Classification of Division Algebras via Cohomology

• Speaker: Sam Homfray

Quantum Ergodic Theory and the Arithmetic Quantum Unique Ergodicity Theorem

• Speaker: Dominic Connors

Fourier Restriction and Incidence Geometry

• Speaker: Kai Wu

• Speaker: A break

Duality of vertex algebras and vertex coalgebras

• Speaker: Thai Le

In this talk, we provide some history on and a gentle introduction to the theory of vertex algebras and vertex coalgebras. We look at their axiomatic definitions and some simple examples arising from algebras and coalgebras over a field of characteristic 0. Then, we conclude by presenting our main theorem which states that the algebraic dual of a vertex coalgebra with an additional truncation condition has a natural vertex algebra structure.

Integrability of Highest Weight Virasoro Representations

• Speaker: Yiran Mao

The Virasoro algebra $\mathfrak{vir}$ is the central extension of a dense subalgebra of the complex vector fields on $S^1$. We show that the orientation-preserving diffeomorphism group $\mathrm{Diff}^+(S^1)$ admits a natural semigroup complexification, namely the \emph{semigroup of annuli}, denoted by $\overline{\mathrm{Ann}}$. Using tools from the theories of \emph{symplectic Hilbert spaces} and \emph{bosonic Fock spaces}, we construct a projective representation of $\overline{\mathrm{Ann}}$. and show that every unitary highest weight representation of the Virasoro algebra integrates to a projective representation of $\overline{\mathrm{Ann}}$.

Riemann Sums and Convex Sets in F-spaces

• Speaker: James Carraro

The familiar and simple concept of a Riemann sum requires much less structure on the codomain than the real numbers give us. Stripping away these unnecessary conditions leads us naturally to the idea of an F-space. We shall look at the role of convexity in describing the behaviour of convergence in these space and apply it to see how the convergence of Riemann sums is intrinsically tied to the boundedness of convex hulls.