The E-polynomial of $M_n(\mathbb{C})$ was determined by Hausel and Rodriguez-Villegas by counting points in $M_n(\mathbb{F}_q)$ for finite fields $\mathbb{F}_q$. I will show how the same methods can be used to calculate a generating function for the E-polynomial of $M_n^\tau(\mathbb{C})$ using the representation theory of $GL_n(\mathbb{F}_q)$. This is part of an ongoing collaboration with Michael Lennox Wong.

The quantum K-theory ring of a flag variety is a generalization of the quantum cohomology ring that encodes information about the arithmetic genera of Gromov-Witten varieties in its structure constants. I will speak about a proof that these structure constants have alternating signs for minuscule flag varieties, including Grassmannians of type A, maximal orthogonal Grassmannians, even dimensional quadrics, and two exceptional flag varieties. This is joint work with Chaput, Mihalcea, and Perrin.

Concrete expressions analogous to the cohomological case (due to the speaker, Goertsches, He, and Mare) only arise in the case when the fundamental group of a stabilizer is free abelian. The computation of the additive structure is mainly representation theory and Lie theory, and the multiplicative structure, surprisingly, follows from the Mayer--Vietoris sequence.

I will discuss invariant-theoretic aspects of the Kostant-Toda lattice, emphasizing two recent developments. The first is a partial compactification of the Kostant-Toda lattice by means of Hessenberg varieties, Slodowy slices, and Mishchenko-Fomenko algebras, and it represents joint work with Hiraku Abe. The second development concerns a Toda-type integrable system on the universal centralizer, a hyperkähler manifold arising in certain representation-theoretic contexts.

I will present an analogous theorem for connected orbit types, where one only looks at the isotropy Lie algebras: The number of connected orbit types is finite if $X$ has finite Betti numbers. The proof rests on fundamental properties of a suitably defined equivariant homology theory.

In current work with Anton Alekseev, Benjamin Hoffman, and Yanpeng Li, we show that for $K$ semisimple and any regular coadjoint orbit $\mathcal{O}_\lambda$, one can construct a family of volume exhausting symplectic embeddings of toric manifolds into $\mathcal{O}_\lambda$. Moreover, these embeddings are equivariant with respect to a Hamiltonian action of a maximal torus of $K$. Our construction combines elements of Poisson-Lie theory, cluster algebras, and a scaling limit of Poisson structures called "tropicalization." In this talk I will endeavour to explain these results as well as our hopes for future work.

Alexander Woo and Alexander Yong carried out a computational-algebraic study of Kazhdan-Lusztig varieties in type A. They produced Grobner bases for their defining ideals, obtained singularity results, and produced combinatorial formulas for their K-polynomials (equivariant K-classes).

After recalling some background, I will discuss work in progress with Laura Escobar, Alex Fink, and Alexander Woo on computational-algebraic properties of certain type C Kazhdan-Lusztig varieties.

This is joint work with Steven Rayan.

The answer is "yes": its natural diffeological group structure. It turns out this group is not just some pathological example, but has many interesting associated structures, and is of interest to many areas of mathematics. In particular, it shows up in geometric quantisation and the integration of certain Lie algebroids as the structure group of certain principal bundles, the main topic of this talk.

We will perform Milnor's construction in the realm of diffeology to obtain a diffeological classifying space for a diffeological group $G$, such as the irrational torus. After mentioning a few hoped-for properties, we then construct a connection 1-form on the $G$-bundle $EG \to BG$, which will naturally pull back to a connection 1-form on sufficiently nice principal $G$-bundles. We then look at what this can tell us about irrational torus bundles.

In the present talk, we explain how to extend the Landweber-Novikov operations on algebraic cobordism to the setup of equivariant generalized cohomology theories via the localization techniques of Kostant-Kumar. The operations we obtain we call localized operations. These operations can be viewed as operations on global sections of the so called structure sheaves on moment graphs (corresponding to arbitrary Coxeter groups). They satisfy several natural properties, e.g. they commute with characteristic map and restrict to usual Landweber-Novikov operations.