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A Hausdorff and etale groupoid is said to be C*-simple if its reduced groupoid C*-algebra is simple. Work on C*-simplicity goes back to the work of Kalantar and Kennedy in 2014, who classified the C*-simplicity of discrete groups by associating to the group a dynamical system. Since then, the study of C*-simplicity has received interest from group theorists and operator algebraists alike. More recently, the works of Kawabe and Borys demonstrate that the groupoid case may be tractible to such dynamical characterizations. In this talk, we present the dynamical characterization of when a groupoid is C*-simple and work out some basic examples. This is joint work with Xin Li, Matt Kennedy, Sven Raum, and Dan Ursu. No previous knowledge of groupoids will be assumed.

In the late 1980s, Berger and Coburn showed that the Hankel operator $H_f$ on the Segal-Bargmann space of Gaussian square-integrable entire functions is compact if and only if $H_{\bar f}$ is compact using C*-algebra and Hilbert space techniques. I will briefly discuss this and three other proofs, and then consider the question of whether an analogous phenomenon holds for Schatten class Hankel operators. 

It is a widely accepted philosophy in additive number theory that convex sets ought not to exhibit much additive structure. We could measure this by estimating the sizes of their sumsets. In this talk, we will hopefully move from the philosophical to the concrete, by giving ways to see that convex sets and functions have poor additive structure. We will also discuss some recent developments in the area.

In a world of polynomial Pell’s equations, where the integers are replaced by polynomials with complex coefficients, and its smallest solution is used to generate all other solutions $(u_{n},v_{n})$, $n\in\mathbb{Z}$. One junior number theory group will embark on a journey in search of the properties of the factors of $v_{n}(t)$. There will be Galois extensions, there will be estimations and of course there will be loglogs.

A nice collection of problems in additive combinatorics focus on analysing solutions to additive equations over sequences that exhibit some flavour of convexity. This, for instance, includes genuine convex sequences as well as images of arbitrary sets under convex functions. In this talk, I will survey some of the literature surrounding these type of questions, along with some motivation from analytic number theory as well as the current best known results towards these problems.

The Jacquet-Langlands correspondence gives a relationship between automorphic representations on $GL_2$ and its twisted forms, which are the unit groups of quaternion algebras. Writing this out in more classical language gives a combinatorial way of producing the eigenvalues of Hecke operators acting on modular forms. In this talk, we will first go over notions of modular forms and quaternion algebras, and then dive into an explicit example by computing some eigenvalues of the lowest level quaternionic modular form of weight $2$ over $\mathbb{Q}$.

In 1973, Serre defined $p$-adic modular forms as limits of modular forms, and constructed the Leopoldt-Kubota $L$-function as the constant term of a limit of Eisenstein series. This was extended by Deligne-Ribet to totally real number fields, and Lauder and Vonk have developed an algorithm for interpolating $p$-adic $L$-functions of such fields using Serre's idea. We explain what an $L$-function is and why you should care, and then move on to giving an overview of the algorithm, extensions, and applications.

In this talk I will review some recent results obtained in collaboration with E. Runa and A. Kerschbaum on the one-dimensionality of the minimizers
of a family of continuous local/nonlocal interaction functionals in general dimension. Such functionals have a local term, typically the perimeter or its Modica-Mortola approximation, which penalizes interfaces, and a nonlocal term favouring oscillations which are high in frequency and in amplitude. The competition between the two terms is expected by experiments and simulations to give rise to periodic patterns at equilibrium. Functionals of this type are used  to model pattern formation, either in material science or in biology. The difficulty in proving the emergence of such structures is due to the fact that the functionals are symmetric with respect to permutation of coordinates, while in more than one space dimensions minimizers are one-dimesnional, thus losing the symmetry property of the functionals. We will present new techniques and results showing that for two classes of functionals (used to model generalized anti-ferromagnetic systems, respectively  colloidal suspensions), both in sharp interface and in diffuse interface models, minimizers are one-dimensional and periodic, in general dimension and also while imposing a nontrivial volume constraint.