Classically, symmetries of algebras are described by actions of finite groups or Lie algebras. The study of actions on polynomial rings, as well as their subrings of invariants, is a deep and beautiful theory. Noncommutative algebras admit a richer notion of "quantum symmetry", which is captured by actions of Hopf algebras. The quantum symmetries of noncommutative analogues of polynomial rings is an active area of research. In this talk, we explore whether weak Hopf algebras can be seen as capturing an even more general notion of symmetry.

Chemotaxis models describe the movement of cells or organisms in response to chemical signals. In this talk, I will discuss a parabolic-parabolic chemotaxis system with a logistic source and chemical consumption. For both linear and nonlinear diffusion, we prove global existence and boundedness of solutions that are not necessarily integrable. In the linear diffusion case, we show that chemicals generally do not slow down the spreading of cells and, under certain conditions, do not enhance the spreading as well. A key analytical insight is a new relation between cell density and chemical concentration. Numerical simulations also reveal a striking phase transition driven by the chemical sensitivity constant. These are joint work with Zulaihat Hassan (PhD student) and Wenxian Shen.

Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.

[pre-talk at 3:00PM]

We will discuss recent advances in the compression of pre-trained neural networks using both novel and existing computationally efficient   algorithms. The approaches we consider leverage sparsity, low-rank approximations of weight matrices, and weight quantization to achieve significant reductions in model size, while maintaining performance. We provide rigorous theoretical error guarantees as well as numerical experiments.

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.

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. 

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.

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!