A manifold is called formal if it has the same rational homotopy type
as the cohomology ring. We first consider the formality of a sphere bundle
over a formal manifold. In this case the formality is entirely determined by
the Bianchi-Massey tensor, which is a 4-tensor on a subspace of the
cohomology ring, introduced by Crowley and Nordstrom. As a special case, we
will see that if a manifold and its unit tangent bundle are both formal,
then the manifold has either Euler characteristic zero or rational
cohomology ring generated by one element. Finally we discuss the case of
a general base manifold, and give an obstruction to formality.

Symplectic geometry is a relatively new branch of geometry.
However, a string theory-inspired duality known as “mirror symmetry” reveals
more about symplectic geometry from its mirror counterparts in complex
geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry
called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS
results were then proved for symplectic mirrors to Calabi-Yau and Fano
manifolds. Those mirror to general type manifolds have been studied in more
recent years, including my research. In this talk, we will introduce HMS
through the example of the 2-torus T^2. We will then outline how it relates
to HMS for a hypersurface of a 4-torus T^4, in joint work with Haniya Azam,
Heather Lee, and Chiu-Chu Melissa Liu. From there, we generalize to
hypersurfaces of higher dimensional tori, otherwise known as “theta
divisors.” This is also joint with Azam, Lee, and Liu.

Joint with Differential Geometry Seminar.

The Jones polynomial is an invariant of knots in $\mathbb{R}^3$. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin--Turaev using quantum groups.

Khovanov homology is a categorification of the Jones polynomial of a knot in $\mathbb{R}^3$, analogously to how ordinary homology is a categorification of the Euler characteristic of a space. It is a major open problem to extend Khovanov homology to knots in 3-manifolds.

In this talk, I will explain forthcoming work towards solving this problem, joint with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. Roughly speaking, our contribution amounts to the first instance of a braiding on 2-representations of a categorified quantum group. More precisely, we construct a braided $(\infty,2)$-category that simultaneously incorporates all of Rouquier's braid group actions on Hecke categories in type A, articulating a novel compatibility among them.

In the 13th of his list of mathematical problems, Hilbert conjectured that the general degree 7 polynomial cannot be solved using only arithmeticoperations and algebraic functions of 2 or fewer variables. In the language of resolvent degree, Hilbert conjectures that RD(S_7) = 3. Reichstein has recently extended the notion of resolvent degree to general algebraic groups G. In this context, a conjecture of Tits asserts that RD(G) = 1 for any connected complex linear algebraic group. Reichstein proves unconditionally that RD(G)\le 5 for such G, and he offers this as possible evidence against Hilbert's conjecture. The goal of this talk is to offer analogous evidence *for* Hilbert's conjecture by extending Reichstein's definition to a notion of resolvent degree for arithmetic groups, variations of Hodge structure, and related moduli problems. We then use geometric techniques to give examples of problems F with RD(F) arbitrarily large. From this perspective, one can paraphrase Hilbert's 13th as asking which is a finite group more like: a connected complex linear algebraic group or an arithmetic lattice? This is joint work with Benson Farb and Mark Kisin.