The study of group actions on probability measure spaces plays a central role in modern mathematics. The (generalized) Neshveyev-Stormer conjecture states that the group action on a probability measure space can be completely understood by studying the inclusion of the group von Neumann algebra inside the group measure space construction. In my talk I shall show that the Neshveyev-Stormer conjecture is true for a large class of actions. This talk is based on a joint work with Ionut Chifan.

The algorithmic refinement of triangular meshes is an important component in numerical simulation codes. Newest vertex bisection is one of the most popular methods for geometrically stable local refinement. Its complexity analysis, however, is a fairly intricate recent result
and many combinatorial aspects of this method are not yet fully understood. In this talk, we access newest vertex bisection from the perspective of theoretical computer science. A new result is the amortized complexity analysis over generalized triangulations. An immediate application is the convergence and complexity analysis of adaptive finite element methods over embedded surfaces and singular surfaces. This is joint work with Michael Holst and Zhao Lyu.

A hypersurface in a compact manifold $M$ with boundary is called a free boundary minimal hypersurface (FBMH) if it is minimal and meets the boundary of $M$ orthogonally. Such hypersurfaces arise naturally as critical points of the area functional in $M$. If we do not assume any boundary convexity of $M$, then the FBMH may be improper, i.e., the interior of the FBMH may touch $\partial M$. We will discuss the compactness and existence results for FBMHs in this general setting.

We propose a novel framework for constructing and computing various stable network observables. Our approach is based on sampling a random homomorphism from a small motif of choice into a given network. Integrals of the law of the random homomorphism induces various network observables, which include well-known quantities such as homomorphism density and average clustering coefficient. We show that these network observables are stable with respect to renormalized cut distance between networks. For their efficient computation, we also propose two Markov chain Monte Carlo algorithms and analyze their convergence and mixing times. We demonstrate how our techniques can be applied to network data analysis, especially for hypothesis testing and hierarchical clustering, through analyzing both synthetic and real world network data.

Semi-supervised classification refers to learning a function that assigns classes to input data using two sets of observations, one where the input and the associated class is recorded and another where only the inputs are observed. This is a very canonical problem in machine learning with strong links to different approaches in Statistics and Mathematics, e.g. probit regression, spectral clustering or the Ginzburg-Landau classifier. In several applications quantifying the uncertainty associated with classification is as important as the classification itself. In semi-supervised learning uncertainty quantification can be used to improve classificationby active-learning, which amounts to manually classifying the subjects for which classification is most uncertain, and then relearning the classification function; this is also known as human-in-the-loop. In the talk I present a recent framework that connects the different approaches to classification and comes automatically with uncertainty quantification. It is based on Gaussian process classification where the covariance operator of the Gaussian process is constructed using information from the graph Laplacian. The plan of the talk is to provide an accessible overview of the key ideas in this paradigm.

In this talk we will discuss an arithmetic analogue of the
gonality of a curve over a number field: the smallest positive integer e
such that the points of residue degree bounded by e are infinite. By work
of Faltings, Harris--Silverman and Abramovich--Harris, it is
well-understood when this invariant is 1, 2, or 3; by work of
Debarre--Fahlaoui these criteria do not generalize to e at least 4. We
will study this invariant using the auxiliary geometry of a surface
containing the curve and devote particular attention to scenarios under
which we can guarantee that this invariant is actually equal to the
gonality. This is joint work with Geoffrey Smith.

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general mathematical audience.

An $(n,k,\ell)$-design is a a family of $k$-sets of $[n]$ such that every $\ell$-set is covered precisely once. The problem of determining whether or not there exists a design for a given set of parameters is a classical and difficult question in combinatorics. We ask a variant of this problem. Namely, given $k,\ell$, can one find a family of $k$-sets of $[n]$ covering every $\ell$-set \textit{at least} once that has ``approximately'' as many sets as an $(n,k,\ell)$-design would have?

In this talk we will solve the above problem using the technique known as the R$\ddot{\text{o}}$
dl nibble. As time permits we will also discuss other problems in design theory, as well as other applications of the R$\ddot{\text{o}}$dl nibble technique.