Related papers: Quantum Bruhat graphs and tilted Richardson variet…

Motivated by the geometry of certain hyperplane arrangements, Manin and Schechtman defined for each positive integer n a hierarchy of finite partially ordered sets B(n, k), indexed by positive integers k, called the higher Bruhat orders.…

This paper provides a detailed exposition of the two main models for equivariant cohomology -- the Cartan and Weil models -- and their explicit isomorphism via the Kalkman (Mathai--Quillen) transformation. We then connect this framework to…

Let g be a complex, simple Lie algebra and t a Cartan subalgebra of g. A new unitary, flat connection on t with values in any finite-dimensional g-module V and simple poles along the root hyperplanes was recently introduced by J. Millson…

Let $G$ be a simple, simply connected, simply laced algebraic group. We construct a monoidal category of representations of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ whose Grothendieck ring contains a cluster algebra with…

In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which…

We apply the quotient graph theory described by Band, Berkolaiko, Joyner and Liu to particular graphs symmetric with respect to $S_3$ and $C_3$ symmetry groups. We find the quotient graphs for the three-edge star quantum graph with Neumann…

The classical permutohedron Perm is the convex hull of the points (w(1),...,w(n)) in R^n where w ranges over all permutations in the symmetric group. This polytope has many beautiful properties -- for example it provides a way to visualize…

Billey and Braden defined a geometric pattern map on flag manifolds which extends the generalized pattern map of Billey and Postnikov on Weyl groups. The interaction of this torus equivariant map with the Bruhat order and its action on line…

We show that a topological quantum computer based on the evaluation of a Witten-Reshetikhin-Turaev TQFT invariant of knots can always be arranged so that the knot diagrams with which one computes are diagrams of hyperbolic knots. The…

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite…

We propose a natural quantized character theory for inductive systems of compact quantum groups based on KMS states on AF-algebras following Stratila-Voiculescu's work (Stratila-Voiculescu, 1975) (or (Enomoto-Izumi, 2016)), and give its…

We prove that Schubert and Richardson varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Bialynicki-Birula cells under…

The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac-Moody group. We give an explicit description…

For any Bott-Samelson resolution $q_{I}:\hat{X_{I}}\rightarrow G/B$ of the flag variety $G/B$, and any torus equivariant oriented cohomology $h_T$, we compute the restriction formula of certain basis $\eta_L$ of $h_T(\hat{X_{I}})$…

Let $p$ be a prime number and let $k$ be an algebraically closed field of characteristic $p$. A $BT_1$ group scheme over $k$ is a finite commutative group scheme which arises as the kernel of $p$ on a $p$-divisible (Barsotti--Tate) group.…

Given a vector bundle $F$ on a smooth Deligne-Mumford stack $\X$ and an invertible multiplicative characteristic class $\bc$, we define the orbifold Gromov-Witten invariants of $\X$ twisted by $F$ and $\bc$. We prove a "quantum Riemann-Roch…

Understanding the metric structure of permutation families is fundamental to combinatorics and has applications in social choice theory, bioinformatics, and coding theory. We study permutation families defined by restriction…

We initiate a study of varieties of minimal degree in weighted projective spaces. We call a weighted projective space $\mathbf{P}(w_0,\dots,w_n)$ divisible if $w_i \mid w_{i+1}$ for all $i$. We provide sharp bounds for when a non-degenerate…

The notion of local weak convergence, or Benjamini--Schramm convergence, was introduced by Benjamini and Schramm. The local weak limit of sparse Erd\H os--R\'enyi graphs is the Galton--Watson measure with Poisson offspring almost surely.…

In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the…