In various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law \([x,y]=1\) holds with probability larger than \(5/8\), must be abelian. In the talk I'll discuss a probabilistic approach to laws on infinite groups, using random walks, and present results, joint with Greenfeld and Olshanskii, answering a few questions of Amir, Blachar, Gerasimova, and Kozma.

A sequence of tuples of random matrices $(X_1^N, \dots, X_d^N)$ converges strongly to a tuple of operators $ (x_1, . . . , x_d)$ in a $C^*$-algebra if for any noncommutative polynomial $P$, $\|P(X_1^N, \dots, X_d^N)\|$ converges (say, in probability) to $\|P(x_1, . . . , x_d)\|$ as $N$ goes to infinity. This phenomenon plays a central role in breakthrough results  in operator algebras, as well as in the construction of: expander graphs, hyperbolic surfaces with nearly optimal spectral gaps and minimal surfaces. Given its far-reaching implications, it is no surprise that strong convergence is notoriously difficult to prove and has generally required delicate problem-specific methods.
In this talk I will discuss recent joint work with Chi-Fang Chen, Joel Tropp and Ramon van Handel, where we introduce a new flexible and elementary technique for proving strong convergence. This technique can be applied to random matrix models that have a lot of symmetry, for example, random permutation matrices, classical Gaussian and unitary matrices (i.e. GOE, GUE, GSE, $O(N)$, $U(N)$, and $Sp(N)$), and some others, constructed via representations of the symmetric and unitary group, for which other methods seem to break. In all of these models, the technique yields the sharpest quantitative results known so far. 

A real number $\sigma$ is called a \textit{linear threshold} of a bipartite graph $H$ if every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^\sigma$ and without a copy of $H$ must have a linear number of edges $|E| \lesssim |V|$. We prove that $\sigma_s = 2 - 1/s$ is a linear threshold of the \textit{complete bipartite subdivision} graph $K_{s,t}'$. Moreover, we show that any $\sigma < \sigma_s$ is not a linear threshold of $K_{s,t}'$ for sufficiently large $t$ (depending on $s$ and $\sigma$). Some applications of our result in incidence geometry are discussed.

The Euler product expression of the Dirichlet series of Fourier coefficients of an elliptic modular eigenform follows from a formal identity in the Hecke algebra for GL(2) with full level. In the case of Siegel modular forms of degree two with paramodular level, the situation is more delicate. In this talk, I will present two rationality results. The first concerns the Dirichlet series of radial Fourier coefficients for an eigenform of paramodular level divisible by the square of a prime. This result is an application of the theory of stable Klingen vectors (joint work with Brooks Roberts and Ralf Schmidt). While we are able to calculate the action of certain Hecke operators on eigenforms, the structure of the Hecke algebra of deep level is not known in general. However, in the case of prime level, there is a robust description of the local Hecke algebra which yields a rationality result for a formal power series of Hecke operators (joint work with Joshua Parker and Brooks Roberts). In both cases, we obtain the expected local L-factor as the denominator of the rational function.

[pre-talk at 3:00pm]

How do linear projections affect the dimensions of subsets of Cartesian space? Marstrand's result from 1954 demonstrates that each Borel set behaves generically under projections onto almost every linear subspace. Recent developments in Fourier analysis have permitted these results to be expanded significantly to apply to much more restricted families of projections, and even effectively bound the dimension of the set of exceptional projections.

We discuss the special case of projecting subsets of three dimensions onto lines, working over non-Archimedean local fields of characteristic not equal to 2. We will briefly discuss the relevancy to polynomial effective equidistribution in homogeneous dynamics. The main technical input will be a refined decoupling theorem for non-Archimedean local fields. This work mirrors the work in the real setting by the authors Gan, Guo, Guth, Harris, Iosevich, Maldague, Ou, and Wang, and builds on previous work by the speaker and Zuo Lin.

Phase separation underpins a wide range of phenomena from the formation of membraneless intracellular compartments known as biomolecular condensates to the collective behavior of bacteria.  Unlike simpler systems like oil and water, however, phase separation in these systems is often complicated by mechanical interactions, nonequilibrium activities, and heterogeneity.

In my talk, I will delve into how I navigated these complexities to uncover new insights into three distinct systems. I will first address biomolecular condensates within chromatin-packed cell nuclei, highlighting how the competition between elastic and capillary forces crucially shapes the structure and mechanics of the chromatin networks. Next, I will share my discovery of emergent phenomena in active systems such as bacteria and active colloids, due to the interplay between movements along chemical gradients and motility-induced phase separation. Lastly, I'll discuss how image-based learning uncovered the physics of phase-separating particles driven out of equilibrium by electrochemical reactions, revealing their reaction kinetics, free energy landscape, and heterogeneity.

In the presence of a large group action, even non-noetherian rings sometimes behave like noetherian rings. For instance, Cohen proved that every symmetric ideal in the infinite variable polynomial ring is generated by the symmetric group orbit of finitely many polynomials. In this talk, I will provide a brief introduction to GL-algebras—commutative algebras equipped with an action of the infinite general linear group—and recall the rich history of the noetherianity problem for GL-algebras. I will then present recent work where I construct a counterexample to the algebraic noetherianity problem for GL-algebras in characteristic two.

Let $D$ be a relatively compact $C^2$ domain in a complex manifold $X$ of dimension $n$. Assume that $H^1(D,\Theta)$ vanishes, where $\Theta$ is the sheaf of germs of holomorphic tangent fields of $D$. Suppose that the Levi-form of the boundary $b D$ has at least $3$ negative eigenvalues or at least $n-1$ positive eigenvalues pointwise. We will show that if a formally integrable  almost complex structure $H$ of the Holder class $C^r$ with $r>5/2$ on $D$ is sufficiently close to the complex structure on $D$, there is a embedding $F$ from $D$ into $X$ that transforms the almost complex structure into the complex structure on $F(D)$, where  $F $ has class $C^s$ for all $s<r+1/2$. This result was due to R. Hamilton in the 1970s when both $b D$ and $H$ are of class $C^\infty$.

Let $X$ be a normal variety with a projective contraction $f:X\to S$. Assume $U$ is an open subset of $S$ and $L_U$ is a Cartier divisor on $X_U:=X\times_S U$ such that $L_U$ is numerical trivial over $U$.

We will discuss about when it is possible and how to extend $L_U$ to a global Cartier divisor $L$ on $X$ such that $L\equiv_f 0$.