Mathematical Physics Seminar, Mondays

The seminar on mathematical physics will be held on select Mondays and Wednesdays from 12 – 1pm in CMSA Building, 20 Garden Street, Room G10. This year’s Seminar will be organized by Artan Sheshmani and Yang Zhou.

The list of speakers for the upcoming academic year will be posted below and updated as details are confirmed. Titles and abstracts for the talks will be added as they are received.

Date Speaker Title/Abstract
9/10/2018 Xiaomeng Xu, MIT Title: Stokes phenomenon, Yang-Baxter equations and Gromov-Witten theory.

Abstract: This talk will include a general introduction to a linear differential system with singularities, and its relation with symplectic geometry, Yang-Baxter equations, quantum groups and 2d topological field theories.

9/17/2018 Gaetan Borot, Max Planck Institute

Video

Title: A generalization of Mirzakhani’s identity, and geometric recursion

Abstract: McShane obtained in 1991 an identity expressing the function 1 on the Teichmueller space of the once-punctured torus as a sum over simple closed curves. It was generalized to bordered surfaces of all topologies by Mirzakhani in 2005, from which she deduced a topological recursion for the Weil-Petersson volumes. I will present new identities which represent linear statistics of the simple length spectrum as a sum over homotopy class of pairs of pants in a hyperbolic surface, from which one can deduce a topological recursion for their average over the moduli space. This is an example of application of a geometric recursion developed with Andersen and Orantin.

9/24/2018 Yi Xie, Simons Center Title: sl(3) Khovanov module and the detection of planar theta-graph

Abstract:  In this talk we will show that Khovanov’s sl(3) link homology together with its module structure can be generalized for spatial webs (bipartite trivalent graphs).We will also introduce a variant called pointed sl(3) Khovanov homology. Those two combinatorial invariants  are related to Kronheimer-Mrowka’s instanton invariants $J^\sharp$ and $I^\sharp$ for spatial webs by two spectral sequences. As an application, we will prove that sl(3) Khovanov module and pointed sl(3) Khovanov homology both detect the planar theta graph.

10/01/2018 Dori Bejleri, MIT Title: Stable pair compactifications of the moduli space of degree one del Pezzo surfaces via elliptic fibrations

Abstract: A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we can construct a stable pair compactification of the moduli space of anti-canonically polarized degree one del Pezzo surfaces. The KSBA theory of stable pairs (X,D) is the natural extension to dimension 2 of the Deligne-Mumford-Knudsen theory of stable curves. I will discuss the construction of the space of interest as a limit of a space of weighted stable elliptic surface pairs and explain how it relates to some previous compactifications of the space of degree one del Pezzo surfaces. This is joint work with Kenny Ascher.

10/08/2018 Pei-Ken Hung, MIT Title: The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part

Abstract: We study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d’Alembertian equation. In particular, we prove the solution decays at rate $\tau^{-1+\delta}$ to a linearlized Kerr solution.

10/15/2018 Chris Gerig, Harvard Title: A geometric interpretation of the Seiberg-Witten invariants

Abstract: Whenever the Seiberg-Witten (SW) invariants of a 4-manifold X are defined, there exist certain 2-forms on X which are symplectic away from some circles. When there are no circles, i.e. X is symplectic, Taubes’ “SW=Gr” theorem asserts that the SW invariants are equal to well-defined counts of J-holomorphic curves (Taubes’ Gromov invariants). In this talk I will describe an extension of Taubes’ theorem to non-symplectic X: there are well-defined counts of J-holomorphic curves in the complement of these circles, which recover the SW invariants. This “Gromov invariant” interpretation was originally conjectured by Taubes in 1995. This talk will involve contact forms and spin-c structures.

10/22/2018

*Room G02*

Sze Ning Mak, Brown Title: Tetrahedral geometry in holoraumy spaces of 4D, $\mathcal{N}=1$ and $\mathcal{N}=2$ minimal supermultiplets

Abstract: In this talk, I will review the supersymmetry algebra. For Lie algebras, the concepts of weights and roots play an important role in the classification of representations. The lack of linear “eigen-equations” in supersymmetry leads to the failure to realize the Jordan-Chevalley decomposition of ordinary Lie algebras on the supersymmetry algebra. Therefore, we introduce the concept “holoraumy” for the 4D, $\mathcal{N}$-extended supersymmetry algebras, which allows us to explore the possible representations of supersymmetric systems of a specified size. The coefficients of the holoraumy tensors for different representations of the same size form a lattice space. For 4D, $\mathcal{N}=1$ minimal supermultiplets (4 bosons + 4 fermions), a tetrahedron is found in a 3D subspace of the 4D lattice parameter space. For 4D, $\mathcal{N}=2$ minimal supermultiplets (8 bosons + 8 fermions), 4 tetrahedrons are found in 4 different 3D subspaces of a 16D lattice parameter space.

10/29/2018 Francois Greer, Simons Center Title: Rigid Varieties with Lagrangian Spheres

Abstract: Let X be a smooth complex projective variety with its induced Kahler structure.  If X admits an algebraic degeneration to a nodal variety, then X contains a Lagrangian sphere as the vanishing cycle.  Donaldson asked whether the converse holds. We answer this question in the negative by constructing rigid complex threefolds with Lagrangian spheres using Teichmuller curves in genus 2.

11/05/2018 Siqi He, Simons Center Title: The Kapustin-Witten Equations, Opers and Khovanov Homology

Abstract: We will discuss a Witten’s gauge theory program to define Jones polynomial and Khovanov homology for knots inside of general 3-manifolds by counting singular solutions to the Kapustin-Witten or Haydys-Witten equations. We will prove that the dimension reduction of the solutions moduli space to the Kapustin-Witten equations can be identified with Beilinson-Drinfeld Opers moduli space. We will also discuss the relationship between the Opers and a symplectic geometry approach to define the Khovanov homology for 3-manifolds. This is joint work with Rafe Mazzeo.

11/12/2018 No Seminar
11/19/2018 Yusuf Barış Kartal, MIT Title: Distinguishing symplectic fillings using dynamics of Fukaya categories

Abstract: The purpose of this talk is to produce examples of symplectic fillings that cannot be distinguished by the dynamical invariants at a geometric level, but that can be distinguished by the dynamics and deformation theory of (wrapped) Fukaya categories. More precisely, given a Weinstein domain $M$ and a compactly supported symplectomorphism $\phi$, one can produce another Weinstein domain $T_\phi$-\textbf{the open symplectic maping torus}. Its contact boundary is independent of $\phi$ and it is the same as the boundary of $T_0\times M$, where $T_0$ is the once punctured torus. We will outline a method to distinguish $T_\phi$ from $T_0\times M$. This will involve the construction of a mirror symmetry inspired algebro-geometric model related to Tate curve for the Fukaya category of $T_\phi$ and exploitation of dynamics on these models to distinguish them.

11/26/2018 Charles Doran (fill-in)Andreas Malmendier, Utah State (originally)

Video

Speaker: Charles Doran

Title:  Feynman Amplitudes from Calabi-Yau Fibrations

Abstract: This talk is a last-minute replacement for the originally scheduled seminar by Andreas Malmendier.

After briefly reviewing the interpretation of Feynman amplitudes as periods of graph hypersurfaces, we will focus on a class of graphs called the n-loop sunset (or banana) graphs.  For these graphs, the underlying geometry consists of very special families of (n-1)-dimensional Calabi-Yau hypersurfaces of degree n+1 in projective n-space. We will present a reformulation using fibrations induced from toric geometry, which implies a simple, iterative construction of the corresponding Feynman integrals to all loop orders.  We will then reinterpret the mass-parameter dependence in the case of the 3-loop sunset in terms of moduli of lattice-polarized elliptic fibered K3 surfaces, and describe a method to construct their Picard-Fuchs equations. (As it turns out, the 3-loop sunset K3 surfaces are all specializations of those constructed by Clingher-Malmendier in the originally scheduled talk!)  This is joint work with Andrey Novoseltsev and Pierre Vanhove

—————-

Speaker: Andreas Malmendier

Title: (1,2) polarized Kummer surfaces and the CHL string

Abstract: A smooth K3 surface obtained as the blow-up of the quotient of a four-torus by the involution automorphism at all 16 fixed points is called a Kummer surface. Kummer surface need not be algebraic, just as the original torus need not be.  However, algebraic Kummer surfaces obtained from abelian varieties provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail.

In this talk, we give an explicit description for the relation between algebraic Kummer surfaces of Jacobians of genus-two curves with principal polarization and those associated to (1, 2)-polarized abelian surfaces from three different angles: the point of view of 1) the binational geometry of quartic surfaces in P^3 using even-eights, 2) elliptic fibrations on K3 surfaces of Picard-rank 17 over P^1 using Nikulin involutions, 3) theta-functions of genus-two using two-isogeny.  Finally, we will explain how these (1,2)-polarized Kummer surfaces naturally appear as F-theory backgrounds for the so-called CHL string. (This is joint work with Adrian Clingher.)

12/03/2018 Monica Pate, Harvard Title: Gravitational Memory in Higher Dimensions

Abstract: A precise equivalence among Weinberg’s soft graviton theorem, supertranslation conservation laws and the gravitational memory effect was previously established in theories of asymptotically flat gravity in four dimensions. Moreover, this triangle of equivalence was proposed to be a universal feature of generic theories of gauge and gravity.  In theories of gravity in even dimensions greater than four, I will show that there exists a universal gravitational memory effect which is precisely equivalent to the soft graviton theorem in higher dimensions and a set of conservation laws associated to infinite-dimensional asymptotic symmetries.

12/10/2018 Fenglong You, University of Alberta Title: Relative and orbifold Gromov-Witten theory

Abstract: Given a smooth projective variety X and a smooth divisor D \subset X, one can study the enumerative geometry of counting curves in X with tangency conditions along D. There are two theories associated to it: relative Gromov-Witten invariants of (X,D) and orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, Abramovich-Cadman-Wise proved that genus zero relative invariants are equal to the genus zero orbifold invariants of root stacks (with a counterexample in genus 1). We prove that higher genus orbifold Gromov-Witten invariants of X_{D,r} are polynomials in r and the constant terms are exactly higher genus relative Gromov-Witten invariants of (X,D). If time permits, I will also talk about further results in genus zero which allows us to study structures of genus zero relative Gromov-Witten theory. This is based on joint work with Hisan-Hua Tseng, Honglu Fan and Longting Wu.

1/28/2019 Per Berglund (University of New Hampshire) Title: A Generalized Construction of Calabi-Yau Manifolds and Mirror Symmetry

Abstract: We extend the construction of Calabi-Yau manifolds to hypersurfaces in non-Fano toric varieties. This provides a generalization of Batyrev’s original work, allowing us to construct new pairs of mirror manifolds. In particular,  we find novel K3-fibered Calabi-Yau manifolds, relevant for type IIA/heterotic duality in d=4, N=2, string compactifications. We also calculate the three-point functions in the A-model following Morrison-Plesser, and find perfect agreement with the B-model result using the Picard-Fuchs equations on the mirror manifold.

2/4/2019 Netanel (Nati) Rubin-Blaier (Cambridge) Title: Abelian cycles, and homology of symplectomorphism groups

Abstract: Based on work of Kawazumi-Morita, Church-Farb, and N. Salter in the classical case of Riemann surfaces, I will describe a technique which allows one to detect some higher homology classes in the symplectic Torelli group using parametrized Gromov-Witten theory. As an application, we will consider the complete intersection of two quadrics in $P^5$, and produce a non-trivial lower bound for the dimension of the 2nd group homology of the symplectic Torelli group (relative to a fixed line) with rational coefficients.

2/11/2019 Tristan Collins (MIT) Title: Stability and Nonlinear PDE in mirror symmetry

Abstract: A longstanding problem in mirror symmetry has been to understand the relationship between the existence of solutions to certain geometric nonlinear PDES (the special Lagrangian equation, and the deformed Hermitian-Yang-Mills equation) and algebraic notions of stability, mainly in the sense of Bridgeland.  I will discuss progress in this direction through ideas originating in infinite dimensional GIT. This is joint work with S.-T. Yau.

2/25/2019 Hossein Movasati (IMPA) Title: Modular vector fields

Abstract: Using the notion of infinitesimal variation of Hodge structures I will define an R-variety which generalizes Calabi-Yau and  abelian varieties, cubic four, seven and ten folds, etc. Then I will prove a theorem concerning the existence of certain vector fields in the moduli of enhanced R-varieties. These are algebraic incarnation of differential equations of the generating functions of GW invariants (Lian-Yau 1995), Ramanujan’s differential equation between Eisenstein series (Darboux 1887, Halphen 1886, Ramanujan 1911), differential equations of Siegel modular forms (Resnikoff 1970, Bertrand-Zudilin 2005).

3/4/2019 Zhenkun Li (MIT) Title: Cobordism and gluing maps in sutured monopoles and applications.

Abstract: The sutured monopole Floer homology was constructed by Kronheimer and Mrowka on balanced sutured manifolds. Floer homologies on closed three manifolds are functors from oriented cobordism category to the category of modules over suitable rings. It is natural to ask whether the sutured monopole Floer homology can be viewed as a functor similarly. In the talk we will answer this question affirmatively.

In order to study the above problem, we will need to use an important tool called the gluing maps. Gluing maps were constructed in the Heegaard Floer theory by Honda, Kazez and Matić , while were previously unknown in the monopole theory. In the talk we will also explain how to construct such gluing maps in monopoles and how to use them to define a minus version of knot monopole Floer homology.

3/11/2019 Yu Pan (MIT) Title: Augmentations and exact Lagrangian cobordisms.

Abstract: Augmentations are tightly connected to embedded exact Lagrangian fillings. However, not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we introduce immersed exact Lagrangian fillings into the picture and show that all the augmentations come from possibly immersed exact Lagrangian fillings. In this way, we realize augmentations, which is an algebraic object, fully geometrically. This is a joint work with Dan Rutherford working in progress.

3/25/2019 Eduardo Gonzalez (UMass Boston) Title: Stratifications in gauged Gromov-Witten theory.

Abstract:  Let G be a reductive group and X be a smooth projective G-variety. In classical geometric invariant theory (GIT), there are stratifications of X that can be used to understand the geometry of the GIT quotients X//G and their dependence on choices. In this talk, after introducing basic theory, I will discuss the moduli of gauged maps, their relation to the Gromov-Witten theory of GIT quotients X//G and work in progress regarding stratifications of the moduli space of gauged maps as well as possible applications to quantum K-theory. This is joint work with D. Halpern-Leistner, P. Solis and C. Woodward.

4/1/2019 Athanassios S. Fokas (University of Cambridge) Title: Asymptotics: the unified transform, a new approach to the Lindelöf Hypothesis,and the ultra-relativistic limit of the Minkowskian approximation of general relativity

Abstract: Employing standard, as well as novel techniques of asymptotics, three different problems will be discussed: (i) The computation of the large time asymptotics of initial-boundary value problems via the unified transform (also known as the Fokas method, www.wikipedia.org/wiki/Fokas_method)[1]. (ii) The evaluation of the large t-asymptotics to all orders of the Riemann zeta function[2], and the introduction of a new approach to the Lindelöf Hypothesis[3]. (iii) The proof that the ultra relativistic limit of the Minkowskian approximation of general relativity [4] yields a force with characteristics of the strong force, including confinement and asymptotic freedom[5].

[1] J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation

with t-Periodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925

(2015).

J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with

t-Periodic Data: II. Perturbative Results, Proc. R. Soc. A 471,

20140926 (2015).

[2] A.S. Fokas and J. Lenells, On the Asymptotics to All Orders of the

Riemann Zeta Function and of a Two-Parameter Generalization of the

Riemann Zeta Function, Mem. Amer. Math. Soc. (to appear).

[3] A.S. Fokas, A Novel Approach to the Lindelof Hypothesis,

Transactions of Mathematics and its Applications (to appear).

[4] L. Blanchet and A.S. Fokas, Equations of Motion of

Self-Gravitating N-Body Systems in the First Post-Minkowskian

Approximation, Phys. Rev. D 98, 084005 (2018).

[5] A.S. Fokas, Super Relativistic Gravity has Properties Associated

with the Strong Force, Eur. Phys. J. C (to appear).

4/8/2019 Yoosik Kim (Boston University) Title: String polytopes and Gelfand-Cetlin polytopes

Abstract: The string polytope was introduced by Littelmann and Berenstein–Zelevinsky as a generalization of the Gelfand-Cetlin polytope in representation theory.  For a connected reductive algebraic group $G$ over $\mathbb{C}$ and a dominant integral weight $\lambda$, a choice of a reduced word of the longest element in the Weyl group of G determines a string polytope. Depending on a reduced word of the longest element in the Weyl group, combinatorially distinct string polytopes arise in general. In this talk, I will explain how to classify the string polytopes that are unimodularly equivalent to Gelfand-Cetlin polytopes when $G = \mathrm{SL}_{n+1}(\mathbb{C})$ and $\lambda$ is a regular dominant integral weight. Also, I will explain a conjectural way obtaining SYZ mirrors respecting a cluster structure invented by Fomin–Zelevinsky. This talk is based on joint work with Yunhyung Cho, Eunjeong Lee, and Kyeong-Dong Park.

4/15/2019

Room G02

Junliang Shen (MIT) Title: Perverse sheaves in hyper-Kähler geometry

Abstract: I will discuss the role played by perverse sheaves in the study of topology and geometry of hyper-Kähler manifolds. Motivated by the P=W conjecture, we establish a connection between topology of Lagrangian fibrations and Hodge theory using perverse filtrations. Our method gives new structural results for topology of Lagrangian fibrations associated with hyper-Kähler varieties. If time permits, I will also discuss connections to enumerative geometry of Calabi-Yau 3-folds. Based on joint work with Qizheng Yin.

4/22/2019 Yang Zhou (CMSA) Title: Quasimap wall-crossing for GIT quotients

Abstract: For a large class of GIT quotients X=W//G, Ciocan-Fontanine–Kim–Maulik have developed the

theory of epsilon-stable quasimap invariants. They are conjecturally equivalent to the Gromov–Witten invariants of X

via explicit wall-crossing formulae, which have been proved in many cases, including targets with good torus action

and complete intersections in a product of projective spaces.

In this talk, we will give a proof for all targets in all genera. The main ingredient is the construction of some moduli space

with C^* action whose fixed-point loci precisely correspond to the terms in the wall-crossing formulae.

4/29/2019

Room G02

Zili Zhang(University of Michigan) Title: P=W, a strange identity for Dynkin diagrams

Abstract: Start with a compact Riemann surface X with marked points and a complex reductive group G. According to Hitchin-Simpson’s nonabelian Hodge theory, the pair (X,G) comes with two new complex varieties: the character variety M_B and the Higgs moduli M_D. I will present some aspects of this story and discuss an identity P=W indexed by affine Dynkin diagrams – occurring in the singular cohomology groups of M_D and M_B, where P and W dwell. Based on joint work with

Junliang Shen.

5/6/2019 Dennis Borisov (CMSA)
Abstract: I will explain the notion of shifted symplectic structures due to Pantev, Toen, Vaquie and Vaquie, and then show that a derived scheme with a -2-shifted symplectic structure can be realized as critical locus of a globally defined -1-shifted potential.

Joint work with Artan Sheshmani

For a listing of previous Mathematical Physics Seminars, please click here.

Related Posts