Quot schemes are fundamental objects in moduli theory of algebraic geometry. We show that the generating series of certain virtual invariants of Quot schemes of surfaces are expressed by the universal series and Seiberg-Witten invariants.
We apply this to several cases including homological and K-theoretic descendent series, reduced invariants of K3 surfaces, virtual Segre and Verlinde series. In particular, descendent series are shown to be rational functions whenever the curve class is of Seiberg-Witten length N.

Branching Brownian motion (BBM) is a random particle system where each particle diffuses as Brownian motion and branches into a random number of particles at a constant rate. In this talk, we will focus on two variant models of BBM, BBM with absorption and BBM with inhomogeneous breeding potential. In the first model, we derive the long run expected number of particles conditioned on survival in the near critical case. In the second model, we study the entire configuration of particles.

In this talk I will explain an interpretation (due to Lewis Bowen) of the f-invariant as a variant of sofic entropy: it is the exponential growth rate of the expected number of ``good models'' for an action over a random sofic approximation. I will then introduce the relative f-invariant and provide a similar interpretation of this quantity. This provides a formula for the growth rate of the expected number of good models over a type of stochastic block model.

I will talk about joint work with Siegfried Echterhoff and Rufus Willett in which we introduce and study amenable actions of locally compact groups on C*-algebras, building on previous similar notions by Anantharaman-Delaroche for actions of discrete groups. Among the new results we prove an extension of Matsumura's theorem giving a characterisation of the weak containment property (coincidence of full and reduced crossed products) for actions on commutative C*-algebras and give examples showing that this result does not extend to general noncommutative C*-algebras.

Corks are special contractible 4-manifolds that play a key role in the study of exotic smooth 4-manifolds. In this talk, I will describe applications of corks to the study of surfaces in 4-manifolds. I'll begin with some badly behaved 2-spheres in 4-space, based on joint work with Piccirillo. Then I'll use a twist on these ideas to construct smoothly (indeed, holomorphically) embedded disks in the 4-ball that are isotopic through ambient homeomorphisms but not through diffeomorphisms. There will be lots of cartoons.

In the 90s', Witten gave a physical derivation of an isomorphism
between the Verlinde algebra of GL(n) of level l and the quantum
cohomology ring of the Grassmannian Gr(n,n+l). In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten's work by relating the $GL_n$ Verlinde numbers to
the level l quantum K-invariants of the Grassmannian Gr(n,n+l), and refer
to it as the Verlinde/Grassmannian correspondence. The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2
case (n=2) there.
\\
\\
In this talk, I will first explain the background of this correspondence
and its interpretation in physics. Then I will discuss the main idea of
the proof for arbitrary rank. A new technical ingredient is the virtual
nonabelian localization formula developed by Daniel Halpern-Leistner.