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Combinatorics of CAT(0) cubical complexes, crossing complexes and coskeletons
This thesis consists of three papers about cubical complexes: Chapter 1 is [Rowlands 22a], Chapter 2 is [Rowlands 23], and Chapter 3 is [Rowlands 22b]. Chapter 1 extends a result by Dancis to cubical complexes: Dancis ... 
On an inverse problem for fractional connection Laplacians
Classical inverse problems seek to determine the unknown coefficients of a PDE from boundary or local measurements of solutions. In the past few years, there has been a sharp increase in attention paid to inverse problems ... 
Computations Related to the Construction of Finite Genus Solutions to the KadomtsevPetviashvili Equation
Krichever's method of integrating certain partial differential equations using algebrogeometric techniques provides an explicit approach to the construction of finitegenus solutions to the KadomtsevPetviashvili (KP) ... 
The Polyhedral Geometry of Graphical Designs
A graphical design is a quadrature rule for a graph. That is, a graphical design is a subset of graph vertices for which the global averages of certain Laplacian eigenvectors are equal to weighted averages of these vectors ... 
On the integral Chow rings of various moduli stacks of curves
The contents of this thesis are focused on the intersection theory of the stack of marked(stable or smooth) elliptic curves. We first recount some results on equivariant intersection theory, then give an exposition of some ... 
A HyperresolutionFree Characterization of the DeligneDu Bois Complex
Let $Y$ be a reduced, finite type scheme over $\mathbb{C}$, $X$ a closed subscheme of $Y$ and $\pi:\widetilde{Y} \to Y$ a projective morphism which is an isomorphism outside of $X$ with $E=(\pi^{1}(X))_\text{red}$. In ... 
Pattern Avoidance Criteria for Smoothness of Positroid Varieties Via Decorated Permutations, Spirographs, and Johnson Graphs
Positroids are certain representable matroids originally studied by Postnikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal ... 
Active Phase for the Stochastic Sandpile on Z
We prove that the critical value of the onedimensional Stochastic Sandpile Model is less than one. This verifies a conjecture of Rolla and Sidoravicius. 
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 FubiniBruhat Order
Fubini words are generalized permutations, allowing for repeated letters, and theyare in onetoone 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 variablecoeﬃcient Plateau problem
We study almostminimizers 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 squaretiled surface ... 
On Inverse Problems and Machine Learning
This document is related to IllPosed 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 FaulhuberSteinerberger \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, maxmin allocation and schedulingjobs with precedent constraints on machines with communication delays. New approximation algorithms are given in Chapter 2, 4 ...