# 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.

### Spring 2020

Date Speaker Title/Abstract
1/27/2020 Lawrence Barrott (Boston College) Title: Log Gromov–Witten invariants via degenerations

Abstract: A classical question in algebraic geometry asks to count the number of plane curves of degree d meeting a smooth elliptic curve in a single point tangent to order 3d. This question is best reformulated in terms of log Gromov–Witten invariants which I will introduce. By considering the degeneration of the elliptic curve to the toric boundary Navid Nabijou and I provide a localisation formalism to count these curves. We uncover a refined set of enumerative invariants which we believe are related to certain scattering diagram calculations. If time permits I will explain what happens in higher dimension.

2/3/2020 Ignacio Barros (Northeastern University) Title: On product identities and the Chow rings of holomorphic symplectic varieties

Abstract: For a moduli space $M$ of stable sheaves over a K3 surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\star (M \times X^\ell),\, \ell \geq 1,$ generalizing the classic Beauville–Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_\star (M) \subset CH_\star (M).$ We prove the proposed identities when $M$ is the Hilbert scheme of points on a K3 surface. This is based on joint work with L. Flapan, A. Marian and R. Silversmith.

2/10/2020 Dan Mangoubi

(Einstein Institute of Mathematics)

Title: On eigenvalues and eigenfunctions of the clamped plate

Abstract: A celebrated theorem of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two. We study the fourth order clamped plate problem, showing that the multiplicity of eigenvalues is uniformly bounded. Our method is based on new recursion formulas and Siegel–Shidlovskii theory. If time permits, we discuss possible applications also to nodal geometry. The talk is based on a joint work with Yuri Lvovsky.

2/17/2020 President’s Day
2/24/2020 Yingdi Qin (Harvard) Title: Coisotropic branes on symplectic tori and homological mirror symmetry

Abstract: Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves on the mirror complex manifold. Without suitable enlargement (split closure) of the Fukaya category, certain objects of it are missing to prevent HMS from being true. Kapustin and Orlov conjecture that coisotropic branes should be included into the Fukaya category from a physics view point. In this talk, I will construct for linear symplectic tori a version of the Fukaya category including coisotropic branes and show that the usual Fukaya category embeds fully faithfully into it. I will also explain the motivation of the construction through the perspective of Homological mirror symmetry.

3/2/2020 Mauricio Romo (Tsinghua University)

Video

Title: Enumerative invariants and exponential networks

Abstract: I will define and review the basics of exponential networks associated to CY 3-folds described by conic bundles. I will focus mostly on the mathematical aspects and general ideas behind this construction as well as its conjectural connection with generalized Donaldson–Thomas invariants. This is based on joint work with S. Banerjee and P. Longhi.

3/9/2020 Laure Flapan (MIT) Title: Fano Lagrangian submanifolds of hyperkahler manifolds

Abstract: For any polarized hyperkahler manifold of K3 type whose dimension is divisible by 8, we produce a Lagrangian submanifold which is Fano arising as a connected component of the fixed locus of an involution on the hyperkahler manifold. This is an ongoing joint work with E. Macrì, K. O’Grady, and G. Saccà.

3/16/2020 Spring Break
3/23/2020 Bogdan Stoica (CMSA) This meeting will be taking place virtually on Zoom.

Title: Bit Threads: Understanding Gravitation from Quantum Entanglement

Abstract: The AdS/CFT correspondence stipulates that gravitational evolution in a bulk spacetime is dual to a boundary description that has no gravity. In the AdS/CFT picture the bulk spacetime evolves gravitationally against an anti-de Sitter space background, and the boundary dual theory is a conformal gauge theory in a spacetime of one dimension less. Recent insights by Ryu and Takayanagi have conjectured that quantum entangled boundary states quantitatively give rise to geometry in the bulk. They do so by explicitly referring to “minimal surfaces” in the bulk, connecting them to the entropy of a related area in the boundary. I will present a conceptually and technically powerful complementary holographic entanglement picture, reformulating Ryu–Takayanagi to no longer refer to minimal surfaces, and suggesting a new way to think about the holographic principle and the connection between spacetime gravitation and information. I will introduce the idea of bit threads, and show how they can be used for fun and profit.

3/30/2020

11:00am

Timothy Large (MIT) This meeting will be taking place virtually on Zoom.

Title: Floer K-theory and exotic Liouville manifolds

Abstract: In this talk, I will discuss how to define the (wrapped) Fukaya category of an exact symplectic manifold with coefficients in extraordinary cohomology theories, following the ideas of Cohen–Jones–Segal. I will then explain how to construct an exotic symplectic ball, which has vanishing ordinary symplectic homology, but can be distinguished from the standard ball by using Floer homology with coefficients in complex K-theory.

4/6/2020

3:00pm

Daniel Harlow (MIT) This meeting will be taking place virtually on Zoom.

Title: What does it mean to classify ‘t Hooft anomalies?

Abstract: Recent discussions of topological phases of matter have sometimes been phrased in the language of classifying anomalies.  In this talk I will review what is really meant by an ‘t Hooft anomaly, and then point out that this is not what is actually classified in most of these discussions.  I also will discuss some progress towards filling this gap. Based on work with Hirosi Ooguri (see section 2.7 of https://arxiv.org/abs/1810.05338).

4/13/2020 Djordje Radicevic (Brandeis University)

Video

This meeting will be taking place virtually on Zoom.

Abstract: The goal of this talk is to precisely describe how certain operator properties of continuum QFT (e.g. operator product expansions, current algebras, vertex operator algebras) emerge from an underlying lattice theory.  The main lesson will be that a “continuum limit” must always involve two or more cutoffs being taken to zero in a specific order.  In other words, the naive statement that continuum theories are obtained from lattice ones by letting a “lattice spacing” go to zero is never sufficient to describe the lattice-continuum correspondence.  Using these insights, I will show in detail how the Kac-Moody algebra arises from a nonperturbatively well defined, fully regularized model of free fermions, and I will comment on generalizations and applications to bosonization.  Time permitting, I will describe more intricate examples involving scalar fields, and I will discuss several open questions.

4/20/2020 Kuan-Wen Lai (UMass Amherst) This meeting will be taking place virtually on Zoom

Title: Fourier–Mukai equivalences arising from Cremona transformations I: K3 surfaces

Abstract: The derived equivalences of K3 surfaces and the K3 categories of certain cubic fourfolds are known to be realizable as Hodge isometries, i.e. lattice isometries preserving Hodge structures. On the other hand, Hodge isometries are also known to appear when one factorizes a birational map between varieties and tracks the actions on the middle cohomologies. When does a Hodge isometry induced from the derived equivalence of K3 surfaces/categories arise from a birational map? This is the first of two related talks discussing this question. In this talk, I will exhibit such examples for general K3 surfaces of degree 12. As a corollary, I will introduce how the construction gives an interesting relation in the Grothendieck ring of algebraic varieties. This is joint work with Brendan Hassett.

4/27/2020 Yu-Wei Fan (UC Berkeley)

Video

This meeting will be taking place virtually on Zoom.

Abstract: It is conjectured that two cubic fourfolds are birational if their associated K3 categories are equivalent. We prove this conjecture for very general cubic fourfolds of discriminant 20, where the birational maps are produced via certain Cremona transformations defined by Veronese surfaces. Using these birational maps, we find new rational cubic fourfolds. Joint work with Kuan-Wen Lai.

5/4/2020

10:30-11:30am

John Alexander Cruz Morales (Universidad Nacional de Colombia) This meeting will be taking place virtually on Zoom.

Abstract: We will revisit the computations of Stokes matrices for tt*-structures done by Cecotti and Vafa in the 90’s in the context of Frobenius manifolds and the so-called monodromy identity. We will argue that those cases provide examples of non-commutative Hodge structures of exponential type in the sense of Katzarkov, Kontsevich and Pantev.

5/15/2020 James Sully (University of British Columbia)

Video

This meeting will be taking place virtually on Zoom.

Abstract: It has long been believed that progress in understanding the black hole information paradox would require coming to terms with microscopic details of quantum gravity, beyond the reach of semiclassical effective field theory. In that light, one of the most surprising discoveries of the last year has been that signature features of the unitary evaporation of black holes can already be seen within effective field theory, albeit with the inclusion of ‘euclidean wormholes’. However, these novel contributions are best understood when the gravitational theory is not a single microscopic theory, but an average over many different theories. To save unitarity must we then simultaneously throw it away? I will explain how the same story can be recovered within a single microscopic theory by thinking carefully about the right effective theory for finite-lifetime observers.

### Fall 2019

Date Speaker Title/Abstract
9/9/2019 Daniel Pomerleano (UMass Boston) Title: Intrinsic mirror symmetry via symplectic topology

Abstract: Given a maximally degenerate log Calabi–Yau variety $X$, I will describe how one can recover the birational class of the mirror manifold from a Floer theoretic invariant of $X$ (symplectic cohomology). I will then explain how this result relates to recent constructions in mirror symmetry due to Gross–Hacking–Keel and Gross–Siebert.

9/16/2019 Eirik Eik Svanes (King’s College London and ICTP) Title: On coupled moduli problems and effective topological theories

Abstract: I will discuss recent developments in understanding coupled moduli spaces for geometries which appear naturally in string theory. Focusing on heterotic geometries and $SU(3)$ and $G2$ structure compactifications in particular, which also come equipped with a gauge sector, I will describe how the moduli are captured by effective quasi-topological theories derived from the heterotic supergravity. In the case of $SU(3)$ structure compactifications the topological theory in question is a natural generalization of holomorphic Chern–Simons theory or Donaldson–Thomas theory. In the case of heterotic $G2$ structures we will see that the moduli problem is a lot more coupled, and the moduli space has no intrinsic fibration structure in general.

9/25/2019
Wednesday
Fenglong You (University of Alberta) Title: Gluing Periods for DHT Mirrors

Abstract: Let $X$ be a Calabi–Yau manifold that admits a Tyurin degeneration to a union of two quasi-Fano varieties $X_1$ and $X_2$ intersecting along a smooth anticanonical divisor $D$. Doran–Harder–Thompson conjectured that the Landau–Ginzburg mirrors of $(X_1,D)$ and $(X_2,D)$ can be glued to obtain the mirror of $X$. In this talk, I will explain how periods on the mirrors of $(X_1,D)$ and $(X_2,D)$ are related to periods on the mirror of $X$. The relation among periods relates different Gromov–Witten invariants via their respective mirror maps. This is joint work with Charles Doran and Jordan Kostiuk.

9/30/2019 Dmitry Tonkonog (Harvard) Title: Floer theory and rigid subsets of symplectic manifolds

Abstract: Suppose we are given a symplectic manifold. What general things can we say about the dynamics of its symplectomorphisms? A classical way to explore this question is to find rigid subsets: subsets that cannot be displaced from themselves by any symplectomorphism. This has inspired many developments in Floer theory, including recent ones. I will survey the topic, and prove that Lagrangian skeleta of divisor complements of Calabi–Yau manifolds are rigid. This partially reports on joint work with Umut Varolgunes, as well as other things I learned from him.

10/7/2019 Xiao Zheng (Boston University)
Abstract: In this talk, I will introduce an equivariant mirror construction using a Morse model of equivariant Lagrangian Floer theory, formulated in a joint work with Kim and Lau. In case of semi-Fano toric manifold, our construction recovers the $T$-equivariant Landau–Ginzburg mirror found by Givental. For toric Calabi–Yau manifold, the equivariant disc potentials of certain immersed Lagrangians are closely related to the open Gromov–Witten invaraints of Aganagic–Vafa branes, which were studied by Katz–Liu, Graber–Zaslow, Fang–Liu–Zong and many others using localization techniques. The later result is a work in progress joint with Hong, Kim and Lau.
10/14/2019 Columbus Day
10/21/2019

12:10pm

Man-Wai Cheung (Harvard) Title: Compactification for cluster varieties without frozen variables of finite type

Abstract: Cluster varieties are blow up of toric varieties. They come in pairs $(A,X)$, with $A$ and $X$ built from dual tori. Compactifications of $A$, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the $A$ and the $X$ cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists a positive polytope for the type $A$ cluster varieties which give us a hint to the Batyrev–Borisov construction.

10/28/2019

G02

Max Zimet

(BHI)

Title: K3 metrics from little string theory

Abstract: Calabi–Yau manifolds have played a central role in both string theory and mathematics for decades, but in spite of this no Ricci-flat metric on a compact non-toroidal Calabi–Yau manifold is known. I will discuss a new physically motivated approach toward the determination of such metrics for K3 surfaces. The key remaining step is the determination of a BPS spectrum of a heterotic little string theory on $T^2$. I will use string dualities to provide a number of mathematical reformulations of this problem, ranging from open string reduced Gromov–Witten theory for the mirror K3 surface (in accordance with the SYZ conjecture) to Donaldson–Thomas theory for auxiliary Calabi–Yau threefolds. Finally, I will discuss new approximations to K3 metrics near the semi-flat limit that require only a minimal knowledge of this BPS spectrum.

11/4/2019 Elana Kalashnikov  (Harvard)
Abstract: I will discuss joint work with Chiodo investigating the mirror symmetry of Calabi–Yau hypersurfaces in weighted projective spaces. I will show how given such a hypersurface endowed with a finite order automorphism of a specific type, the traditional cohomological mirror statement can be both specialised and broadened to take into account the weights of the action of the automorphism and the cohomology of its fixed locus. The main tool is Berglund–Hubsch–Krawitz duality. When the automorphism is an involution, this allows us to construct generalisations of Borcea–Voisin orbifolds in any dimension and with any number of factors (joint work with Chiodo and Veniani). For odd prime order automorphisms and dimension 2 orbifolds, this implies mirror symmetry for the associated lattice polarised K3 surfaces.
11/13/2019 Cancelled
11/18/2019 Lampros Lamprou (MIT) Title: Holographic order from modular chaos

Abstract: What quantum mechanical principle underlies the emergence of the local geometric structure of the holographic Universe? I will develop a notion of chaos associated to the entanglement pattern of a QFT state by introducing a bound characterizing modular flow of a subregion. In AdS/CFT,  saturation of modular chaos is intimately linked to the local Poincare symmetry and curvature about a bulk Ryu-Takayanagi surface, suggesting an appealing candidate answer to the question posed above.

11/25/2019 Time: 12:00 – 1:00pm

Speaker: David Svoboda (Perimeter Institute)

Abstract: In recent years, para-Hermitian geometry has been used to describe T-duality covariant spacetimes for string theory. In my talk, I will present applications of para-Hermitian geometry to 2D (2,2) SUSY sigma models and show that this geometry gives rise to a new, yet unexplored, notion of mirror symmetry.

Time: 3:00 – 4:00pm

Speaker: Philsang Yoo (Yale)

Title: From Quantum Field Theory to Geometric Representation Theory

Abstract: Quantum Field Theory is a framework of fundamental physics, which in particular has played important roles in the modern development of various subjects in mathematics, including enumerative geometry, knot theory, and low-dimensional topology. On the other hand, Geometric Representation Theory is a subject in mathematics that studies a linear model of various types of symmetries using powerful techniques of algebraic geometry. In recent years, there has been much progress relating the two subjects, enriching the subject of Geometric Representation Theory. In this colloquium style talk, we will review some recent advancements on the topic. No prior knowledge of either Quantum Field Theory or Geometric Representation Theory will be assumed.

12/2/2019 W. A. Zuniga-Galindo (CINVESTAV) Title: Strings and Quantum Fields Over p-Adic Spacetimes

Abstract: I will discuss some recent results on the connections between local zeta functions with the regularization of p-adic Koba-Nielsen string amplitudes, and with the construction of quantum fields over p-adic spacetimes. The theory of local zeta functions was started in the 50s by Gel’fand and Weil. In the 70s Igusa developed a uniform theory of local zeta functions over local fields of characteristic zero. In the last years the theory has suffered a tremendous expansion due to the introduction of the motivic Igusa zeta functions by Denef and Loeser. By using the theory of local zeta functions is possible to establish (in rigorous way) the regularization of Koba-Nielsen amplitudes on R, C or Q_{p}. There is empirical evidence that in the limit p approaches to one, the p-adic strings are related with the ordinary ones. The theory of local zeta functions allows to define rigorously the limit p tends to one of p-adic Koba-Nielsen amplitudes. Since the 50s is known that the existence of fundamental solutions (Green functions) is consequence of the existence of meromorphic continuation for the Archimedean local zeta functions. This fact is also true in the p-adic case, for this reason the p-adic local zeta functions play a central role in the construction of quantum fields over p-adic spacetimes.

12/9/2019 Jie-qiang Wu (MIT) Title: Covariant phase space with boundaries

Abstract: The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian ﬁeld theories without breaking covariance. However, in the original literature, the authors didn’t systematically treat the total derivatives and boundary terms. In this talk, I will introduce a systematic treatment of the covariant phase space formalism for ﬁeld theories with spatial boundaries. With this formalism, we can give an explicit algorithm for the Hamiltonian, only assuming that the configurations satisfy the equations of motion and boundary conditions. Our formalism also produce an extra boundary term in the Hamiltonian, which is non zero even in Einstein’s general relativity with permissive gauge fixing at the boundary. To illustrate this covariant phase space formalism, we study an interesting example the Jackiw-Teitelboim gravity, where we can explicitly solve the phase space and symplectic form. Reference: arXiv:1906.08616

### 2018-2019

 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 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

### 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.

### 2017

 Date Name Title/Abstract 01-30-17 Yu Qiu, CUHK Title: Spherical twists on 3-Calabi-Yau categories of quivers with potentials from surfaces and spaces of stability conditions Abstract: We study the 3-Calabi-Yau category D(S) associated to a marked surface S. In the case when S is unpunctured, we show that the spherical twist group, which is a subgroup of auto-equivalence group of D(S), is isomorphic to a subgroup of the mapping class group of S_Delta–the decorated version of S. In the case when S is an annulus, we prove that the space Stab of stability conditions on D(S) is contractible. We also present working progress on proving the simply connectedness of Stab for any unpunctured case and on studying Stab for the punctured case. 02-06-17 Christoph Keller, Harvard School of Applied Science and Engineering Title: Mathieu Moonshine and Symmetry Surfing 02-13-17 Artan Sheshmani, Aarhus University/CMSA Title: The theory of Nested Hilbert schemes on surfaces Abstract: In joint work with Amin Gholampour and Shing-Tung Yau we construct natural virtual fundamental classes for nested Hilbert schemes on a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as  Poincare invariants of D\”{urr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In the case of the nested Hilbert scheme of points, we can express these invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. The virtual fundamental classes of the nested Hilbert schemes play a crucial role in the local Donaldson-Thomas theory of threefolds that I will talk about, in talk 2. 02-20-17 Holiday — NO SEMINAR 02-27-17 Wenbin Yan, CMSA Title: Argyres-Douglas Theories, Vertex Operator Algebras and Wild Hitchin Characters Abstract: We discuss some interesting relations among 4d Argyles-Douglas (AD) theories, vertex operator algebras (VOA) and wild Hitchin system. We use the Coulomb branch index of AD theories to study geometric and topological data of moduli spaces of wild Hitchin system. These data show an one to one map between fixed points on the moduli space and irreducible modules of the VOA. Moreover, a limit of the Coulomb branch index of AD theories can be identified with matrix elements of the modular transform ST^kS in certain two-dimensional VOAs. The appearance of VOAs, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising. 03-06-17 Tom Rudelius, Harvard University Title: 6D SCFTs and Group Theory Abstract: We will explore the surprising connection between certain classes of homomorphisms and certain classes of non-compact Calabi-Yau manifolds using 6D superconformal field theories as an intermediate link. 03-13-17 Spring Break — NO SEMINAR 03-20-17 Philippe Sosoe, CMSA Title: New bounds for the chemical distance in 2D critical percolation Abstract: We consider the problem of estimating the length, in lattice spacings, of the shortest open connection between the two vertical sides of a square of side length N in critical percolation, when N tends to infinity. This is known as the chemical distance between the sides. Kesten and Zhang asked if this length is asymptotically negligible compared to the length of the ”lowest crossing”, whose length can be expressed in terms of arm exponents and thus calculated quite precisely on the hexagonal lattice. With M. Damron and J. Hanson, we answered this question in 2015. In this talk, we present improved estimates on the chemical distance, using a new iteration technique. 03-27-17 Agnese Bissi, Harvard University Title: Loops in AdS from conformal symmetry Abstract: In this talk I will discuss a new use for conformal field theory crossing equation in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. I will revisit this problem and the dual 1/N expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in 1/N^2, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. The second approach involves Mellin space. As an example, I’ll show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of \phi^4 theory in AdS, the crossing solution reproduces a previous computation of the one-loop bubble diagram. I will end with a discussion on open problems and new developments. 04-03-17 Nathan Haouzi, University of California, Berkeley Title: Little Strings and Classification of surface defects  Abstract: The so-called 6d (2,0) conformal field theory in six dimensions, labeled by an ADE Lie algebra, has become of great interest in recent years. Most notably, it gave new insights into lower dimensional supersymmetric field theories, for instance in four dimensions, after compactification. In this talk, I will talk about a deformation of this CFT, the six-dimensional (2,0) little string theory: its origin lies in type IIB string theory, compactified on an ADE singularity. We further compactify the 6d little string on a Riemann surface with punctures. The resulting defects are D-branes that wrap the 2-cycles of the singularity. This construction has many applications, and I will focus on one: I will provide the little string origin of the classification of surface defects of the 6d (2,0) CFT, for ADE Lie algebras. Furthermore, I will give the physical realization of the so-called Bala-Carter labels that classify nilpotent orbits of these Lie algebras. 04-10-17 Burkhard Schwab, Harvard CMSA Title: Large Gauge symmetries in Supergravity Abstract: In the recent literature, a class of new symmetries — collectively known as “large gauge symmetries” — has emerged that governs the scattering of massless particles of very low energy on asymptotically flat space times. I will show that this statement extends to supergravity where an infinite family of fermionic symmetries can be derived. The algebra of these fermionic symmetries close in the BMS group and their Ward identity is the factorization of soft gravitinos in the S-matrix. 04-17-17 Ingmar Saberi, Universität Heidelberg Title: Holographic lattice field theories Abstract: Recent developments in tensor network models (which are, roughly speaking, quantum circuits designed to produce analogues of the ground state in a conformal field theory) have led to speculation that such networks provide a natural discretization of the AdS/CFT correspondence. This raises many questions: just to begin, is there any sort of dynamical model or lattice field theory underlying this connection? And how much of the usual AdS/CFT dictionary really makes sense in a discrete setting? I’ll describe some recent work that proposes a setting in which such questions can perhaps be addressed: a discrete spacetime whose bulk isometries nevertheless match its boundary conformal symmetries. Many of the first steps in the AdS/CFT dictionary carry over without much alteration to lattice field theories in this background, and one can even consider natural analogues of BTZ black hole geometries. 04-24-17 Patrick Jefferson, Harvard University Title:  Towards a classification of 5d N = 1 SCFTs Abstract:  I will discuss a new proposal for classifying five-dimensional SCFTs with N = 1 supersymmetry and a simple gauge algebra. This classification program entails studying supersymmetry-protected quantities on the Coulomb branch of moduli space using only representation-theoretic data, and subsumes all known predictions in the literature while predicting the existence of novel theories. Geometric constructions of 5d N = 1 theories via string compactifications interpret the supersymmetric protected data as geometric data associated to a local Calabi-Yau threefold, suggesting the possibility of translating this program into a partial cataloguing of Calabi-Yau geometries. 05-01-17 NO SEMINAR 05-08-17 NO SEMINAR 05-15-17 05-22-17 05-29-17

### 2016

 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