Plenary Talks

Julie Bergner (University of Virginia)

Title: 2-Segal Sets, Algebraic K-theory, and Hall Algebras

Abstract: The notion of 2-Segal space was defined by Dyckerhoff and Kapranov, and independently under the name of decomposition space by Gálvez-Carrillo, Kock, and Tonks. These structures encode algebraic objects for which composition need not always exist or be unique, yet still satisfy associativity.  There are many examples of 2-Segal spaces, but two main applications stand out.  First, 2-Segal spaces arise from the Waldhausen S-construction in algebraic K-theory.  Second, they give rise to Hall algebra constructions, of interest in representation theory.  In this talk, we'll look at a specific family of discrete 2-Segal spaces, or 2-Segal sets, associated to finite graphs, and how we can understand these two very general constructions in this setting.  In particular, we'll show that many of the associated Hall algebras can be identified with cohomology algebras of familiar topological spaces.

Ciprian Manolescu (Stanford University)

Title: Generalizations of Rasmussen’s Invariant

Abstract: Over the last 20 years, the Rasmussen invariant of knots in S^3 has had a number of interesting applications to questions about surfaces in B^4. In this talk I will survey some recent extensions of the invariant to knots in other three-manifolds: in connected sums of S^1 x S^2 (joint work with Marengon, Sarkar, and Willis), in RP^3 (joint work with Willis, and also separate work of Chen), and in a general setting (work by Morrison, Walker and Wedrich; as well as Ren and Willis). I will describe how these invariants give bounds on the genus of smooth surfaces in 4-manifolds such as CP^2 - B^4, S^1 x B^3, S^2 x B^2, RP^3 x I, and the unit disk bundle of S^2. By work of Ren and Willis, the general invariant can even detect exotic smooth structures on some compact 4-manifolds with boundary. 

Gábor Székelyhidi (Northwestern University)

Title: The Lagrangian Mean Curvature Flow

Abstract: An interesting problem in differential geometry is to find special Lagrangian submanifolds, but so far we have limited tools for this. The Lagrangian mean curvature flow gives a possible approach, aiming to deform Lagrangian submanifolds to special Lagrangian ones. I will give an introduction to this topic, and explain what we know so far about singularities that can form along the flow, in connection with conjectures by Thomas-Yau and Joyce. The talk is partly based on joint work with Jason Lotay and Felix Schulze.