Related papers: Nonsymmetric Rogers-Ramanujan sums and thick Demaz…
We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this…
We extend the definitions of Chern-Schwartz-MacPherson (CSM) cycles of matroids to tropical manifolds. To do this, we provide an alternate description of CSM cycles of matroids which is invariant under integer affine transformations.…
We prove a new Fermionic quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson-Slater $q$-series identity of Rogers-Ramanujan type and to Nahm sums for the matrix $\left(…
We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of…
The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier…
A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of…
A basic problem for any class of nonassociative algebras is to determine the polynomial identities satisfied by the symmetrization and the skew-symmetrization of the original product. We consider the symmetrization of the product in the…
We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence…
Non-global logarithms (NGLs) are the leading manifestation of correlations between distinct phase space regions in QCD and gauge theories and have proven a challenge to understand using traditional resummation techniques. Recently, the…
We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of…
Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. We describe {standard graded} $\mathfrak{g}$-modules $V$, which we use to construct a completion $\widehat{V}$ and pro-unipotent group $\widehat{U}$ in $\GL(\widehat{V})$. These…
We develop a theory of levels for irreducible representations of symmetric groups of degree $n$ analogous to the theory of levels for finite classical groups. A key property of level is that the level of a character, provided it is not too…
In the present article we introduce and study a class of topological reflection spaces that we call Kac-Moody symmetric spaces. These generalize Riemannian symmetric spaces of non-compact type. We observe that in a non-spherical Kac-Moody…
In their seminal work J. Lepowsky and R. L. Wilson gave a vertex-operator theoretic interpretation of Gordon-Andrews-Bressoud's generalization of Rogers-Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the…
We prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is…
We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then…
An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm\"{u}ller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower…
In this paper, we study the relationship between polynomial integrals on the unitary group and the conjugacy class expansion of symmetric functions in Jucys-Murphy elements. Our main result is an explicit formula for the top coefficients in…
We introduce and study a family of power series, which we call Theta series, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel--Hernandez, which are…
In this paper we study modular tensor categories (braided rigid balanced tensor categories with additional finiteness and non-degeneracy conditions), in particular, representations of quantum groups at roots of unity. We show that the…