This dissertation lies at the intersection of extremal combinatorics and probabilistic combinatorics.  Roughly speaking, extremal combinatorics studies how large a combinatorial object can be.  For example, a classical result of Mantel's says that every $n$-vertex triangle-free graph has at most $\frac{1}{4} n^2$ edges.  The area of probabilistic combinatorics encompasses both the application of probability to combinatorial problems, as well as the study of random combinatorial objects such as random graphs and random permutations.  In this dissertation we study problems related to extremal properties of random objects.  In particular we study a certain card guessing game, $F$-free subgraphs of random hypergraphs, and thresholds of random hypergraphs.  Minimal prerequisites will be assumed.

We give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure preserving equivalence relation. The main philosophy is to understand the behavior of the Radon-Nikodym cocycle in terms of the geometry of the amenable subrelation within the tree. This allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. The two main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.

We establish sufficient conditions for a locally-warped product structure to propagate backward in time under the Ricci flow. As an application, we show that if a gradient shrinking soliton is asymptotic to a cone whose cross-section is a locally warped product of Einstein manifolds, the soliton must itself be a warped product over the same manifolds.

The p-widths of a closed Riemannian manifold are a nonlinear analog of the spectrum of its Laplace--Beltrami operator, which was defined by Gromov in the 1980s and correspond to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any ≥ 2-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich--Marques--Neves Weyl law constant in dimension 2.

In classical real and complex dynamics, one studies topological and analytic properties of orbits of points under iteration of self-maps of $\mathbb R$ or $\mathbb C$ (or more generally self-maps of a real or complex manifold). In arithmetic dynamics, a more recent subject, one likewise studies properties of orbits of self-maps, but with a number theoretic flavor. Many of the motivating problems in arithmetic dynamics come via analogy with classical problems in arithmetic geometry: rational and integral points on varieties correspond to rational and integral points in orbits; torsion points on abelian varieties correspond to periodic and preperiodic points of rational maps; and abelian varieties with complex multiplication correspond to post-critically finite rational maps.

This analogy focuses on forward iteration, but sometimes surprising and interesting results can be found by thinking instead about pre-images of rational points under iteration. In this talk, we will give some background and motivation for the field of arithmetic dynamics in order to describe some of these "backwards iteration" results, including uniform boundedness for rational pre-images and open image results for Galois representations associated to dynamical systems.

We develop an efficient and accurate level set method to study numerically a crawling eukaryotic cell using a minimal model. This model describes the cell polarity and movement using a reaction-diffusion system coupled with a sharp-interface model. 

We employ an efficient finite difference method for the reaction-diffusion equations with no-flux boundary conditions. This results in a symmetric positive definite system, which can be solved by the conjugate gradient method accelerated by preconditioners. To track the long-time dynamics, we employ techniques of the moving computational window to keep the efficiency. Our level-set simulations capture well the cell crawling, the straight line trajectory, the circular trajectory, and other features. 

Our efficient and accurate computational techniques can be extended to a broad class of biochemical descriptions of cell motility, for which problems are posed on moving domains with complex geometry and fast simulations are very important. This is a joint work with Li-Tien Cheng and Bo Li.

We give a method for sampling points from an affine algebraic variety over a local field with a prescribed probability distribution. In the spirit of the previous work by Breiding and Marigliano on real algebraic manifolds, our method is based on slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of our sampling method and discuss a few applications, in particular we sample from algebraic p-adic matrix groups and modular curves.

Tissue  growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics.

We will give an overview of the modeling aspects and focus on the links between those mathematical models. Then, we will focus on the `compressible' description  describing the cell population density based on systems of porous medium type equations with reaction terms.  A  more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form.

The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the  pressure gradient and emergence of instabilities.

Tissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics. We will give an overview of the modeling aspects and focuss on the links between those mathematical models. Then, we will focus on the `compressible' description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form. The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the pressure gradient and emergence of instabilities.

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different. 

In this talk, I will give an introduction to optimal transport, which has evolved as one of the major frameworks to meaningfully compare distributional data. The focus will mostly be on machine learning, and how optimal transport can be used efficiently for clustering and supervised learning tasks. Applications of interest include image classification as well as medical data such as gene expression profiles.

Analyzing a linear model is a fundamental topic in statistical inference and has been well-studied. However, the complex nature of modern data brings new challenges to statisticians, i.e., the existing theories and methods may fail to provide consistent results. Focusing on a high dimensional linear model with i.i.d. errors or heteroskedastic and dependent errors, this talk introduces a new ridge regression method called `the debiased and thresholded ridge regression' that fits the linear model. After that, it introduces new bootstrap algorithms that generate consistent simultaneous confidence intervals/performs hypothesis testing for the linear model. This talk also applies bootstrap algorithm to construct the simultaneous prediction intervals for future observations. 

Another topic of this talk is about properties of a residual-based bootstrap prediction interval. It derives the asymptotic distribution of the difference between the conditional coverage probability of a nominal prediction interval and the conditional coverage probability of a prediction interval obtained via a residual-based bootstrap. This result shows that the residual-based bootstrap prediction interval has about $50\%$ possibility of yielding conditional under-coverage. Moreover, it introduces a new bootstrap prediction interval that has the desired asymptotic conditional coverage probability and the possibility of conditional under-coverage.