Speaker: Equivariant Floer theory and SYZ mirror symmetry

Abstract: In this talk, we will first review a symplectic realization of the SYZ program and some of its applications. Then I will explain some recent works on equivariant Lagrangian Floer theory and disc potentials of immersed SYZ fibers. They are joint works with Hansol Hong, Yoosik Kim and Xiao Zheng.

Title: Legendrian graphs and their invariants

Abstract: Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian submanifolds. I will introduce various invariant of Legendrian graphs including DGA type, polynomial type, sheaf theoretic one, and their relationship. This is joint work with Byunghee An, and partially with Tamas Kalman and Tao Su.

Title: Loops in hypertoric varieties and symplectic duality

Abstract: Hypertoric varieties are algebraic symplectic varieties associated to graphs, or more generally certain hyperplane arrangements. They make many appearances in modern geometric representation theory. I will discuss certain infinite dimensional or infinite type generalizations of hypertoric varieties which occur in the study of enumerative invariants, focusing on some elementary examples. Joint work with Artan Sheshmani and Shing-Tung Yau.

Title: The geometry of Maurer–Cartan equation near degenerate Calabi–Yau varieties

Abstract: In this talk, we construct a \(dgBV algebra PV*(X)\) associated to a possibly degenerate Calabi–Yau variety X equipped with local thickening data. This gives a version of the Kodaira–Spencer dgLa which is applicable to degenerated spaces including both log smooth or maximally degenerated Calabi–Yau. We use this to prove an unobstructedness result about the smoothing of degenerated Log Calabi–Yau varieties X satisfying Hodge–deRham degeneracy property for cohomology of X, in the spirit of Kontsevich–Katzarkov–Pantev. This is a joint work with Kwokwai Chan and Naichung Conan Leung.

Title: 3-manifolds, q-series, and topological strings

Abstract: \(\hat{Z}\) is an invariant of 3-manifolds valued in q-series (i.e. power series in q with integer coefficients), which has interesting modular properties. While originally from physics, this invariant has been mathematically constructed for a big class of 3-manifolds, and conjecturally it can be extended to all 3-manifolds. In this talk, I will give a gentle introduction to \(\hat{Z}\) and what is known about it, as well as highlighting some recent developments, including the use of R-matrix, generalization to higher rank, large N-limit and interpretation as open topological string partition functions.

Title: Collective integrable systems and global action-angle coordinates

Abstract: A “collective integrable system” on a symplectic manifold is a commutative integrable system constructed from a Hamiltonian action of a non-commutative Lie group. Motivated by the example of Gelfand-Zeitlin systems, we give a construction of collective integrable systems that generate a Hamiltonian torus action on a dense subset of any Hamiltonian K-manifold, where K is any compact connected Lie group. In the case where the Hamiltonian K-manifold is compact and multiplicity free, the resulting Hamiltonian torus action is completely integrable and yields global action angle coordinates.  Moreover, the image of the moment map is a (non-simple) convex polytope.

Title: Berry phase in quantum field theory

Abstract: We will discuss Berry phase in family of quantum field theories using effective field theory. The family is labelled by parameters which we promote to be spacetime-dependent sigma model background fields. The Berry phase is equivalent to Wess-Zumino-Witten action for the sigma model. We use Berry phase to study diabolic points in the phase diagram of the quantum field theory and discuss applications to deconfined quantum criticality and new tests for boson/fermion dualities in \((2+1)d\).

Title: A geometric construction of orbifold Jacobian algebras

Abstract: We review the definition of a twisted Jacobian algebra of a Landau-Ginzburg orbifold due to Kaufmann et al. Then we construct an A-infinity algebra of a weakly unobstructed Lagrangian submanifold in a symplectic orbifold. We work on an elliptic orbifold sphere and see that above two algebras are isomorphic, and furthermore their structure constants are related by a modular identity which was used to prove the mirror symmetry of closed string pairings. This is a joint work with Cheol-Hyun Cho.

Title: Parabolic de Rham bundles: motivic vs periodic

Abstract: Let \($C$\) be a complex smooth projective curve. We consider the set of parabolic de Rham bundles over \($C$\) (with rational weights in parabolic structure). Many examples arise from geometry: let \($f: X\to U$\) be a smooth projective morphism over some nonempty Zariski open subset \($U\subset C$\). Then the Deligne–Iyer–Simpson canonical parabolic extension of the Gauss–Manin systems associated to \($f$\) provides such examples. We call a parabolic de Rham bundle \emph{motivic}, if it appears as a direct summand of such an example of geometric origin. It is a deep question in the theory of linear ordinary differential equations and in Hodge theory, to get a characterization of motivic parabolic de Rham bundles. In this talk, I introduce another subcategory of parabolic de Rham bundles, the so-called \emph{periodic} parabolic de Rham bundles. It is based on the work of Lan–Sheng–Zuo on Higgs-de Rham flows, with aim towards linking the Simpson correspondence over the field of complex numbers and the Ogus–Vologodsky correspondence over the finite fields. We show that motivic parabolic de Rham bundles are periodic, and conjecture that they are all periodic parabolic de Rham bundles. The conjecture for rank one case follows from the solution of Grothendieck–Katz p-curvature conjecture, and for some versions of rigid cases should follow from Katz’s work on rigid local systems. The conjecture implies that in a spread-out of any complex elliptic curve, there will be infinitely many supersingular primes, a result of N. Elkies for rational elliptic curves. Among other implications of the conjecture, we would like to single out the conjectural arithmetic Simpson correspondence, which asserts that the grading functor is an equivalence of categories from the category of periodic parabolic de Rham bundles to the category of periodic parabolic Higgs bundles. This is a joint work in progress with R. Krishnamoorthy.

Title: Kapustin–Witten TFT on 3-manifolds and skein modules

Abstract: Kapustin and Witten have studied a one-parameter family of topological twists of \(4d N=4\) super Yang–Mills. They have shown that the categories of boundary conditions on a surface are exactly the categories participating in the geometric Langlands program of Beilinson and Drinfeld. Moreover, S-duality is manifested as a quantum geometric Langlands duality after the topological twist. In this talk I will describe some mathematical formalizations of Hilbert spaces of states on a 3-manifold. I will outline an equivalence between two such possible formalizations: complexified Floer homology of Abouzaid–Manolescu and skein modules. This is a report on work in progress joint with Sam Gunningham.

Title: Lifting cobordisms and Kontsevich-type recursions for counts of real curves

Abstract: Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne-Mumford moduli spaces of stable real curves (which is different from Solomon’s approach).

Title: M-theoretic genesis of topological phases

Abstract:  I will talk about a novel way of constructing \((2+1)d\) topological phases using M-theory. They emerge as macroscopic world-volume theories of M5-branes wrapped on non-hyperbolic 3-manifolds. After explaining the algorithm of extracting modular structures of the topological phase  from topological data of the 3-manifold, I will discuss the possibility of full classification of topological orders via the geometrical construction.

Title: Algebras and traces at the boundary of \(4d N=4\) SYM

Abstract: I will describe how the structure of supersymmetric boundary correlators in \(4d N=4\) SYM can be encoded in a class of associative algebras equipped with twisted traces. In the case of interfaces, this yields a new connection to integrability.