### Recent Submissions

• #### Rough Collisions ﻿

A rough collision law describes the limiting contact dynamics of a pair of rough rigid bodies, as the scale of the rough features (asperities) on the surface of each body goes to zero. The class of rough collision laws is ...
• #### Face Numbers of Polytopes, Posets, and Complexes ﻿

A key tool that combinatorialists use to study simplicial complexes and polytopes is the {\bf $f$-vector} (or face vector), which records the number of faces of each dimension. In order to better understand the face numbers, ...
• #### Flavors of the Fubini-Bruhat Order ﻿

Fubini words are generalized permutations, allowing for repeated letters, and theyare in one-to-one correspondence with ordered set partitions. Brendan Pawlowski and Brendon Rhoades extended permutation matrices to pattern ...
• #### Representations and Support Theory for Bosonized Quantum Complete Intersections ﻿

Support theories are frequently used by representation theorists when trying to understand module categories with complicated structure. We associate to an algebra A a variety where the topological structure is determined ...
• #### Epidemics on critical random graphs: limits and continuum descriptions ﻿

Understanding how diseases spread through populations is vital for mitigation efforts. For any disease at hand, the specifics of how a disease spreads through a community depends on many factors: how the disease is ...
• #### Regularity results for the variable-coeﬃcient Plateau problem ﻿

We study almost-minimizers of anisotropic surface energies deﬁned by a Holder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove ...
• #### Rational Point on Conic Bundles ﻿

In this paper, we focus on obstructions to the existence of rational points for a special class of algebraic varieties. In particular, we consider the case where $\pi \colon X \rightarrow \PPP_k^1$ is a smooth conic bundle ...
• #### Quantitative density statements for translation surfaces ﻿

The main results in this thesis are quantitative descriptions of the orbits of two dynamical systems on translation surfaces. First, we study the action of a discrete subgroup of $SL_2(\R)$ on a closed square-tiled surface ...
• #### On Inverse Problems and Machine Learning ﻿

This document is related to Ill-Posed and Inverse problems particularly focused on economicmeasurements. In 2015, I proposed to myself to work both analytically and numerically on a very fresh and surprising idea: to predict ...
• #### Determinantal Representations and the Image of the Principal Minor Map ﻿

Research in algebraic geometry has interfaces with other fields, such as matrix theory, combinatorics, and convex geometry. It is a branch of mathematics that studies solution to systems of polynomial equations and ...
• #### An Extremal Property of the Square Lattice ﻿

\nI{Motivated} by a 2019 result of Faulhuber-Steinerberger \cite{extremal} on the hexagonal lattice $\Lambda$, we demonstrate that the square lattice $\Z^2$ exhibits the same local extremal property as $\Lambda$, where ...
• #### Approximation Algorithms for Scheduling and Fair Allocations ﻿

In this thesis, we will have discussions on two main topics, max-min allocation and schedulingjobs with precedent constraints on machines with communication delays. New approximation algorithms are given in Chapter 2, 4 ...
• #### Cubes, Codes, and Graphical Designs ﻿

Graphical designs are an extension of spherical designs to functions on graphs. We connect linear codes to graphical designs on cube graphs, and show that the Hamming code in particular is a highly effective graphical ...
• #### Morphisms, Minors, and Minimal Obstructions to Convexity of Neural Codes ﻿

We study open and closed convex codes from a geometric and combinatorial point of view. We prove constructive geometric results that establish new upper bounds on the open and closed embedding dimensions of intersection ...
• #### Nonlinear PDEs: regularity, rigidity, and an inverse problem ﻿

Based on joint work with Arunima Bhattacharya, we obtain a sharp regularity result for Lagrangian mean curvature type equations with possibly H\"older continuous Lagrangian phases. Along the way, the constant rank theorem ...
• #### Brownian Motion, Quasiconformal Mappings and the Beltrami Equation ﻿

Consider a Jordan domain $\Omega$ in the plane with $3$ distinct points marked on its boundary. These $3$ points split $\partial \Omega$ into $3$ arcs. For each $z \in \Omega$, we can assign it the harmonic coordinates by ...
• #### How to weld: Energies, weldings, and driving functions ﻿

We prove a variant of the welding zipper algorithm converges for curves $\gamma \subset \nH \cup \{0\}$ that have Loewner driving functions $\xi \in C^{3/2+\epsilon}$. Convergence holds whether one zips up'' with straight ...
• #### Intersection Rigidity ﻿

We consider three inverse problems related to geodesic intersections. First, we consider theproblem of recovering the geometry of a Riemannian manifold with boundary from the knowledge of all pairs of inward pointing ...
• #### Designing Scheduling Algorithms via a Mathematical Perspective ﻿

This document will discuss three problems that I worked on during my Ph.D. Chapter \ref{chapter: SC} contains my work on the Santa Claus problem, and Chapters \ref{chapter: S1} and \ref{chapter: S2} contain my work on ...
• #### Counting social interactions for discrete subsets of the plane ﻿

We will use dynamical, geometric, and analytic techniques to study translation surfaces. A translation surface is, informally, a collection of polygons in the plane with parallel sides identified by translation to form a ...