Abstract

"In this talk I will revisit a few key aspects of the relation between \hat Z -invariants and modular-type objects, including the role of the SL_2(Z) representation, the mock modular invariants, and the connection to vertex operator algebras."

"Radial limit conjectures state that we can write quantum invariants of 3-manifolds as radial limits of q-series. They are important to they relates to modular forms. Such conjectures are established by Gukov-Pei-Putrov-Vafa and Gukov-Manolescu from a point of view of physics, topology and number theory. In this talk, I prove these conjectures for some cases with three key ideas: (1) To develop a new asymptotic formula by the Euler–Maclaurin summation formula; (2) To prove that the conjectures is deduced from the holomorphy of a certain rational function; (3) To prove the holomorphy by the induction on pruning of a plumbing graph."

"I will discuss joint work with Peter Johnson and Slava Krushkal which unifies two invariants of negative definite plumbed 3-manifolds: lattice homology, due to Némethi, and the Gukov-Pei-Putrov-Vafa $\widehat{Z}$ $q$-series. The degree zero part of lattice homology is encoded by a graph called the graded root and is isomorphic to Heegaard Floer homology for a large class of such 3-manifolds. Our invariant takes the form of a two-variable Laurent polynomial weight assigned to each node of the graded root, and the weights stabilize, in an appropriate sense, to a 2-variable refinement of $\widehat{Z}$. Both theories have extensions to plumbed knot complements, namely knot lattice homology, due to Ozsváth-Stipsicz-Szabó, and the Gukov-Manolescu $q$-series for knot complements. I will also discuss knot lattice homology in preparation for part 2."

"Following Part 1 of the talk to be given by Ross, I will discuss a joint work with Peter Johnson and Ross Akhmechet, which unifies knot lattice homology, due to Ozsvath, Stipsicz, and Szabo, and the BPS q-series of Gukov and Manolescu. This invariant is a natural extension of weighted graded roots of negative definite plumbed 3-manifolds introduced earlier by Akhmechet, Johnson, and Krushkal. We prove an integer surgery formula relating our invariant with the weighted graded root of the surgered 3-manifold."

TBD

"I will discuss the definition and properties of the universal pairing of manifolds in various dimensions, and the notion of positivity of the universal pairing. After surveying work of Freedman et. al. with the focus on 4-manifolds, I will discuss recent work with M. Khovanov and J. Nicholson on the universal pairing for 2-complexes. The talk will be entirely self-contained, and it will bring together ideas in topology of 4-manifolds, group theory, and quantum topology."

"A homology plane is an algebraic complex smooth surface with the same integral homology groups as the complex plane. A Mazur type manifold is a compact contractible smooth (real) 4-manifold built only with 0-,1- and 2- handles. We call a homology 3-sphere a Kirby-Ramanujam sphere if it bounds both a homology plane and a Mazur type manifold. In this talk, we present several infinite families of Kirby-Ramanujam spheres and some related topics, if time permits, as splice diagrams, representation varieties and future directions. This is joint work with Oğuz Şavk."

I will revisit known and conjectured relations of the q-series GPPV invariants with torsion of 3-manifolds. The vast literature on combinatorial and analytic torsions gives us clues about the structural properties of GPPV invariants. In particular, I will concentrate on the role of spin^c structures in the story. Joint work with Sergei Gukov.

TBD

Mock 3 File

Resurgence theory is a wide-ranging extension of analyticity and of Borel summability, known to apply in many problems of natural origin. I will talk about some elements of this theory, how it applies to crossing the boundary $|q|=1$ and the numerical procedures it leads to. Work in collaboration with G. Dunne, A. Gruen and S. Gukov.