# Previous Mathematical Physics seminars

Below is a list of Mathematical Physics Seminars held prior to the current academic year. For information on the current academic year’s seminars, please click here.

2018:

 Date……… Speaker…………. Title/Abstract 2-5-2018 Hyungchul Kim (Pohang University of Science and Technology) Seiberg duality and superconformal index in 3d Abstract: I will discuss 3d N=2 supersymmetric gauge theories with a unitary gauge group and two matter fields in the adjoint representation. The low energy spectrum of BPS states of the theory can be studied from the superconformal index. The information on the low energy spectrum including monopole operators is essential to construct a Seiberg-type dual theory. Superconformal indices for a dual pair of the theories should be the same, which is a physical basis for a mathematical identity. 2-12-2018 Matthew Stroffregen (MIT) Equivariant Khovanov Spaces Abstract: Associated to a link L in the three-sphere, Lipshitz-Sarkar constructed a topological space, well-defined up to stable homotopy, whose homology is the (even) Khovanov homology of L.  We extend this to construct an “odd Khovanov space” of L, whose homology recovers odd Khovanov homology. We also equip the odd Khovanov space with a natural involution whose fixed point set is the (even) Khovanov space of Lipshitz-Sarkar, and show that the even Khovanov space admits its own natural involution.  We outline some conjectures relating the even and odd Khovanov spaces. This is joint work with Sucharit Sarkar and Chris Scaduto. 2-26-2018 Jordan Keller (Harvard) Linear Stability of Schwarzschild Black Holes Abstract: The Schwarzschild black holes comprise a static, spherically symmetric family of black hole solutions to the vacuum Einstein equations.  The physical relevance of such solutions is intimately related to their stability under gravitational perturbations. We present results on the linear stability of the Schwarzschild black holes, joint work with Pei-Ken Hung and Mu-Tao Wang. 3-5-2018 Shinobu Hosono (Gakushuin University) Movable vs monodromy nilpotent cones of Calabi-Yau manifolds abstract: I will show two interesting examples of mirror symmetry of Calabi-Yau complete intersections which have birational automorphisms of infinite order. I will first describe/observe mirror correspondences between the movable cones in birational geometry and the monodromy nilpotent cones which are defined at each boundary points (called LCSLs) in the moduli spaces and naturally glued together. In doing this, I will identify  “Picard-Lefschetzs monodromy transformations for flopping curves” in the mirror families. If time permits, I will show one more example of Calabi-Yau complete intersections for which we observe similar correspondence between the birational geometry and monodromy nilpotent cones. However, in this example, we observe that the correspondence becomes complete when we include a non-toric boundary point in the mirror family. This is based on a recent paper with H. Takagi (arXiv:170 3-5-2018 Emanuel Scheidegger (Albert Ludwigs University of Freiburg) Periods and quasiperiods of modular forms and the mirror quintic at the conifold. Abstract: We review the theory of periods of modular forms and extend it to quasiperiods. General motivic conjectures predict a relation between periods and quasiperiods of certain weight 4 Hecke eigenforms associated to hypergeometric one-parameter families of Calabi-Yau threefolds. We verify this prediction and discuss some of its implications. 3-19-2018 Room G02 Emanuel Scheidegger (Albert Ludwigs University of Freiburg) From Gauged Linear Sigma Models to Landau-Ginzburg orbifolds via central charge functions Abstract: We review the categorical description of the Calabi-Yau/Landau-Ginzburg correspondence in terms of equivariant matrix factorizations in the gauged linear sigma model. We present the hemisphere partition function as a central charge function in the gauged linear sigma model. We study the relation of this function in the LG phase to the Chern character of equivariant matrix factorizations of the LG potential and generating functions of FJRW invariants. 3-26-2018 Yi Xie SCGP Surgery, Polygons and Instanton Floer homology Abstract: Many classical numerical invariants (including Casson invariant, Alexander polynomial and Jones polynomial) for 3-manifolds or links satisfy surgery formulas relating three different 3-manifolds or links. All those invariants are categorified by certain Floer homologies (or Khovanov homology) which also satisfy so-called surgery exact triangles. In this talk I will discuss the notion of “surgery exact polygons” which appears in the SU(N)-instanton Floer homology theory. Roughly speaking, an “n-gon” is a relation among the Floer homology groups of 3-manifolds obtained by n different Dehn surgeries on a fixed knot. It generalizes the surgery exact triangle in SU(2)-instanton Floer homology. If time permits, I will also talk about a homological-mirror-symmetry-type  conjecture which motivates this work. This is joint work with Lucas Culler and Aliakbar Daemi. 4-2-2018 Cheuk-Yu Mak Cambridge University Discrete Legendre transform and tropical multiplicity from symplectic geometry Abstract:There is a long history of enumerative invariants and related problems in mirror symmetry. One powerful approach to understand it is given by counting tropical curves with multiplicities. Tropical multiplicity formula in dimension two can be easily understood but the generalization to higher dimensions is less transparent. In this talk, we explain the relation between tropical multiplicities and the torsion of the first homology group of the associated Lagrangian submanifolds/cell complexes. It is a preparation talk for the talk “Tropically constructed Lagrangians in mirror quintic threefolds”, which explains the construction of associated Lagrangian submanifolds using degeneration of hypersurface in toric orbifold.It is a joint work with Helge Ruddat. 4-9-2018 Room G02 Brandon B. Meredith Embry-Riddle Aeronautical University Mirror Symmetry on Toric Surfaces via Tropical Geometry Abstract: Mirror symmetry is a curious duality, first noticed by physicists and then excitedly embraced by mathematicians, between certain manifolds and their “mirror” spaces. This talk considers mirror symmetry on toric surfaces, which are varieties with certain convenient combinatorial properties and include many well-known surfaces. These surfaces are especially suited to being exploited by tropical geometry, which is a form of algebraic geometry over the “tropical semi-ring.” This talk will discuss the generalization of mirror symmetry to all toric surfaces (expanded from just the Fano case) following the Gross-Siebert Program wherein singularities are added to the tropical picture in order to pull more curves into view. 4-16-2018 Yuan Gao Stony Brook Title: On the extension of the Viterbo functor Abstract: In this talk I will describe deformations of wrapped Fukaya categories that arise from cobordisms, using which the Viterbo restriction functor can be extended to any exact cylindrical Lagrangian submanifold. I will also discuss how this extension can be viewed from the perspective of Lagrangian correspondences. 4-23-2018 Room G02 Baohua Fu Chinese Academy of Science Title: Equivariant compactifications of vector groups Abstract: In 1954, Hirzebruch raised the problem to classify smooth compactifications of vector spaces with second Betti number 1, which is known till now up to dim 3. In 1999, Hassett-Tschinkel considered the equivariant version of this problem and obtained the classification up to dim. 3. I’ll report recent progress on this (equivariant) problem. In particular, we obtain the classification up to dimension 5. 4-30-2018 Dmitry Tonkonog UC Berkeley Title: Geometry of symplectic flux Abstract: Symplectic flux measures the areas of cylinders swept in the process of an isotopy of a Lagrangian submanifold. This is a classical invariant which captures quantitative aspects of symplectic manifolds. I will report on joint work with Egor Shelukhin and Renato Vianna in which we study the geometry of flux using a technique inspired by mirror symmetry.

2017:

 Date………. Name Title/Abstract 09-04-17 No Talk 09-11-2017 Yu-Shen Lin (Harvad CMSA) From the Decomposition of Picard-Lefschetz Transformation to Tropical Geometry Abstract: Picard-Lefschetz transformation tells the monodromy of a fibration with “good” singular fibres. In the case of fibres are Lagrangian in a symplectic $4$-manifold, there is a natural decomposition of Picard-Lefschetz transformation into two elementary transformations from Floer theory. The idea will help to develop the tropical geometry for some hyperKahler surfaces. 09-18-17 Yoosik Kim (Boston University) Monotone Lagrangian tori in cotangent bundles. Abstract: As an attempt to classify Lagrangian submanifolds and due to their importance in Floer theory, monotone Lagrangian tori have been got attention. In this talk, we provide a way producing monotone Lagrangian tori in the cotangent bundles of some manifolds including spheres or unitary groups. The construction is based on the classification of Lagrangian fibers of a certain completely integrable system on a partial flag manifolds of various types. We then discuss when their Floer cohomologies (under a certain deformation by non-unitary flat line bundles) do not vanish. This talk is based on joint work with Yunhyung Cho and Yong-Geun Oh. 09-27-17 Yu-Wei Fan (Harvard) Weil-Petersson geometry on the space of Bridgeland stability conditions Abstract: Inspired by mirror symmetry, we define Weil-Petersson geometry on the space of Bridgeland stability conditions on a Calabi-Yau category. The goal is to further understand the stringy Kahler moduli space of Calabi-Yau manifolds. This is a joint work with A. Kanazawa and S.-T. Yau. 10-04-17 Dingxin Zhang (Brandeis) <1 part of slopes under degeneration Abstract: For a smooth family of projective varieties over a field of characteristic p > 0, it is known that the Newton polygon of fibers goes up under specialization. In this talk, we will show that when the family acquires singular members, the less than one part of the slopes of the Newton polygon goes up under specialization. This could be viewed as a characteristic p analogue of a simple phenomenon in Hodge theory. 10-11-17 No Talk 10-18-2017 Nati Blaier (Harvard CMSA) Geometry of the symplectic Torelli group Abstract: This talk has two parts. In the first part of the talk, I will introduce the group of symplectomorphism and try to convince you that it is a very important object in symplectic topology by surveying some known structural results and drawing a comparison with the situation in the smooth and Kahler geometries as well as the world of low-dimensional topology. In the second part, I’ll discuss the symplectic Torelli group for higher dimensional symplectic manifolds, and an ongoing project to use Gromov-Witten theory to detect interesting elements. 10-23-2017 *Monday* Florian Beck (Universität Hamburg) Hitchin systems in terms of Calabi-Yau threefolds. Abstract: Integrable systems are often constructed from geometric and/or Lie-theoretic data. Two important example classes are Hitchin systems and Calabi-Yau integrable systems. A Hitchin system is constructed from a compact Riemann surface  together with a complex Lie group with mild extra conditions. In contrast, Calabi-Yau integrable systems are constructed from a priori purely geometric data, namely certain families of Calabi-Yau threefolds. Despite their different origins there is a non-trivial relation between Hitchin and Calabi-Yau integrable systems. More precisely, we will see in this talk that any Hitchin system for a simply-connected or adjoint simple complex Lie group is isomorphicto a Calabi-Yau integrable system (away from singular fibers). 11-01-2017 *12:30pm-1:30pm* Chenglong Yu (Harvard Math) Picard-Fuchs systems of zero loci of vector bundle sections Abstract: We propose an explicit construction for Picard-Fuchs systems of zero loci of vector bundle sections. When the vector bundle admits large symmetry, the system we constructed is holonomic. This is a joint work with Huang, Lian and Yau. 11-06-2017 *Monday* Pietro Benetti Genolini(Univ. of Oxford) Topological AdS/CFT Abstract: I will describe a holographic dual to the Donaldson-Witten topological twist of gauge theories on a Riemannian four-manifold. Specifically, I will consider asymptotically locally hyperbolic solutions to Romans’ gauged supergravity in five dimensions with the four-manifold as conformal boundary, and show that the renormalised supergravity action is independent of the choice of boundary metric. This is a first step in the direction of combining topological quantum field theory with the AdS/CFT correspondence. 11-13-2017 *Monday 12:30pm* *Room G02* Yusuf Baris Kartal(MIT) Dynamical invariants of categories associated to mapping tori Abstract: One can construct the symplectic mapping torus for a given a symplectic manifold with a symplectomorphism and use the flux invariant to distinguish the mapping tori of maps of different order. The essential argument is that the flow in a certain direction have different periods depending on the order of the symplectomorphism. In this talk, we will introduce an abstract categorical version of the mapping torus- associated to an $A_\infty$ category and an auto-equivalence. Then, we will construct a family of bimodules analogous to the flow and discuss how to characterize it intrinsically and how to use it to distinguish different categorical mapping tori. 11-22-2017 No Talk 11-29-2017 Amitai Zernik (IAS) Computing the A∞ algebra of RP2m CP2m using open fixed-point localization. Abstract: I’ll explain how to compute the equivariant quantum A∞ algebra A associated with the Lagrangian embedding of RP2m in CP2m, using a new fixed-point localization technique that takes into account contributions from all the corner strata. It turns out that A is rigid, so its structure constants are independent of all choices. When m = 1 and in the non-equivariant limit, they specialize to give Welschinger’s counts of real rational planar curves passing through some generic, conjugation invariant congurations of points in CP2m. So we get a diagrammatic expression for computing Welschinger invariants, which I’ll demonstrate with some examples.Time permitting, I’ll discuss a formal extension to higher genus which satises string and dilaton. 12-06-2017 Sarah Venkatesh (Columbia) Closed-string mirror symmetry for subdomains Abstract: We construct a symplectic cohomology theory for Liouville cobordisms that detects non-trivial elements of the Fukaya category.  This theory is conjecturally mirror to the Jacobian ring of a Landau-Ginzburg superpotential on an affinoid subdomain.  We illustrate this manifestation of mirror symmetry by examining cobordisms contained in negative line bundles.

 Date Name Title/Abstract 09-12-16 Chong Wang, Harvard Title: A duality web in 2+1 dimensions Abstract: I will discuss a web of field theory dualities in 2+1 dimensions that generalize the known particle/vortex duality. Some of these dualities relate theories of fermions to theories of bosons. Others relate different theories of fermions. Assuming some of these dualities, other dualities can be derived. I will present several consistency checks of the dualities and relate them to S-dualities in 3+1 dimensions. 09-19-16 Johannes Kleiner, University of Regensburg Title: A New Candidate for a Unified Physical Theory Abstract: The CFS theory is a new approach to describe fundamental physics. Giving quantum mechanics, general relativity and quantum field theory as limiting cases, it is a candidate for a unified physical theory. The goal of my talk is to explain the basic concepts and the general physical picture behind the approach. In particular, I will focus on the connection to contemporary physics. 09-26-16 Can Kozcaz, CMSA Cheshire Cat Resurgence We explore a one parameter ζ-deformation of the quantum-mechanical Sine-Gordon and Double-Well potentials which we call the Double Sine-Gordon (DSG) and the Tilted Double Well (TDW), respectively. In these systems, for positive integer values of ζ, the lowest ζ states turn out to be exactly solvable for DSG – a feature known as Quasi-Exact-Solvability (QES) – and solvable to all orders in perturbation theory for TDW. For DSG such states do not show any instanton-like depen- dence on the coupling constant, although the action has real saddles. On the other hand, although it has no real saddles, the TDW admits all-orders perturbative states that are not normalizable, and hence, requires a non-perturbative energy shift. Both of these puzzles are solved by including complex saddles. 10-03-16 Masahito Yamazaki, IMPU abstract: (for physicists) I will discuss analytic structures of the conformal block as a function of the scaling dimension. This will lead us torecursion relations for conformal blocks, which are also efficient for numerics.  (for mathematicians) I will discuss representation theory of parabolic Verma modules for basic Lie superalgebras. In particular I will introduce a new determinant formula for the contravariant form. 10-17-16 Fabian Haiden, Harvard I will discuss some recent results which came out of the study of the flow on metrized quiver representations. This flow is a finite-dimensional toy model for non-linear heat-type flows. In joint work with Katzarkov, Kontsevich, and Pantev, we find that the asymptotics of the flow on a given quiver representation define a filtration (indexed by R^\infty) which has a purely algebraic interpretation. A novel feature is the existence of non-linear walls, on which asymptotics of the metric are described by nested logarithms. 10-24-16 Arnav Tripathy, Harvard University Spinning BPS states and motivic Donaldson-Thomas invariants I’ll describe a new chapter in the enumerative geometry of the K3 surface and its product with an elliptic curve in a long line of extensions starting from the classic Yau-Zaslow formula for counts of rational nodal curves. In particular, I’ll give a string-theoretic derivation of the threefold’s motivic Donaldson-Thomas invariants given the Hodge-elliptic genus of the K3, a new quantity interpolating between the Hodge polynomial and the elliptic genus. 10-31-16 Joseph Minahan, Uppsala University Supersymmetric gauge theories on $d$-dimensional spheres Abstract: In this talk I discuss localizing super Yang-Mills theories on spheres in various dimensions.  Our results can be continued to non-integer dimensions, at least perturbatively,  and can thus be used to regulate UV divergences.  I will also show how this can provide a way to localize theories with less supersymmetry. 11-07-16 Seung-Joo Lee, Virginia Tech Multiple Fibrations in Calabi-Yau Geometry and String Dualities Abstract: We study the ubiquity of multiple fibration structures in known constructions of Calabi-Yau manifolds and explore the role they play for string dualities. Upon introducing new tools for resolved Calabi-Yau varieties, we analyze a set of F-theory effective theories associated to the different elliptic fibrations and relate them via the M-/F-theory correspondence. Explicit geometric examples will include higher-rank Mordell-Weil groups and non-flat fibrations. In addition, in the context of heterotic/F-theory duality, we also investigate the role played by multiple nested structures of K3- and elliptic fibrations in known and novel string dualities in various dimensions. 11-14-16 Thomas Walpuski, MIT Title: Singular PHYM connections (on ACyl Kähler manifolds) Abstract: The celebrated Donaldson–Uhlenbeck–Yau Theorem asserts that a holomorphic vector bundle over a compact Kähler manifolds admits a projectively Hermitian Yang–Mills (PHYM) metric if and only if it is μ–polystable.  Using a geometric regularization scheme, Bando–Siu extended the DUY Theorem to reflexive sheaves; however, they leave the singularities of the PHYM metrics unstudied. In the first part of this talk I will discuss a version of the DUY/BS Theorem for asymptotically cylindrical Kähler manifolds.  I will briefly explain our motivation coming from G2 gauge theory and then sketch the crucial step of proof, which is how to use μ–stability at infinity to obtain a priori C^0 estimates.  The second part of this talk focuses on understanding the singularities of PHYM metrics.  In particular, I will explain a simple proof of uniqueness of tangent cones for singular projectively Hermitian Yang–Mills connections on reflexive sheaves at isolated singularities modelled on μ–polystable holomorphic bundles over \P^{n-1}. This is joint work with A. Jacob and H. Sá Earp. 11-21-16 Hee Cheol Kim, Harvard Physics Abstract : In this talk I will discuss various BPS defects in 5d SUSY field theories. In the first part, I will talk about co-dimension 4 defects and their interaction with instanton particles. I will show that the partition function of this co-dimension 4 defect is related to Nekrasov’s qq-character. In the second part, I will talk about co-dimension 2 defects and instanton partition functions. I will also explain that the partition functions of the co-dimension 2 defects give rise to eigenfunctions of associated integral Hamiltonians. 11-28-16 NO MEETING THIS WEEK 12-05-16 Hansol Hong, CMSA Abstract: I’ll briefly review algebraic structures on categories that appear in homological mirror symmetry, and explain how the deformation of this algebraic structure on a Fukaya category can arise a mirror space. 12-12-16