Related papers: A combinatorial proof of the $\lambda_g$ conjectur…

We give a geometric proof of a conjecture of W. Fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for GL(r).

We show that Generic Green's conjecture holds for generic binary curves, through a detailed analysis of the family of scrolls containing fixed rational normal curves.

We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of…

We propose a natural generalization of a conjecture by Garsia, originally concerning the realization of conformal classes of genus-1 surfaces via embeddings in three-dimensional Euclidean space. This generalized conjecture is formulated…

We observe that a lemma used in the study of even sets of nodes on surfaces applies almost verbatim to prove a celebrated formula of Gauss on the 2-torsion of the class group of a quadratic field.

The Razumov-Stroganov conjecture relates the ground-state coefficients in the even-length dense O(1) loop model to the enumeration of fully-packed loop configuration on the square, with alternating boundary conditions, refined according to…

We formulate a version of the Breuil--Mezard conjecture for quaternion algebras, and show that it follows from the Breuil--Mezard conjecture for GL_2. In the course of the proof we establish a mod p analogue of the Jacquet--Langlands…

Let $C$ be a smooth projective curve of genus $g\ge2$ and let $N$ be the moduli space of stable rank $2$ vector bundles on $C$ of odd degree. We construct a semi-orthogonal decomposition of the bounded derived category of $N$ conjectured by…

In a recent paper [8], it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus…

We give a new proof of the string topology structure of a compact oriented surface of genus g greater than or equal to 2, using elementary algebraic topology. This reproves the result of Vaintrob.

The gonality conjecture, proved by Ein--Lazarsfeld, asserts that the gonality of a nonsingular projective curve of genus $g$ can be detected from its syzygies in the embedding given by a line bundle of sufficiently large degree. An…

We establish the exact overlaps conjecture for iterated functions systems on the real line with algebraic contractions and arbitrary translations.

For a graph $G$, and a nonnegative integer $g$, let $a_g(G)$ be the number of $2$-cell embeddings of $G$ in an orientable surface of genus $g$ (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker…

We consider some string invariants at genus two that appear in the analysis of the $D^8\mathcal{R}^4$ and $D^6\mathcal{R}^5$ interactions in type II string theory. We conjecture a Poisson equation involving them and the Kawazumi--Zhang…

We show that many tame modules of the quantum toroidal $\mathfrak{gl}_2$ algebra can be explicitly constructed in a purely combinatorial way using the theory of $q$-characters. The examples include families of evaluation modules obtained…

Let G' be a connected reductive group over the complex numbers. We show that the set of conjugacy classes of G' is in natural bijection with the set of two-sided cells associated to a certain algebra.

We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1- and 2-form gauge theory. In particular, we utilize string diagrams, tools similar to tensor…

Developing observation made in \cite{commut} we show that simple identity of the commutator type on an associative algebra is in one-to-one correspondence to 2D (infinite) Toda chain. We introduce representation of elements of associative…

We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the…

We prove a combinatorial rule for a complete decomposition, in terms of Langlands parameters, for representations of p-adic $GL_n$ that appear as parabolic induction from a large family (ladder representations). Our rule obviates the need…