Related papers: The monotone secant conjecture in the real Schuber…
The HRT conjecture states that any finite collection of time-frequency shifts of a non-zero square-integrable function on the real line is linearly independent. In this paper, we establish the linear independence of finite systems of…
We develop a theory of the field of double Laurent series, iterated Laurent series, and Malcev-Neumann series that applies to most constant term evaluation problems. These include (i) MacMahon's partition analysis, counting solutions of…
In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is…
We establish that a mode-coupling approximation for the dynamics of multi-component systems obeying Smoluchowski dynamics preserves a subtle yet fundamental property: the matrices of partial density correlation functions are completely…
We reduce some key calculations of compositions of morphisms between Soergel bimodules ("Soergel calculus") to calculations in the nil Hecke ring ("Schubert calculus"). This formula has several applications in modular representation theory.
Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and…
Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard…
We provide counter-examples to Mulmuley's strong saturation conjecture (strong SH) for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker…
We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary…
In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We…
Traditional formulations of geometric problems from the Schubert calculus, either in Plucker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that are typically far from complete intersections and…
Tsfasman-Boguslavsky Conjecture predicts the maximum number of zeros that a system of linearly independent homogeneous polynomials of the same positive degree with coefficients in a finite field can have in the corresponding projective…
We consider the theory of algebraically closed fields of characteristic zero with multivalued operations $x\mapsto x^r$ (raising to powers). It is in fact the theory of equations in exponential sums. In an earlier paper we have described…
We propose to study the quantum Schubert calculus for Schubert varieties, and investigate the smooth Schubert divisors X of the complete flag variety Fl_n. We provide a Borel-type ring presentation of the quantum cohomology of X. We derive…
For any fixed irrational frequency and trigonometric-polynomial potential, we show that every type I energy with positive Lyapunov exponent that satisfies the gap-labelling condition is a boundary of an open spectral gap. As a corollary,…
We explore systems of polynomial equations where we seek complex solutions with absolute value 1. Geometrically, this amounts to understanding intersections of algebraic varieties with tori -- Cartesian powers of the unit circle. We study…
We introduce polynomials that represent general degeneracy loci for maps of vector bundles. These polynomials specialize to the known classical and quantum forms of single and double Schubert polynomials. This is the final version of the…
In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with…
The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n-d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of…
We describe a direct connection between the representation theory of the general linear group and classical Schubert calculus on the Grassmannian, which goes via the Chern-Weil theory of characteristic classes. We also explain why the…