BU-Keio-Tsinghua 2022 Registration

BOSTON UNIVERSITY-KEIO UNIVERSITY-TSINGHUA WORKSHOP 2022

Geometry and Mathematical Physics

June 27-July 1, 2022

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Abstracts for talks: Click on talk titles for notes/recordings.

Lino Amorim (Kansas State) Enumerative invariants from Calabi-Yau categories

Kontsevich suggested that enumerative predictions of Mirror Symmetry should follow directly from Homological Mirror Symmetry. This requires a natural construction of analogues of Gromov-Witten invariants associated to any dg or A-infinity Calabi-Yau category (with some extra choices). I will survey two approaches to this construction: 1) categorical primitive forms, a non-commutative version of Saito's theory of primitive forms for singularities, which gives only genus zero invariants; 2) Costello's enumerative invariants which conjecturally give invariants in all genera.

Sam Bardwell-Evans (Boston University) Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

We construct special Lagrangian fibrations for log Calabi-Yau surfaces and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the algebro-geometric scattering diagrams of Gross-Pandharipande-Siebert and canonical scattering diagrams of Gross-Hacking-Keel. The argument relies on a holomorphic/tropical disc correspondence to control the behavior of holomorphic discs, allowing us to relate open Gromov-Witten invariants to log Gromov-Witten invariants in this setting without the use of virtual techniques.

David Clausen (UC Irvine) Cohomology and Morse Theory on Symplectic Manifolds

On symplectic manifolds, there are intrinsically symplectic cohomologies of differential forms that are analogous to the Dolbeault cohomology on complex manifolds. These cohomologies are isomorphic to the de Rham cohomologies on odd-dimensional sphere bundles over the symplectic manifold. In this talk, I will describe how we can use this sphere bundle perspective to define a novel Morse-type theory on symplectic manifolds associated with the symplectic cohomologies. This is joint work with Li-Sheng Tseng and Xiang Tang.

A Geometric Approach to Complex Binary Quadratic Forms (For published version, click here.)

Consider a quadratic imaginary field, its ring O of quadratic integers, the Bianchi group B = SL(2,O), and the isometric action of B on hyperbolic 3-space H. Using a B-invariant horoball packing of H, we describe, in finite terms, the action of B on the space of oriented lines in H. This can be interpreted as a classification of complex binary quadratic forms under the action of B, analogous to the classical theory for indefinite real binary quadratic forms under the action of Sl(2,Z). Our method extends to many geometrically finite groups acting on hyperbolic n-space, giving a description, in finite terms, of their geodesic flows.

Masahiro Futaki (Chiba University) Equivariant Homological Mirror Symmetry for CP^1

The mirror of an n-dimensional toric Fano manifold X is known to be (C^*)^n equipped with a Laurent polynomial f and is called the Landau-Ginzburg model. The homological mirror symmetry for toric Fano manifold says that the Fukaya category of X is equivalent to the category of matrix factorizations of f (Cho, Hong and Lau 2019). Givental introduced the equivariant Landau-Ginzburg mirror F by adding logarithmic terms to f. In this talk we formulate and show a version of equivariant homological mirror symmetry for CP^1 by introducing equivariant Floer A-infinity algebra for toric fibers. This is a joint work with Fumihiko Sanda (Gakushuin University) and is based on our preprint arXiv:2112.14622

Noriyuki Hamada (UMass Amherst) Signatures of Lefschetz fibrations and symplectic geography

Lefschetz fibrations arose as a complex version of Morse theory and they have been used as a topological characterization of symplectic manifolds. Particularly in 4--dimension, there is a more handy description of Lefschetz fibrations in terms of monodromy factorizations in the mapping class group, which provides a combinatorial way to construct symplectic 4--manifolds with various features. The main focus in this talk will be on signatures of Lefschetz fibrations. It has been conjectured that signatures of Lefschetz fibrations were always non-positive but I will show there is in fact no constraint on them at all. As applications, we construct symplectic 4--manifolds that realize new lattices on symplectic geography. This is based on joint work with Inanc Baykur (UMass Amherst), arXiv:2010.11916.

Yuichi Ike (U. Tokyo) Interleaving distance for sheaves and symplectic geometry (Recording); (Slides)

The interleaving distance is a canonical pseudo-distance for persistence modules and plays an important role in topological data analysis. Kashiwara and Schapira introduced the interleaving distance on the derived category of sheaves and studied its stability property. In this talk, I show that this distance for sheaves can be used to give a lower bound of the Hamiltonian displacement energy in a cotangent bundle. I would also like to explain our recent result on the completeness of the distance and its application to C^0-symplectic geometry. Joint work with Tomohiro Asano.

George Jeffreys (Boston University) Quantum Finite Automata and Framed Quiver Moduli

The Measurement Problem is an old and important problem in quantum mechanics about modeling the state-changing operations of a quantum system. For the case of Quantum Finite Automata, we formulate noncommutative activation modules for the sake of modeling these operations as a near-ring. This also gives a suitable setting for other types of computing machines such as artificial neural networks. The space of distinct neural networks for a fixed activation module has a framed quiver moduli structure. We develop calculus for this type of moduli stack. For framed quiver moduli in particular, we construct families of explicit Hermitian metrics which allow us to consider positive, negative, and zero curvature variants. We prove a universal approximation theorem for neural networks defined over quiver moduli with a proof that uses a modified tropical geometry construction.

Atsushi Kanazawa (Keio University) Attractor mechanisms of moduli spaces of Calabi-Yau 3-folds

We investigate the complex and Kähler attractor mechanisms of moduli spaces of Calabi-Yau 3-folds. The complex attractor mechanism is concerned with a minimizing problem of the central charges of 3-cycles and defines a new interesting class of Calabi-Yau 3-folds called, the complex attractor varieties. In light of mirror symmetry, we will introduce the Kähler attractor mechanism and define the Kähler attractor varieties. The complex and Kähler attractor varieties are expected to possess very rich structures, in particular certain complex and Kähler rigidities. The talk is based on a joint work with Yu-Wei Fan.

Takeshi Katsura (Keio University) Topological groupoids and C*-algebras

For a (locally) compact Hausdorff space, the algebra of all continuous complex functions on the space remembers the space completely. For a (locally compact) group, the suitable completion of the group algebra remembers unitary representations of the group. These are examples of the construction of C*-algebras from locally compact groupoids. Other such examples include crossed products from group actions on topological spaces. In the talk, we overview the construction of C*-algebras from locally compact groupoids.

Yoosik Kim (Pusan National University) Exotic monotone Lagrangian tori in flag manifolds (Recording); (Slides)

As a part of classification problems of Lagrangian submanifolds in a symplectic manifold, constructing monotone Lagrangian tori that are not Hamiltonian isotopic to each other is important. In this talk, I discuss how to construct infinitely many monotone Lagrangian tori and distinguish them in complete flag manifolds Fl(n) (n > 5) by using toric degenerations and cluster mutations.

Fuyuta Komura (Keio University) Compact abelian group actions on groupoid C*-algebras

The theory of C*-algebras is a branch of functional analysis. One of the main topics in this theory is to investigate the symmetry like group actions on C*-algebras. In this talk, the speaker will explain compact abelian group actions on C*-algebras associated with étale groupoids (i.e. groupoid C*-algebras). More precisely, the speaker will explain that certain compact abelian group actions on groupoid C*-algebras correspond to groupoid homomorphisms from the étale groupoids to the Pontryagin duals of the acting groups. If time permits, the speaker will explain Galois correspondence results for compact abelian group actions on groupoid C*-algebras.

Siu-Cheong Lau (Boston University) Quiver algebroid stack and its applications

Quiver representation emerges from Lie theory and mathematical physics. Its simplicity and beautiful theory have attracted a lot of mathematicians and physicists. On the other hand, algebroid stack is a noncommutative generalization of manifold and arises naturally from moduli theory. In this talk, I will explain a version of algebroid stack that can glue together several quiver algebras with different numbers of vertices. I will also explain our original motivation in mirror symmetry and its applications.

Oleg Lazarev (UMass Boston) Localization and flexibilization in symplectic geometry

Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two symplectic manifolds that are diffeomorphic (plus a bit of tangent bundle data) become symplectomorphic. In this talk, I will explain that it is fruitful to view flexibilization as a localization (away from zero ). Building on work of Abouzaid and Seidel, l will also give examples of new localization functors of symplectic manifolds (up to stabilization and subcriticals) that interpolate between flexible and rigid symplectic geometry and can be viewed as symplectic analogs of topological localization of Sullivan, Quillen, and Bousfield. This talk is based on joint work with Z. Sylvan and H. Tanaka.

Yu-Shen Lin (Boston University) Moduli Spaces of ALH^*-Gravitational Instantons (Recording); (Slides)

Gravitational instantons are non-compact complete hyperKahler 4-manifolds with L² curvatures. They are introduced by Hawkings as the building blocks of his quantum gravity theory. The gravitational instantons are classified by their volume growths into types ALE, ALF, ALG, ALH, and ALG^*, ALH^*. In this talk, I will first give a short proof of the Torelli theorem of gravitational instantons of type ALH^*, namely their corresponding hyperKahler triples are completely determined by the cohomology classes of their hyperKahler triple. The proof is inspired by the SYZ mirror symmetry of log Calabi-Yau surfaces. Then I will explain the characterization for the period domain of the ALH^* gravitational instantons and together give a full description of the moduli space of ALH^* gravitational instantons. The talk is based on joint works with T. Collins, A. Jacob and T.J. Lee.

Angus McAndrew (Boston University) Effective Galois Descent for Motives: The K3 Case (Recording); (Slides)

A theorem of Grothendieck tells us that if the Galois action on the Tate module of an abelian variety factors through a smaller field, then the abelian variety, up to isogeny and finite extension of the base, is itself defined over the smaller field. Inspired by this, we give a Galois descent datum for a motive H over a field by asking that the Galois action on an ℓ-adic realisation factor through a smaller field. We conjecture that this descent datum is effective, that is if a motive H satisfies the above criterion, then it must itself descend to the smaller field.
We prove this conjecture for K3 surfaces, under some hypotheses. The proof is based on Madapusi-Pera’s extension of the Kuga-Satake construction to arbitrary fields.

Yu Qiu (Tsinghua University) Geometric classification of total stability conditions and Reineke's conjecture

We construct a geometric model for root category of any Dynkin diagram Q, which is a h_Q-gon with cores for h_Q is the Coxeter number. We show that the space of total stability conditions on bounded derived category of Q is isomorphic to the moduli space of stable h_Q-gons (up to translation). As an application, by describing the hearts of total stability conditions geometrically, we prove Reineke's conjecture, that for any Dynkin quiver, there is a stability function on its module category such that any indecomposable is stable.

Steve Rosenberg (Boston University) Differential geometric techniques in machine learning

Many problems in machine learning involve using gradient flow to find a function f: ℝⁿ → ℝ^m that best fits some data without overfitting, i.e. curving too much to fit a specific sampled data set. Measuring the amount of curving in a coordinate-free way ideally involves the Riemann curvature tensor, which is too expensive to compute. Using the volume of the graph of the function is an efficient substitute. The choice of topology on the space of functions influences the gradient, and we argue that a Sobolev space metric works better than the standard L² metric. We also discuss the geometry of manifold learning, the art of fitting data to a low dimensional manifold. This is joint work with Qinxun Bai and Dara Gold.

Sema Salur (Rochester) Manifolds with Special Holonomy and Applications

During the past few decades, M-theory, a "theory of everything", has emerged as a leading candidate to unify the four fundamental forces of nature--electromagnetism, gravity, the weak and strong nuclear forces. It is a very active area of research both in mathematics and theoretical physics.
In this talk we will focus on manifolds with special holonomy, spaces whose infinitesimal symmetries play an important role in M-theory compactifications. We will first give brief introductions to Calabi-Yau and G_2 manifolds and then a survey of my recent research on relations between calibrated geometries and dualities of Calabi-Yau manifolds.

Hiro Lee Tanaka (Texas State) Stable Weinstein geometry through localizations

Much of computational math is formula-driven, while much of categorical math is formalism-driven. Mirror symmetry is rich in part because many of its results are driven by both. With the advent of stable-homotopy-theoretic invariants in symplectic geometry--such as Nadler-Shende's microlocal categories and (on the horizon) spectrally enriched wrapped Fukaya categories--there has been a real need for better-behaved formalisms in symplectic geometry. (This is because, nowadays, much of stable homotopy theory is possible only thanks to extremely well-constructed formalisms.) In this talk, we will talk about recent success in constructing the formalism, especially in the setting of certain non-compact symplectic manifolds called Weinstein sectors. The results have concrete geometric consequences, like showing that spaces of embeddings of these manifolds map continuously to spaces of maps between certain invariants. (And in particular, leads to higher-homotopy-group generalizations, in the Weinstein setting, of the Seidel homomorphism, similar to works of Savelyev and Oh-Tanaka.) The main result we'll discuss is that the infinity-category of stabilized sectors can be constructed using the categorically formal process of localization. Most of what we discuss is joint with Oleg Lazarev and Zachary Sylvan.

Tomoki Uchimura (Keio University) New Morphisms Between Groupoids and Between Inverse Semigroup Actions

In studies of groupoid $C^*$-algebras, it is well-known that the groupoid homomorphisms between groupoids are not compatible with the $*$-homomorphisms between groupoid $C^*$-algebras. Considering groups and spaces as simple examples of groupoids, we find that a group homomorphism produces a $*$-homomorphisms between group $C^*$-algebras covariantly and a continuous map between spaces produces a $*$-homomorphisms between commutative $C^*$-algebras contravariantly.
We introduce a generalized morphism between groupoids which produces a $*$-homomorphism between groupoid $C^*$-algebras. Our morphisms are a generalization of algebraic morphisms studied by Buneci and Stachura.

We show that our generalized morphism also produces a morphism between inverse semigroup actions on spaces and there are adjoint functors between the category of groupoids and the other category of inverse semigroup actions. The result on adjoint functors is a generalization of the result proved by Buss, Exel, and Meyer.

Brian Williams (Edinburgh/Boston University) Enhanced symmetry for instantons on CP^2

We propose a symmetry for the space of instantons on CP^2 by an exceptional super Lie algebra called E(3|6). In the case of rank one instantons this factors through a `Heisenberg’ type super Lie algebra and the story is analogous to Nakajima’s action on the Hilbert scheme. Our analysis is motivated by a setup in twisted holography for M-theory. This is based on joint work with Raghavendran and Saberi.

Jie (Frank) Xu (Boston University) Prescribed scalar curvature problems in conformal geometry

In this talk, I discuss first the Yamabe problem on some domain M, or equivalently, the existence of a metric g' = u^p-2 g which admits a constant scalar curvature. This problem is reduced to solving a nonlinear elliptic PDE, called the Yamabe equation, with respect to u. Here p = 2n/(n - 2)}, where n is the dimension of M. We give results for the Yamabe problems on a small Riemannian domain with Dirichlet boundary conditions, on closed manifolds, and on compact manifolds with smooth boundary with Robin boundary conditions involving mean curvature.
We then discuss the Kazdan-Warner problem on closed manifolds and compact manifolds with smooth boundary, which asks if for any given smooth function S, there exists a metric under conformal change for which admits the prescribed scalar curvature S. Some sufficient and necessary conditions will be given.

Finally, we discuss briefly about the connections between the Yamabe problem, especially the elliptic PDE, and the Yamabe flow on closed manifolds.

Junya Yagi (Tsinghua University) Defects, branes and 3D lattice model

The 6-vertex model is one of the most famous integrable 2D lattice models and appears in many different quantum field theories, such as 2D N=(2,2) gauge theories, 4D N=2 gauge theories and 4D Chern-Simons theory. Less known is that the 6-vertex model arises from an integrable 3D lattice model by reduction on a circle. In my previous work with Kevin Costello, we gave a unified understanding to the above-mentioned appearances of the 6-vertex model using branes and string dualities. In this talk, I will explain the construction of the 3D lattice model using branes in M-theory, emphasizing the role played by a structure of TQFT with defects and extra dimensions.

Mayuko Yamashita (RIMS) Differential cohomology and physics

(Generalized) differential cohomology theories are defined on manifolds, and refine generalized cohomology theories with differential geometric data. Their study has been particularly motivated by physics. Various objects in physics, such as gauge fields and their variants, can be understood in terms of differential cohomologies. In this talk, I will review the theory, and explain my works with K.Gomi and K.Yonekura which give new models of generalized differential cohomology theories related to physics.

Jie Zhou (Tsinghua University) Twisted Sectors in Quasi-Homogeneous Polynomial Singularities and Automorphic Forms

Vanishing cohomology of quasi-homogeneous polynomial singularities can be studied using the tool of mixed Hodge structures. For one-parameter deformations of Calabi-Yau type Fermat polynomial singularities along degree-one directions, we show that twisted sectors in the vanishing cohomology are automorphic forms for certain triangular groups, using the Riemann-Hilbert correspondence. We prove consequently that genus zero Gromov-Witten generating series of the corresponding Fermat Calabi-Yau varieties are components of automorphic forms. We also explain some properties of the so-called Yamaguchi-Yau ring which have shown to be useful in studying higher-genus mirror symmetry.
The talk is based on the joint work arxiv: 2112.07182 with Dingxin Zhang.

Zhengyu Zong (Tsinghua University) Torus knots, open Gromov-Witten invariants, and topological recursion (Recording); (Slides)

In this talk, I will discuss interesting relations between torus knots, open Gromov-Witten invariants on the resolved conifold, and the topological recursion on the mirror curve. I will first give a brief introduction to Gromov-Witten theory. Then I will describe the construction of a certain Lagrangian submanifold in the resolved conifold from a given torus knot. In the end, I will discuss the all genus mirror symmetry for the open-closed Gromov-Witten invariants with respect to this Lagrangian submanifold. The B-model will be the topological recursion on the mirror curve. This talk is based on a joint work with Bohan Fang.

Our thanks to the following organizations for workshop funding:

Boston University Department of Mathematics and Statistics
Boston University Graduate School of Arts and Sciences
Keio University
Tsinghua University
National Science Foundation