Shimura varieties are moduli spaces of abelian varieties with extra structures (e.g. algebraic cycles, or more generally Hodge cycles). Over the decades, various mathematicians (e.g. Mumford, Deligne, Rapoport, Kottwitz, etc.) have constructed nice integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type (or more generally abelian type) constructed by Kisin and Kisin-Pappas. I will talk about my recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Ag. I will also mention an application to toroidal compactifications of such integral models. 

If time permits, I will also mention a new result on connected components of affine Deligne–Lusztig varieties, which gives us a new CM (i.e. complex multiplication) lifting result for integral models of Shimura varieties at parahoric levels and serves as an ingredient for my main theorem at parahoric levels.
 

A major problem in four-dimensional topology is to understand the difference between topological and smooth four-manifolds, e.g. four-manifolds which are homeomorphic but not diffeomorphic. Smooth manifolds are usually studied by considering invariants which count solutions to a PDE on the four-manifold, like the instanton or Seiberg-Witten equations. These invariants are well-behaved on manifolds with nice geometric properties, like positive scalar curvature or symplectic, but their use for closed manifolds has mostly plateaued. In this talk, we will discuss a slightly different perspective on invariants of four-manifolds and, if time, more topology-intrinsic constructions of four-manifolds. This is joint work with Adam Levine and Lisa Piccirillo.

Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction.  We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

We discuss work-in-progress constructing a quantum trace map for
the special linear group SL_n.  This is a kind of Reshetikhin-Turaev
invariant for knots in thickened punctured surfaces, coming from an
interaction between higher Teichmüller theory and quantum groups.

Let S be a punctured surface of finite genus.  The SL_2-skein algebra of S
is a non-commutative algebra whose elements are represented by framed links
K in the thickened surface S x [0,1] subject to certain relations.  The
skein algebra is a quantization of the SL_2(C)-character variety of S, where
the deformation depends on a complex parameter q.  Bonahon and Wong
constructed an injective algebra map, called the quantum trace, from the
skein algebra of S into a simpler non-commutative algebra which can be
thought of as a quantum Teichmüller space of S. This map associates to a
link K in S x [0,1] a Laurent q-polynomial in non-commuting variables X_i,
which in the specialization q=1 recovers the classical trace polynomial
expressing the trace of monodromies of hyperbolic structures on S when
written in Thurston's shear-bend coordinates for Teichmüller space.  In the
early 2000s, Fock and Goncharov, among others, developed a higher
Teichmüller theory, which should lead to a SL_n-version of this invariant.