• Speaker: Tatsuki Kuwagaki, Osaka
• Title: Sheaf quantization from spectral network
• Date: 09/08/2020
• Time: 8:00 PM - 9:00 PM
• Place: Online (virtual meeting)

A sheaf quantization is a sheaf associated to a Lagrangian brane. This sheaf conjecturally has information as much as Floer theory of the Lagrangian. On the other hand, exact WKB analysis is an analysis of differential equations containing the Planck constant hbar. In this talk, I will explain how to construct a sheaf quantization over the Novikov ring of the spectral curve of an hbar-differential equation, by using the ideas of exact WKB analysis and spectral network. In the construction, one can see how (conjecturally) the convergence in WKB analysis are related to the convergence of Fukaya category. In degree 2, the sheaf quantization associates a cluster coordinate which is the same as Fock—Goncharov coordinate. I will also mention about some relationships to Riemann—Hilbert correspondence of D’Agnolo—Kashiwara and Kontsevich—Soibelman. https://msu.zoom.us/j/95159415920?pwd=bUlETkdpazdiWGNjZnNkUWNIaXRFQT09

• Speaker: Jeffrey Case, Penn State
• Title: A geometric approach to fractional powers of the Laplacian and sharp Sobolev trace inequalities
• Date: 09/22/2020
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting)

A seminal paper of Caffarelli and Silvestre identifies fractional powers of the Laplacian on Euclidean space as Dirichlet-to-Neumann operators. In this talk, I will use conformal geometry to generalize their approach to Riemannian manifolds. More specifically, I will present multiple (equivalent) definitions of (conformally covariant) operators with principal symbol that of a fractional power of the Laplacian. I will also discuss how these operators lead to a simple derivation of a broad family of sharp Sobolev trace inequalities. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

• Speaker: David Jordan, Edinburgh
• Title: Skein theory in the geometric Langlands TFT
• Date: 09/29/2020
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting)

I will overview several appearances (some recently established, some conjectural) of skein theory in the so-called quantum geometric Langlands fully extended TFT. The talk will be mostly elementary, and I'll highlight an application to a conjecture of Witten concerning the finite-dimensionality of skein modules of 3-manifolds at generic values of the quantum parameter. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

• Speaker: Josh Wang, Harvard
• Title: Floer and Khovanov homologies of band sums
• Date: 10/13/2020
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting)
• Contact: Honghao Gao (gaohongh@msu.edu)

Given a nontrivial band sum of two knots, we may add full twists to the band to obtain a family of knots indexed by the integers. In this talk, I'll show that the knots in this family have the same Heegaard and instanton knot Floer homology but distinct Khovanov homology, generalizing a result of M. Hedden and L. Watson. A key component of the argument is a proof that each of the three knot homologies detects the trivial band. The main application is a verification of the generalized cosmetic crossing conjecture for split links. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

• Speaker: Lenny Ng, Duke
• Title: Infinitely many fillings through augmentations
• Date: 10/27/2020
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting)
• Contact: Honghao Gao (gaohongh@msu.edu)

An interesting problem in contact topology is to understand the Lagrangian surfaces that fill a given Legendrian link in a contact 3-manifold. A key breakthrough this year is that we now know some families of Legendrian links that have infinitely many different fillings. This is due to various work by Honghao Gao, Linhui Shen, Daping Weng, and some people who aren't currently at Michigan State, and the common approach is through microlocal sheaf theory and clusters. I'll describe a different, Floer-theoretic approach to the same sort of result, using augmentations of Legendrian contact homology. The Floer approach recovers some of the results from the sheaf approach and also produces some new examples of links with infinitely many fillings. This is joint work in progress with Roger Casals. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

• Speaker: Eric Samperton, UIUC
• Title: Finite gauge groups, TQFT, and the computational complexity of 3-manifold invariants
• Date: 11/03/2020
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting)
• Contact: Honghao Gao (gaohongh@msu.edu)

I will give an overview of some relations between finite group theory, G-equivariant topological quantum field theory, and the computational complexity of invariants of 3-manifolds, both classical and quantum. We will start with one of the simplest kinds of invariants in knot theory: the coloring invariants, introduced by Fox when giving a talk to undergraduates in the 1950s. We will then build up to the idea of G-equivariant TQFT (aka homotopy QFT with target K(G,1)), which mathematically describes the topological order determined by a symmetry-enriched topological phase of matter. Physicists have studied these in part motivated by the search for new universal topological quantum computing architectures. Our goal will be to convey two complexity-theoretic lessons. First, when G is sufficiently complicated (nonabelian simple), the simple-to-define coloring invariants associated to G are, in fact, very difficult to compute, even on a quantum computer. Second, no matter what finite group G one uses, a 3-dimensional G-equivariant TQFT can not be used for universal topological quantum computation if the underlying non-equivariant theory is not already universal. This talk is based on joint works with Greg Kuperberg and Colleen Delaney. https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

• Speaker: Yu Pan
• Title: Augmentations and immersed exact Lagrangian cobordisms
• Date: 11/17/2020
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting)
• Contact: Honghao Gao (gaohongh@msu.edu)

Not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all augmentations come from possibly immersed exact Lagrangian fillings. We first associate an immersed cobordism with a DGA map from the top Legendrian knot to the DGA of the cobordism. This gives a functor from a Legendrian category whose morphisms involve immersed Lagrangian cobordisms to a DGA category. With this functorality, an immersed filling L together with an augmentation of L induce an augmentation of the top Legendrian knot. This is a joint work with Dan Rutherford. Zoom: https://msu.zoom.us/j/92550015779?pwd=azRER2p1Sm5CWWRML3lHbVQyWDU1QT09

• Speaker: Pyongwon Suh, Northwestern University
• Title: TBA
• Date: 11/24/2020
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting)
• Contact: Honghao Gao (gaohongh@msu.edu)

TBA

• Speaker: Shea Vela-Vick, Louisiana State University
• Title: TBA
• Date: 12/08/2020
• Time: 3:00 PM - 4:00 PM
• Place: Online (virtual meeting)
• Contact: Honghao Gao (gaohongh@msu.edu)

TBA

• Speaker: Junxiao Wang, Northwestern
• Title: TBA
• Date: 12/15/2020
• Time: 11:00 AM - 11:59 AM
• Place: Online (virtual meeting)
• Contact: Honghao Gao (gaohongh@msu.edu)

TBA