#### 12:00PM - 12:30PM refreshments. Science Hall 10712:30PM - 1:30PM talk. Zoom and Science Hall 107

 11/11/2022: Vakhtang Putkaradze Title: Variational Approach to Porous Media: Equations of Motion, Muscle Action and Thermodynamics Abstract: Porous media presents a highly complex example of fluid-structure interactions where a deforming elastic matrix interacts with the fluid. Many biological organisms are comprised of deformable porous media, with additional complexity of a muscle acting on the matrix. Using geometric variational methods, we derive the equations of motion of a for the dynamics of both the passive and active porous media. The use of variational methods allows to incorporate both the muscle action and incompressibility of the fluid and the elastic matrix in a consistent, rigorous framework. We also derive conservation laws for the motion, perform numerical simulations and show the possibility of self-propulsion of a biological organism due to particular running wave-like application of the muscle stress. We also discuss variational derivation for equations of porous media from the point of view of variational thermodynamics, leading to conclusions about thermodynamically consistent functional forms of friction forces and stresses acting on the media. Joint work with F. Gay-Balmaz and T. Farkhutdinov. 11/4/2022: Mihai Popa Title: Entry Permutations, Limit Distributions and Asymptotic Free Independence for Some Relevant Classes of Random Matrices Abstract:  In the study of large random matrices, the notion of free independence is a suitable analogue to the notion of independence from the study of commutative random variables. In particular, since 1980s, large classes of random matrices have been shown to be asymptotically free from random matrices with entries independent from their entries. Some years ago, together with J.A. Mingo, we showed the (at that time, surprising) result that, under reasonable technical assumptions, ensembles of unitarily invariant random matrices are asymptotically free from their transposes. This fact brought the natural question "How special is the transpose?". That is, given an ensemble of random matrices $( A_N)_N$ and a sequence of entry permutations $( \sigma_N)_N$ can we formulate conditions such that $(A_N)_N$, the initial ensemble, and $(A_N^{\sigma_N)_N$, the ensemble consisting of matrices with permuted entries, are asymptotically free? Also, what are the possible limit distributions for the matrices with permuted entries? The lecture will present some recent progresses as well as some still open questions in this topic. 10/28/2022: Quanhua Xu Title: Maximal Inequalities in Noncommutative Analysis Abstract: Maximal inequalities are of paramount importance in analysis. Here “analysis” is understood in a wide sense and includes functional/harmonic analysis, ergodic theory and probability theory. Consider, for instance, the following two fundamental examples: Click here We will consider in this survey talk the analogues of these classical inequalities in noncommutative analysis. Then the usual Lp-spaces are replaced by noncommutative Lp-spaces. The theory of noncommutative martingale/ergodic inequalities was remarkable developed in the past 20 years. Many classical results were successfully transferred to the noncommutative setting. This theory has fruitful interactions with operator spaces, quantum probability and noncommutative harmonic analysis. We will discuss some of these noncommutative results and explain certain substantial difficulties in proving them. 10/21/2022: Andre Kornell Title: Four Categories of Hilbert Spaces Abstract: Hilbert spaces over the real or the complex numbers form monoidal categories in two distinct ways. We can take a morphism of Hilbert spaces to be a bounded operator or a contractive operator. All four categories can be defined axiomatically without any reference to the real or the complex numbers. The proof appeals to Soler's theorem, the Hellinger-Toeplitz theorem, and the Russo-Dye theorem, as well as to a body of research on dagger categories. Joint work Chris Heunen and Nesta van der Schaaf. 10/07/2022: Joel Lucero-Bryan Title: Modal Logic of Topology: An Introduction Abstract: In topological semantics of modal logic, the modal box is interpreted as topological interior (and hence diamond as closure). This is one of the oldest semantics for modal logic, dating back to the late 1930s and early 1940s. One of the key early results is the 1944 theorem of McKinsey and Tarski that the modal logic of any crowded (separable) metric space is Lewis's well known modal system S4. In the last 30 years, utilizing the machinery of relational semantics, new techniques have been developed that paved the way to obtaining further results that both generalize the McKinsey-Tarski theorem in several directions as well as provide interesting connections with other areas of mathematics. Our aim is to review some of these results. The talk will include a hands-on portion addressing one of the key features of topological semantics: obtaining finite configurations as continuous and open images of such well-known topological spaces as the real line. Time permitting, I will also present some new topological invariants that arise via these considerations. 09/30/2022: John C. Dallon Title: Techniques of Analyzing Stochastic Differential Equation Systems Abstract: In this talk I will present a force based mathematical model of mesenchymal cell motion where adhesion sites are randomly placed and have a probabilistically determined lifespan. Numerical simulations and theoretical results will be discussed and compared to experimental data. The goal is to understand how focal adhesion dynamics affect cell motion. 09/16/2022: Tuan Phan Title: Modeling Mesenchymal Cell Motion - Force, Speed and the Role of Focal Adhesion Lifetimes Abstract: In this talk, I will discuss two techniques to analyze theoretically a stochastic differential equation (SDE) system that often arises in modeling biological and medical problems. The first one is to construct stochastic Lyapunov functions for proving the existence of a unique stationary distribution of a SDE system in the interior of its invariant domain. The second one is to analyze the dynamics of a SDE system on the boundary of its invariant domain. The latter technique has some advantages and can provide sharp conditions for the extinction and persistence of a SDE system, which is an important concern in mathematical biology. I will use a cholera epidemic model and human papilloma virus model to demonstrate how to use and combine these two techniques. I will keep technique low and emphasize ideas. 09/02/2022: Prasit Bhattacharya Title: Rabbit holes of Spheres Abstract: Many intricate and bizarre patterns have been discovered while studying the homotopy groups of spheres. These groups have been the subject of study for more than a century yet they are mostly unknown. This talk will discuss many interesting patterns, especially the ones that led to chromatic homotopy theory, a modern subject which sheds light on large scale patterns within the stable homotopy groups of spheres.