Winter 2025

Title: An Application of Class Field Theory in the Proof of the Chebotarev Density Theorem
Speaker: Süeda Şentürk Avcı
Abstract: Prime numbers show up everywhere in mathematics, but how are they actually distributed? In this talk, we’ll explore density theorems that help answer this question. We’ll start with classic results, like Dirichlet’s theorem on primes in arithmetic progressions, and then move into the world of algebraic number theory, where primes behave in fascinating ways within number fields. Along the way, we’ll see how Galois theory and deep results like the Chebotarev Density Theorem help us understand prime distributions on a much larger scale.

This talk assumes some familiarity with real analysis, group theory, and basic number theory, but no advanced background in algebraic number theory is required.

Title: Generalized amenability in C*-algebras and Quantum groups
Speaker: Fouad Naderi
Abstract: Let $A$ be a C*-algebra and $X$ be a dual Banach space that is also an $A$-bimodule. A bounded linear map $D:A \longrightarrow X$ with the Leibniz property $D(ab)=D(a)b+aD(b)$ is called a derivation. Now, the amenability of $A$ means that each derivation $D$ is given by a commutator or a Lie bracket; i.e., $D(a)=ax-xa$ for some $x\in X$.

By a result of A. Connes and U. Haagerup,  a C*-algebra is amenable if and only if it is nuclear.  

Sequential (generalized) amenability means that there exists a sequence (or net) $(x_i)$ in $X$ such that $D(a)=\mathrm{lim}_{i}~(ax_i - x_i a)$.

In this talk, we show that a C*-algebra is sequentially amenable if and only if it is amenable. Our main tools are a new class of functionals on von Neumann algebras, which we call {\it ultratraces}, and the weak* decomposition of non-commutative quantum expectations (developed by the author), though we do not go into the details of the latter.

Title: Series Representations Beyond Disks
Speaker: James Young
Abstract: In the same way that complex power series represent holomorphic functions on disks, Faber series generalize this idea by providing “form-fitting” polynomial series representations for said functions on far more general shapes of domains. In this talk, we give an overview of the construction of Faber series using classical techniques in complex analysis, and list some of their interesting properties.

Title: An introduction to Reproducing Kernel Hilbert Spaces and their applications
Speaker: Maximillian Tornes
Abstract: This talk will give an introduction to the theory of reproducing kernel Hilbert spaces and their applications from the perspective of complex analysis. After covering basic definitions and properties, we will see how viewing certain Hilbert spaces of holomorphic functions as reproducing kernel Hilbert spaces can help answer questions about these spaces and their operators. In particular, we will mention some recent work, which characterizes unitary weighted composition operators on a large class of reproducing kernel Hilbert spaces of holomorphic functions on the Euclidean unit ball of C^n. This is joint work with Michael Hartz.