Searcharxiv⌕ Search

arXiv subjects

Mark Goresky

Publications and source records attributed to Mark Goresky.

At least 19 recordsLinked to original sources
arXiv2026

Cycle intersection form and oscillation of graph eigenfunctions

For a real symmetric matrix $H$ strictly supported on a finite simple graph, it is shown that the inertia of a weighted intersection form on the cycle space of the graph, with weights derived from a non-vanishing eigenvector of $H$, governs oscillation data on the graph. Specifically, the null space of the form controls eigenvalue multiplicity, while its Morse index determines the number of sign changes across edges. In the case of a simple eigenvalue, the Hessian at zero of the eigenvalue branch of the discrete magnetic Schrödinger operator is identified with the dual of the cycle intersection form. Applications are given to stability analysis of coupled oscillator networks, to the local behavior of dispersion relations for (decorated) strained graphene, and to nodal-domain counts.

math.SP↗
arXiv2025

Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs

Motivated by the nodal distribution universality conjecture for discrete operators on graphs and by the spectral analysis of their maximal abelian covers, we consider a family of Hermitian matrices $h_α$ obtained by varying the complex phases of individual matrix elements. This family is parametrized by a $β$-dimensional torus, where $β$ is the first Betti number of the underlying graph. The eigenvalues of each matrix are ordered, enabling us to treat the $k$-th eigenvalue $λ_k$ as a function on the torus. We classify the smooth critical points of $λ_k$, describe their structure and Morse index in terms of the support and nodal count, that is, the number of sign changes between adjacent vertices of the corresponding eigenvector. In general, the families under consideration exhibit critical submanifolds rather than isolated critical points. These critical manifolds appear frequently and cannot be removed through perturbations. We provide an algorithmic way of determining all critical submanifolds by investigating finitely many eigenvalue problems: the $2^β$ real symmetric matrices $h_α$ in the family under consideration as well as their principal minors.

math-ph↗
arXiv2024

Nodal count for a random signing of a graph with disjoint cycles

Let $G$ be a simple, connected graph on $n$ vertices, and further assume that $G$ has disjoint cycles. Let $h$ be a real symmetric matrix supported on $G$ (for example, a discrete Schrödinger operator). The eigenvalues of $h$ are ordered increasingly, $λ_1 \le \cdots \le λ_n$, and if $ϕ$ is the eigenvector corresponding to $λ_k$, the nodal (edge) count $ν(h,k)$ is the number of edges $(rs)$ such that $ h_{rs}ϕ_{r}ϕ_{s}>0$. The nodal surplus is $σ(h,k)= ν(h,k) - (k-1)$. Let $h'$ be a random signing of $h$, that is a real symmetric matrix obtained from $h$ by changing the sign of some of its off-diagonal elements. If $h$ satisfies a certain generic condition, we show for each $k$ that the nodal surplus has a binomial distribution $σ(h',k)\sim Bin(β,\frac{1}{2})$. Part of the proof follows ideas developed by the first author together with Ram Band and Gregory Berkolaiko in a joint unpublished project studying a similar question on quantum graphs.

math-ph↗
arXiv2023

Morse theory for discrete magnetic operators and nodal count distribution for graphs

Given a discrete Schrödinger operator $h$ on a finite connected graph $G$ of $n$ vertices, the nodal count $ϕ(h,k)$ denotes the number of edges on which the $k$-th eigenvector changes sign. A {\em signing} $h'$ of $h$ is any real symmetric matrix constructed by changing the sign of some off-diagonal entries of $h$, and its nodal count is defined according to the signing. The set of signings of $h$ lie in a naturally defined torus $\mathbb{T}_h$ of ``magnetic perturbations" of $h$. G. Berkolaiko discovered that every signing $h'$ of $h$ is a critical point of every eigenvalue $λ_k:\mathbb{T}_h \to \mathbb{R}$, with Morse index equal to the nodal surplus. We add further Morse theoretic information to this result. We show if $h_α \in \mathbb{T}_h$ is a critical point of $λ_k$ and the eigenvector vanishes at a single vertex $v$ of degree $d$, then the critical point lies in a nondegenerate critical submanifold of dimension $d+n-4$, closely related to the configuration space of a planar linkage. We compute its Morse index in terms of spectral data. The average nodal surplus distribution is the distribution of values of $ϕ(h',k)-(k-1)$, averaged over all signings $h'$ of $h$. If all critical points correspond to simple eigenvalues with nowhere-vanishing eigenvectors, then the average nodal surplus distribution is binomial. In general, we conjecture that the nodal surplus distribution converges to a Gaussian in a CLT fashion as the first Betti number of $G$ goes to infinity.

math-ph↗
arXiv2021

Lecture notes on sheaves and perverse sheaves

Informal lecture notes with examples on sheaf theory and the derived category of sheaves; sheaves and Morse theory; perverse sheaves, and some applications to representation theory. Added Oct 2021: cellular perverse sheaves. Proofs are outlined with references for further details.

math.AG↗
arXiv2019

Pfaffian integrals and invariants of singular varieties

Integrals of the Pfaffian form over the nonsingular part of a projective variety compute information closely related to the Mather-Chern class of the variety and to other invariants such as the local Euler obstruction along strata of its singular locus and, in the hypersurface case, Milnor numbers. We obtain simple proofs of these formulas, recovering along the way several classically known results.

math.AG↗
arXiv2017

Real structures on ordinary Abelian varieties

The authors define an "anti-holomorphic" involution (or "real structure") on an ordinary Abelian variety (defined over a finite field k) to be an involution of the associated Deligne module (T,F,V) that exchanges F (the Frobenius) with V (the Verschiebung). The definition extends to include principal polarizations and certain level structures. The authors show there are finitely many isomorphism classes in each dimension, and they give a formula for this number which resembles the Kottwitz "counting formula" (for the number of principally polarized Abelian varieties over k), where the symplectic group (in the Kottwitz formula) has been replaced by the general linear group.

math.NT↗
arXiv2014

Euler flag enumeration of Whitney stratified spaces

The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a cd-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components.

math.CO↗
arXiv2007

Loop Products and Closed Geodesics

We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.

math.AT↗
arXiv2006

Discrete series characters and the Lefschetz formula for Hecke operators

This paper gives combinatorial formulas for discrete series constants, both stable and unstable, on real reductive groups. It also carries out one step of the comparison of the topological trace formula for Hecke operators with Arthur's trace formula for Hecke operators, the step in which the topological terms are combined in such a way that stable discrete series characters appear.

math.RT↗
arXiv2006

Codimensions of root valuation strata

This paper defines and studies a stratification of the adjoint quotient of the Lie algebra of a reductive group over a Laurent power series field. The stratification arises naturally in the context of affine Springer fibers.

math.RT↗
arXiv2003

Regular points in affine Springer fibers

Work of Kazhdan-Lusztig and Bezrukavnikov suggests the importance of points in affine Springer fibers for which the associated conjugacy class in the finite dimensional Lie algebra is regular. Such points are characterized in a different way in this paper, using the same kind of information as goes into the weight factors used to form weighted orbital integrals.

math.RT↗
arXiv2003

Purity of equivalued affine Springer fibers

The affine Springer fiber corresponding to a regular integral equivalued semisimple element admits a paving by vector bundles over Hessenberg varieties and hence its (Borel-Moore) homology is "pure".

math.RT↗
arXiv2003

Homology of affine Springer fibers in the unramified case

Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semi-simple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the "kappa" orbital integral which arises in the fundamental lemma is equal to the Lefschetz trace of Frobenius acting on the etale cohomology of a related variety.

math.RT↗
arXiv2001

Moduli space of real abelian varieties with level structure

The moduli space of principally polarized abelian varieties with real structure and with level $N=4m$ structure (with $m \ge 1$) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over $\mathbb Q$, and to consist of finitely many copies of the quotient of the space $\mathbf{GL}(n, \mathbb R)/\mathbf O(N)$ (of positive definite symmetric matrices) by the principal congruence subgroup of level $N$ in $\mathbf{GL}(n, \mathbb Z).$

math.AG↗
arXiv2001

Abelian surfaces with anti-holomorphic multiplication

For appropriate $N\ge 3$ and $d<0,$ the moduli space of principally polarized abelian surfaces with level $N$ structure and anti-holomorphic multiplication by $\mathcal O_d$ (the ring of integers in $\mathbb Q(\sqrt{d})$) is shown to consist of the real points of a quasi-projective algebraic variety defined over $\mathbb Q$, and to coincide with finitely many copies of the quotient of hyperbolic 3-space by the principal congruence subgroup of level $N$ in $\mathbf{SL}(2, \mathcal O_d).$

math.AG↗