Talk titles and abstracts
Anna Marie Bohmann, Loop spaces, coTHH, and a new spectral sequence
Given a space with its diagonal map, we construct a topological invariant called "topological coHochschild homology." This is in fact a special case of an invariant for a coalgebra. It generalizes a classical algebraic invariant of coalgebras due to Doi. We show this invariant is related to algebraic K-theory and Waldhausen's A-theory. In the case of a space with the diagonal map, we obtain the free loop space, which indicates connections to string topology as well. Additionally, we build a new spectral sequence to calculate topological coHochschild homology, which allows for new computational techniques. This is joint work with Gerhardt, Hogenhaven, Shipley and Ziegenhagen.
Elden Elmanto, Cobordisms in algebraic geometry and the moduli stack of varieties
That the stable moduli space of complex curves is described by the infinite loop space of a Thom spectrum is a celebrated theorem of Madsen-Weiss and later extended by Galatius-Madsen-Tillman-Weiss. Such a theorem is still faraway in algebraic geometry, but we will describe the moduli space of 0-dimensional varieties, and more generally, of negative-dimensional derived schemes as Thom spectra in algebraic geometry. This is joint work with Hoyois, Khan, Sosnilo and Yakerson.
Ryan Grady, L∞ Spaces, Derived Stacks, and σ-models
Motivated by computing manifold invariants via quantum field theory, I will recall the notion of L∞ space. These spaces are one possible presentation of (smooth) derived stacks. Moreover, smooth manifolds, complex manifolds, Lie algebroids, and their nilpotent thickenings all give natural examples of L∞ spaces. I will then show that a shifted symplectic structure on a given Lie algebroid determines a shifted symplectic structure on the corresponding L∞ space. Via the AKSZ machine, these shifted symplectic spaces define topological quantum field theories. This talk is based on joint work with Owen Gwilliam.
Matthew Hedden, Knot concordance and satellite operators
One can form a group from knots in the 3-sphere after quotienting by an equivalence relation called concordance. I'll discuss this group, and describe how a classical construction in knot theory called "the satellite construction" produces a zoo of operators acting upon it. I'll then raise a number of questions and conjectures about the behavior of these operators, and talk about how to use SO(3) gauge theory to make headway on them. This is joint work with Juanita Pinzon-Caicedo.
John Lind, Twisted Motives
Waldhausen's A-theory A(X) is the universal receptacle for the Euler characteristic of a topological space X. Given a fibration of spaces E → B, the Euler characteristics of the fibers may be amalgamated into a single invariant, which a celebrated theorem of Dwyer, Weiss, and Williams identifies as the obstruction to smoothability into a fiber bundle of manifolds. The best known machinery for proving this theorem is a bivariant form A(E → B) of Waldhausen's A-theory. I will use the analogy between categories and topological spaces to describe a generalization of bivariant A-theory to the setting where the fibration is a continuously parametrized family of categories. The untwisted sector of this theory may be identified with the category of noncommutative motives. An illustrative example in the general setting is the "twisted motive" associated to string topology. This project is joint work with Kim Nguyen, George Raptis, and Christoph Schrade.
Laura Strakston, Symplectic curves in the complex projective plane
Algebraic curves in the plane are a classical subject but with rich complexity. We will discuss a symplectic version of this topic. We will see similarities and differences between the symplectic and algebraic categories. We will focus on symplectic rational cuspidal curves. This is joint work with Marco Golla.