Related papers: The Wronski map and shifted tableau theory

We introduce marked relative Pandharipande-Thomas (PT) invariants for a pair $(X,D)$ of a smooth projective threefold and a smooth divisor. These invariants are defined by integration over the moduli space of $r$-marked stable pairs on…

We prove that the space of shifted Poisson structures on a derived scheme $X$ locally of finite presentation is equivalent to the space of shifted Lagrangian thickenings out $X$, solving a conjecture in shifted Poisson geometry. As a…

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2].…

The 2d gauge theory on the lattice is equivalent to the twisted Eguchi-Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large N scaling for the 1- and 2-point function of Wilson loops, as well as the 2-point…

In this paper, we explore the fixed point theory of $n$-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the…

We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-$d$ morphisms from a general genus $g$,…

Let $ R \subset \R $ be a GCD-domain. In this paper, Weinberg's conjecture on the $ n \times n $ matrix algebra $ M_{n}(R) \ (n \geq 2) $ is proved. Moreover, all the lattice orders (up to isomorphisms) on a full $ 2 \times 2 $ matrix…

We study the algebra of Wilson line operators in three-dimensional N=2 supersymmetric U(M) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M,N), and its connection to K-theoretic Gromov-Witten invariants for Gr(M,N).…

Consider k x n matrices with rank conditions placed on intervals of columns. The ranks that are actually achievable correspond naturally to upper triangular partial permutation matrices, and we call the corresponding subvarieties of Gr(k,n)…

Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us…

The Hilbert space of level $q$ Chern-Simons theory of gauge group $G$ of the ADE type quantized on $T^2$ can be represented by points that lie on the weight lattice of the Lie algebra $\mathfrak{g}$ up to some discrete identifications. Of…

Working in any model theoretic structure, we single out a class of definable bipartite graphs that admit definable, close to perfect matchings. We use this result to prove a strengthening of Tarski's theorem for the definable setting.

A classical Theorem of Alexandrov states that the map associating its boundary to a convex polyhdedron of the 3-dimensional Euclidean space is a bijection from the set of convex polyhdedron up to congruence to the set of isometry classes of…

In 2005, A. Knutson--R. Vakil conjectured a puzzle rule for equivariant K-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors' recently…

The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to \R^d identifies points from q disjoint faces. (This has been proved for affine maps, for d=1, and if q is a prime power, but not yet in general.) The…

A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic $\lambda$-hypersurfaces (\cite{…

The ability to align points across two related yet incomparable point clouds (e.g. living in different spaces) plays an important role in machine learning. The Gromov-Wasserstein (GW) framework provides an increasingly popular answer to…

Nazarov-Treil-Volberg recently proved an elegant two-weight T1 theorem for "almost diagonal" operators that played a key role in the proof of the $A_2$ conjecture for dyadic shifts and related operators. In this paper, we obtain a…

We prove the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs $(Y|\partial Y)$ consisting of a toric threefold $Y$ and any torus invariant divisor $\partial…

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a…