z_1)-J_{\Gamma_0(N)}(z_2)$, for genus zero congruence subgroup $\Gamma_0(N)$.

The goal of this work is to present an alternative way of calculating the values of the Green's function associated with the Hirzebruch-Zagier divisor at the points of another Hirzebruch-Zagier divisor $T_m$.

We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational and nodal quartic double solids with at least eleven singular points are rational.

We study the integral Bailey lemma associated with the A_n-root system and identities for elliptic hypergeometric integrals generated thereby. Interpreting integrals as superconformal indices of four-dimensional N = 1 quiver gauge theories with the gauge groups being products of SU(n + 1), we provide evidence for various new dualities. Further con rmation is achieved by explicitly checking that the `t Hooft anomaly matching conditions holds. We discuss a flavour symmetry breaking phenomenon for supersymmetric quantum chromodynamics (SQCD), and by making use of the Bailey lemma we indicate its manifestation in a web of linear quivers dual to SQCD that exhibits full s-confinement.

Given a relatively projective birational morphism f : X → Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators TX,f and S_X,f in D^b(X). We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that D^b(X) has such a filtration L where the lattice is the set of all birational decompositions f : X −→^g Z −→^h Y with smooth Z. The t-structures related to T_X,f and S_X,f are proved to be glued via filtrations left and right dual to L. We realise all such Z as the fine moduli spaces of simple quotients of O_X in the heart of the t-structure for which S_X,g is a relative projective generator over Y . This implements the program of interpreting relevant smooth contractions of X in terms of a suitable system of t-structures on D^b(X).

For a simply connected, connected, semisimple complex algebraic group G, we dene two geometric crystals on the A-cluster variety of double Bruhat cell B\cap \ Bw_0B. These crystals are related by the duality . We dene the graded Donaldson-Thomas correspondence as the crystal bijection between these crystals. We show that this correspondence is equal to the composition of the cluster cham- ber Ansatz, the inverse generalized geometric RSK-correspondence, and transposed twist map due to Berenstein and Zelevinsky.

We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree.

In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a k-linear 1-category for a eld k. Our main result states that if C is a k-linear 1-category which has a compact generator whose groups of self-extensions vanish for suciently high positive degrees, then every formal deformation of C has zero curvature and moreover admits a compact generator.

We consider the conjectures of Katzarkov, Kontsevich, and Pantev about Landau--Ginzburg Hodge numbers associated to tamely compactifiable Landau--Ginzburg models. We test these conjectures in case of dimension two, verifying some and giving a counterexample to the other.

In this paper, we prove that for any A>0 there exist infinitely many primes *p* for which sums of the Legendre symbols modulo *p* over an interval of length (ln p)^A can take large values.

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices $S$ with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.