^{This dissertation concerns the community detection problems under the Hypergraph Stochastic Block Model (HSBM), where the sizes of edges may vary, and each edge appears independently with some given probability, purely determined by the labels of its vertices.} 

^{For regimes where the expected degrees grow with the number of vertices, for the first time in the literature, we prove a wide-ranging, information-theoretic lower bound on the number of misclassified vertices for any algorithm, where the bound is characterized by the generalized Chernoff-Hellinger divergence involving model parameters. Besides that, when the expected degrees grow logarithmically, we establish a sharp threshold for exact recovery for the non-uniform, multiple-community setting, subject to only minor constraints. A key insight reveals that aggregating information across uniform layers enables exact recovery even in cases where this is impossible if each layer were considered alone. We present two efficient algorithms, for minimal and full information scenarios, which successfully achieve exact recovery when above the threshold, and attain the lowest possible mismatch ratio when below the threshold where exact recovery is impossible, confirming their optimality.} 

^{In the regime with bounded expected degrees, we develop a spectral algorithm that achieves partial recovery, correctly classifying a constant fraction of vertices. This fraction is determined by the Signal-to-Noise Ratio (SNR) of the model, and approaches 1 as the SNR grows slowly with the number of vertices. Our algorithm employs a three-stage approach: first, preprocessing the hypergraph to maximize SNR; second, applying spectral methods to obtain an initial partition; and finally, implementing tensor-based refinement to enhance classification accuracy.} 

^{Additionally, we provide novel concentration results for adjacency matrices of sparse random hypergraphs, serving as the foundations of the theoretical analysis of our algorithms, which could be of independent interest. We also address several open problems concerning incomplete model information and parameter estimations.}

Understanding the loss landscape of the deep networks can provide many insights into the theoretical understanding of how the networks learn and why they work so well in practice. In this talk, starting with the observation that the flat minima correspond to continuous symmetries of the loss function, two symmetry breaking methods are proposed to provably remove all the flat minima (and flat stationary points) from the loss landscape for any deep feedforward network as long as the activation function is a smooth function. Examples of activation functions that satisfy the assumptions are sigmoid, hyperbolic tangent, softplus, polynomial, etc., and those of loss functions are cross-entropy, squared loss, etc. The methods can be essentially viewed as generalized regularizations of the loss function. The proposed methods are applied on the polynomial neural networks, where the activation function is a polynomial with arbitrary degree, and a first result on estimates of the number of isolated solutions is provided and we get a first glimpse on the complexity of the loss landscapes even in the absence of flat minima.

Directed graphs, or digraphs, are useful in many areas of theoretical and applied mathematics, including for describing other combinatorial objects. We will review a method for counting closed walks in a digraph using a transfer matrix. Then we will use a group action to count singular cycles (closed walks for which the initial vertex has been forgotten). Finally we will apply these results to count structures in circulant graphs, up to rotational equivalence.

The simple symmetric exclusion process on Z models the dynamics of particles with strong local interaction induced by an exclusion rule: each attempts the motion of a nearest-neighbor symmetric random walk, but jumps to occupied sites are suppressed. While this process has been studied extensively over the past several decades, not much has been known rigorously about the behavior of extremal particles when the system is out of equilibrium. We consider a `step’ initial condition in which infinitely many particles lie below a maximal one. As time tends to infinity, the system becomes indistinguishable from one without particle interaction, in the sense that the point process of particle positions, appropriately scaled, converges in distribution to a Poisson process on the real line with intensity exp(-x)dx. Correspondingly, the position of the maximal particle converges to the Gumbel distribution exp(-exp(-x)), which answers a question left open by Arratia (1983). I will discuss several properties of the symmetric exclusion process that lead to this result, including negative association, self-duality, and the so-called `stirring’ construction, as well as extensions to higher dimensions and to dynamics that allow more than one particle per site. The talk is based on joint work with Sunder Sethuraman and Adrián González Casanova. 

DNA encodes the foundations of life, but it can also be thought of as a physical material, where its information-carrying capacity can be used to encode complexity in the structures it forms. I will talk about our group’s work studying DNA in a material setting. First, I will zoom in on the microscopic dynamics of DNA, and ask how it changes the coarse-grained dynamics of systems of particles when these are coated with single-stranded DNA. We will use stochastic models and homogenization techniques to show that DNA changes the dynamics dramatically, and we confirm our predictions with experiments. Our model bears much in common with many biological systems, such as blood cells and virus particles, and we use our model to make predictions about the dynamics of these systems. Then, we will zoom out and ask how to best use DNA to encode highly specific interactions between particles. Specifically, we wish to understand how to avoid the “kinetic traps” that prevent a system of assembling particles from reaching its equilibrium, or stationary, state. We discovered an unexpected connection to a combinatorial property of graphs, which we use to propose a strategy for designing optimal interactions.
 

DNA encodes the foundations of life, but it can also be thought of as a physical material, where its information-carrying capacity can be used to encode complexity in the structures it forms. I will talk about our group’s work studying DNA in a material setting. First, I will zoom in on the microscopic dynamics of DNA, and ask how it changes the coarse-grained dynamics of systems of particles when these are coated with single-stranded DNA. We will use stochastic models and homogenization techniques to show that DNA changes the dynamics dramatically, and we confirm our predictions with experiments. Our model bears much in common with many biological systems, such as blood cells and virus particles, and we use our model to make predictions about the dynamics of these systems. Then, we will zoom out and ask how to best use DNA to encode highly specific interactions between particles. Specifically, we wish to understand how to avoid the “kinetic traps” that prevent a system of assembling particles from reaching its equilibrium, or stationary, state. We discovered an unexpected connection to a combinatorial property of graphs, which we use to propose a strategy for designing optimal interactions.

Quasi-F-Splittings have proven to be a vital invariant in the study of varieties in positive characteristic, with numerous applications to birational geometry and F-singularities. In this talk I'll provide an overview of Quasi-F-Splittings and introduce a local variant, dubbed Quasi-F-Purity, which extends the theory of Quasi-F-Splittings to arbitrary prime characteristic fields. I will also discuss various permanence properties of Quasi-F-Purity, including stability under completion and étale extension.

The main object of study in algebraic geometry is a variety, which is locally the solution set to polynomial equations. One fundamental research direction is the classification of these objects. In this talk, I'll introduce the idea of a moduli (or parameter) space for algebraic varieties. There will be many examples!

Markov Decision Processes are the major model of controlled stochastic processes in discrete time. Value iteration (VI) is one of the major methods for finding optimal policies. For each discount factor, starting from a finite number of iterations, which is called the turnpike integer, value iteration algorithms always generate decision rules which are deterministic optimal policies for the infinite-horizon problems. This fact justifies the rolling horizon approach for computing infinite-horizon optimal policies by conducting a finite number of value iterations. In this talk, we will first discuss the complexity of using VI to approximately solve MDPs, and then introduce properties of turnpike integers and provide their upper bounds.