We design a monotone meshfree finite difference method for linear elliptic PDEs in non-divergence form on point clouds via a nonlocal relaxation method. The key idea is a combination of a nonlocal integral relaxation of the PDE problem with a robust meshfree discretization on point clouds. Minimal positive stencils are obtained through a linear optimization procedure that automatically guarantees the stability and, therefore, the convergence of the meshfree discretization.  A major theoretical contribution is the existence of consistent and positive stencils for a given point cloud geometry. We provide sufficient conditions for the existence of positive stencils by finding neighbors within an ellipse (2d) or ellipsoid (3d) surrounding each interior point, generalizing the study for Poisson’s equation by Seibold in 2008. It is well-known that wide stencils are in general needed for constructing consistent and monotone finite difference schemes for linear elliptic equat ions. Our result represents a significant improvement in the stencil width estimate for positive-type finite difference methods for linear elliptic equations in the near-degenerate regime (when the ellipticity constant becomes small), compared to previously known works in this area. Numerical algorithms and practical guidance are provided with an eye on the case of small ellipticity constant. Numerical results will be presented in both 2d and 3d, examining a range of ellipticity constants including the near-degenerate regime.

 In this talk, I will discuss injectivity and injective envelope of objects in different categories. I will present our recent work, which attempts to answer the question “What happens to injective objects under particular functors?” This is based on joint work with David Blecher and Mehrdad Kalantar. 

We generalize a result of Mazorchuk and Tenner, showing that the “run” statistic influences the shape of the RSK tableaux of an arbitrary permutation. We define and construct the “canonical reduced word” of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations in terms of “uncrowded sets.” We then extend this work to fully commutative permutations, showing that each fully commutative permutation has a well-defined “boolean core,” related to the right weak order. The contents of the second row of the insertion tableaux of fully commutative permutations are partially ordered as subsets, with respect to the right weak order. We explore the partial order of these subsets, with particular interest in when they change from uncrowded to crowded. This is joint work with Gunawan, Russell and Tenner, based on recent work in arXiv:2207.05119 and arXiv:2212.05002.

The purpose of this talk is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $F$ is at most $k$ over the algebraic closure of $F$, where $k$ is a given positive integer. We estimate the arithmetic complexity of our algorithm. 

 Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner's theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider $\mathbb{Z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.

We first describe the mathematical foundation of DSGRN (Dynamic Signatures Generated by Regulatory Networks), an approach that provides a queryable description of global dynamics of a network over its entire parameter space. We also describe a connection to Boolean network models that allows us to view DSGRN as a platform for bifurcation theory of Boolean maps. Finally, we compare DSGRN to RACIPE, an approach based on sampling parameters for ODE models.  We discuss several applications to systems biology as well as design of robust networks in synthetic biology.

Analogies between number theory and graph theory have been studied for quite some time now.  During the past few years, it has been observed in particular that there is an analogy between classical Iwasawa theory and some phenomena in graph theory.  In this talk, we will explain this analogy and present some of the results that have been obtained so far in this area.

[pre-talk at 1:20PM]

Problems in extremal graph theory typically aim to maximize some graph parameter under local restrictions. In order to prove lower bounds for these kinds of problems, several techniques have been developed. The most popular one, initiated by Paul Erdős, being the probabilistic method. While this technique has enjoyed tremendous success, it does not always provide sharp lower bounds. Frequently, algebraically and geometrically defined graphs outperform random graphs. We will show how historically, geometry over finite fields has been a rich source of such graphs. I will show a broad class of graphs defined from geometry of finite fields, which has found several recent applications in extremal graph theory. Often, certain interesting families of graph had in fact already been discovered and studied, years before their value in extremal graph theory was realized. I will demonstrate some instances of this phenomenon as well, which indicates that there might still be uncharted territory to explore.

In this talk I will discuss two different topics: a) the topic of precise asymptotics for linear waves on black hole spacetimes, and b) the topic of construction of spacetimes containing curvature singularities. If time permits, I will try to make connections with more general problems for quasilinear wave equations (for both topics).