A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in parallel pairs. This description allows us to encode an STS combinatorially by a pair of permutations -- one of which encodes the gluing of the vertical edges and the other the gluing of the horizontal edges. In this talk I will use the combinatorial description of STSs to consider two models for random STSs. The first model will encompass all square-tiled surfaces while the second will encompass a horizontally restricted class of them. I will discuss topological and geometric properties of a random STS from each of these models.

Introduced by Mallows in statistical ranking theory, the Mallows permutation model is a class of non-uniform probability measures on permutations. The general model depends on a distance metric on the symmetric group. This talk focuses on Mallows permutation models with L1 and L2 distances, which possess spatial structure and are also known as “spatial random permutations” in the mathematical physics literature.

A natural question from probabilistic combinatorics is: Picking a random permutation from either of the models, what does it “look like”? This may involve various features of the permutation, such as the length of the longest increasing subsequence and the cycle structure. In this talk, I will explain how multi-scale analysis and the hit and run algorithm—a Markov chain for sampling from both models—can be used to establish limit theorems for these features.

Capillary surfaces model the geometry of liquids meeting a container at an angle, and arise naturally as (constrained) minimizers of the Gauss free energy.  We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least 4 in the surface, and this estimate improves if the capillary angle is close to $0$, $\pi/2$, or $\pi$. For capillary angles that are close to $0$ or $\pi$, our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem.  This is joint work with Otis Chodosh and Chao Li.

Morphogenesis, a biological process by which cells organize to form complex tissues, emerges from a highly dynamic interplay between molecular factors and biomechanical forces. This process is tightly regulated, and even minor aberrations in morphogenesis can have lasting effects on disease susceptibility and lifelong organ function. Furthermore, the molecular and biomechanical factors that drive morphogenesis are often dysregulated during aging and disease. Despite its central role in development, our understanding of how molecular and mechanical factors interact during morphogenesis remains limited. A deeper understanding of morphogenesis may inform interventions to prevent disease onset and guide research in organ regeneration. In this talk, I will present an approach for systematically integrating computational modeling and laboratory experimentation to elucidate the interplay between molecular factors and biomechanical forces in organ formation, with a focus on the lungs and the mammary gland. 

We give a negative solution of a matrix version of Grothendieck’s classical inequality formulated by Blecher and Shlyahktenko/Pisier using non-signaling games considered by Shor generalizing the famous CSHS inequality.

Constructing accurate data-driven models of dynamical systems in the face of data-sparsity, measurement errors, and uncertainty is of crucial importance across a wide range of scientific disciplines. In this talk, we propose a variety of techniques rooted in the concept of measure-transport, designed to be robust against such data imperfections. In the first half of the talk, we introduce a novel approach for performing system identification in which synthetic invariant measures, approximated as fixed points of a Fokker—Planck equation, are aligned with invariant measures extracted from observed trajectory data during optimization. We then use Takens' embedding theory to introduce a critical data-dependent coordinate transformation which can guarantee unique system identifiability from the invariant measure alone. In the second half of the talk, we consider the problem of forecasting the full state of a dynamical system from partial measurement data. While Takens' theorem provides the justification for a host of computational methods for data-driven attractor reconstruction, the classical theory assumes the dynamics are deterministic and that observations are noise-free. Motivated by this limitation, we leverage recent advances in optimal transportation theory to establish a measure-theoretic generalization and robust computational framework that recasts the embedding map as a pushforward between probability spaces. Throughout, we showcase the effectiveness of our proposed methods on synthetic test examples, including the Lorenz-63 system and Kuramoto—Sivashinsky equation, as well as large-scale, real-world applications, including Hall-effect thruster dynamics, a NOAA sea surface temperature dataset, and the ERA5 wind field dataset.

About two years ago, the world was rocked by the discovery of an aperiodic monotile dubbed the 'Hat' and its chiral cousin the 'Spectre'. Perhaps the really interesting thing about this discovery is that it came with a novel proof of aperiodicity which does not follow the standard arguments. One might expect that these new ideas would lead to the discovery of more aperiodic tiles, but even the Spectre was not analyzed this way! So why are there no more aperiodic monotiles, and why are the only two we know so closely related? No one seems to know. In this talk we demand answers, exploring the proof and trying to imagine how it could be adapted into a search strategy for more aperiodic tiles.

A complex manifold X is said to be Brody hyperbolic if it admits no entire curves, which are non-constant holomorphic maps from the complex numbers. When X is a smooth complex projective variety, Demailly introduced an algebraic analogue of this property known as algebraic hyperbolicity. We propose a conjecture on the algebraic hyperbolicity of generic sections of adjoint bundles on X, motivated by Fujita’s freeness conjecture and recent results by Bangere and Lacini on syzygies of adjoint bundles. We present some old and new evidence supporting this conjecture, including when X is any smooth projective toric variety or Gorenstein toric threefold. This is based on joint work with Joaquín Moraga.

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.

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.

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.