Related papers: Random eigenfunctions on flat tori: universality f…
We present a notion of a random toric surface modeled on a notion of a random graph. We then study some threshold phenomena related to the smoothness of the resulting surfaces.
We study crossing numbers for systoles of congruence surfaces. Taken as a family of curves on a family of surfaces, we show that the growth rate of their intersection is optimally small among all sets of curves of the same cardinality lying…
Given a set of points in P^2, we consider the common zeros of the set of curves of a given degree passing through those points. For general sets of points, these zero sets have the expected dimension and are smooth. In fact, given graded…
Random intersection graphs have received much interest and been used in diverse applications. They are naturally induced in modeling secure sensor networks under random key predistribution schemes, as well as in modeling the topologies of…
We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution…
We introduce the notion of a generalized intersection pairing for an Artin stack with a proper good moduli space and nonempty stable part. For the moduli stack of semistable bundles over a smooth projective curve, there are four known…
In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.
We show that the linear statistics of eigenvalues of circulant matrix obey the Gaussian central limit theorem for a large class of input sequences.
Although intersection homology lacks a ring structure, certain expressions (called uniform) in the intersection homology of an irreducible projective variety $X$ always give the same value, when computed via the decomposition theorem on any…
We demonstrate that the melting curves of various model systems of interacting particles collapse to (or are located very close to) a universal master curve on a plane of appropriately chosen scaled variables. The physics behind this…
Given the ensemble of random Gaussian Laplace eigenfunctions on the three-dimensional torus (`3d arithmetic random waves'), we investigate the $1$-dimensional Hausdorff measure of the nodal intersection curve against a compact regular toral…
Uniform probability distributions on $\ell_p$ balls and spheres have been studied extensively and are known to behave like product measures in high dimensions. In this note we consider the uniform distribution on the intersection of a…
The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance)…
We prove the following results: (1) For every generic closed smooth curve in $\mathbb{R}^3$ there is a point with at least $6$ emanating normals to the curve. (2) For every generic closed piecewise linear curve in $\mathbb{R}^3$ there is a…
By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length of the adjacency matrices of Erd\H{o}s-R\'enyi (ER) {\it fully}…
In this short note, we expose some of the works on Serre intersection multiplicity conjecture. I provide a proof of the vanishing of Serre intersection multiplicity in non-proper intersection over a regular ring based on the intersection…
We prove a generalization of Fulton's conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.
The universality of small scales, a cornerstone of turbulence, has been nominally confirmed for low-order mean-field statistics, such as the energy spectrum. However, small scales exhibit strong intermittency, exemplified by formation of…
We study the universality of the eigenvalue statistics of the covariance matrices $\frac{1}{n}M^* M$ where $M$ is a large $p\times n$ matrix obeying condition $\bf{C1}$. In particular, as an application, we prove a variant of universality…
We show that for any sufficiently rich compact family $\mathcal{H}$ of $C^1$ diffeomorphisms of a closed Riemannanian manifold $M$, the average geometric intersection number over $h \in \mathcal{H}$ between $h(V)$ and $W$, for $V, W$ any…