## BOSTON UNIVERSITY/KEIO UNIVERSITY WORKSHOP 2017## Geometry and Mathematical Physics## Boston University — June 26-30, 2017 |

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** Abstracts and lecture notes for contributed talks:**

- Lino Amorim (BU):
*Quantum cohomology of orbifold spheres*- In this talk we will describe the quantum cohomology of a sphere with three orbifold points. This is done by defining an isomorphism between the quantum cohomology ring and the Jacobian ring of a certain power series built from the Lagrangian Floer theory of an immersed circle.

- Yu-Wei Fan (Harvard):
*Entropy of an autoequivalence on Calabi-Yau manifolds*- We will present counterexamples to a conjecture on categorical entropy by Kikuta and Takahashi. We will also explain the reason to expect counterexamples from homological mirror symmetry.

- David Fried (BU):
*Geometric methods for closed geodesics in negative curvature*- We will describe the closed geodesics on a negatively curved manifold using geometric structures such as ideal tilings or horoball packings. In the case of the modular surface we recover results of Klein and Hurwitz on binary quadratic forms and continued fractions. Using work of Ruelle our results apply to dynamical zeta functions for geodesic flows.

- Kota Hattori (Keio):
*The nonuniqueness of tangent cones at infinity of Ricci flat manifolds*- For a metric space (X,d), the Gromov-Hausdorff limit of (X, a_n d) as a_n goes to zero is called the tangent cone at infinity of (X,d). Colding and Minicozzi showed the uniqueness of the tangent cone at infinity of Ricci flat manifolds satisfying some additional conditions. In this talk, I construct an example of noncompact complete hyper-Kaehler manifold which has several tangent cones at infinity, and determine the moduli space of them.

- Kenta Hayano (Keio)):
*Construction of Lefschetz fibrations and pencils via mapping class groups*- In this talk we will discuss Lefschetz fibrations and pencils on four-manifolds. Although these fibrations were originally studied in complex/algebraic geometry, they have also attracted a lot of attentions in differential topology since Donaldson and Gompf clarified relation between Lefschetz fibrations/pencils and symplectic strucutres. We will construct several examples of Lefschetz fibrations and pencils in a combinatorial way via mapping class groups of surfaces. This is a joint work with Refik Inanc Baykur.

- Shun'ichi Honda (Hokkaido):
*Talk: Developable surfaces along framed base curves*- As application of framed curve theory, I'll introduce some developable surfaces which come from a framed curve. In particular, I'll focus on their come from a Frenet type framed curve. A Frenet type framed curve has a moving frame with geometrically important information.

*Poster: Framed curves in Euclidean space*- As preparation for my presentation, I'll introduce the notion of framed curve. A framed curves is a curve with a moving frame. It is a generalization not only of a regular curve with a linear independence condition, but also of a Legendre curve in the unit tangent bundle. A framed curve may have singularities. I'll introduce the existence and the uniqueness for framed curves by using their curvature.

- Yusuke Inagaki (Osaka):
*Talk: The SL(n)-representation of fundamental groups and related topics*- SL(2)-representations are well studied in the context of the moduli spaces of hyperbolic structures for 2 or 3-manifolds. In this talk, we review and discuss the higher dimensional representation theory of fundamental groups as generalizations of the SL(2)-representation theory.

*Poster: The SL(n)-representation of fundamental groups and related topics*- This poster explains the definition of the higher Teichmuller space and the Bonahon-Dreyer coordinate of its space. To define this coordinate, we use laminations on surfaces and the Anosov property of Hitchin representations which are elements of the higher Teichmuller space. Moreover we see how the Fuchsian locus which is some canonical locus of the higher Teichmuller space is described for a pair of pants.

- Hiroyasu Izeki (Keio):
*A fixed-point property of random groups*- A group G is said to have a fixed-point property for a metric space Y if every isometric action of G on Y has a fixed point. For example, fixed-point property for Hilbert spaces is known to be equivalent to Kazhdan's property (T). Recent results suggest that such property is, in a sense, common among finitely generated groups. In this talk I will present a slightly stronger result in this direction: for certain random groups, every uniformly Lipschitz affine action on a Hilbert space has a fixed point.

- Takeshi Katsura (Keio):
*Noncommutative topology and quantum groups*- Theory of C*-algebras can be considered as noncommutative analogue of theory of locally compact spaces. C*-algebras also give quantum version of locally compact groups in two ways (via group algebras and via Hopf algebras). The talk starts by explaining of these rough statements with examples including noncommutative tori, and finishes discussion on two classes of C*-algebras I study recently; noncommutative 3-spheres obtained from Hopf fibration, and free version of symmetric groups.

- Takayuki Kobayashi (Keio):
*Poster: The Rasmussen invariant of torus knots*- The Milnor conjecture is a claim that the genus of (p,q)-torus knot is calculated by (p,q). It was proved by P. Kronheimer and T. Mrowka using gauge theory (1993). The Rasmussen invariant defined by J. Rasmussen gives another proof of Milnor conjecture in a combinatorial way (2004). I'll introduce the definition of Rasmussen invariant and the proof of Milnor conjecture by J. Rasmussen.

- Fuyuta Komura (Keio):
*Poster: On the relation between amenability of discrete groups and nuclearity of group C*-algebras*- Nuclearity is an important concept in the theory of C*-algebras. Generally, it is difficult to determine whether a given C*-algebra is nuclear or not. In the case of group C*-algebras, it is known that nuclearity of a group C*-algebra is equivalent to amenability of the group. In my poster, I'll describe amenability of groups, nuclearity of C*-algebras and the relation between them. As a byproduct, we observe some examples of nonnuclear C*-algebras.

- Yosuke Kubota (RIKEN):
*Dualizing the relative higher index and almost flat bundles*- The Baum-Connes assembly map is a central subject in modern index theory in connection with big problems in geometry such as the Novikov conjecture and the existence of positive scalar curvature metrics. Recently, the notion of relative assembly for manifolds with boundaries is introduced by Chang-Weinberger-Yu. It is also studied by Deely-Goffeng from another viewpoint. In this talk, I introduce an equivalent definition of the relative assembly map as a generalization of the Mishchenko-Fomenko higher index. This new definition enables us to consider the dual of the assembly map, which is related to the pairing with almost flat vector bundles.

- Leandro Lichtenfelz (Notre Dame):
*Singularities of the L^2 Exponential Map on Diffeomorphism Groups*- In this talk, we will discuss recent results on conjugate points and normal forms for the L^2 Riemannian exponential map on the diffeomorphism group of a compact 2D manifold. This map is strongly tied to the motion of a perfect incompressible fluid on the manifold. We will begin by reviewing this connection, which goes back to the work of Arnold.

- Siu-Cheong Lau (BU):
*Geometric transition in toric degenerations and SYZ mirror symmetry*- Morrison asserted that the process of geometric transition is reversed under mirror symmetry. Locally it was realized using the SYZ program by my previous work and also my joint work with Kanazawa. In the case of conifold transitions, global construction was done by the work of Castano-Bernard-Matessi via affine geometry and the Gross-Siebert program of toric degenerations. In this talk I will review the local geometry and carry out the global construction for more general geometric transitions.

- Takayuki Morifuji (Keio):
*Twisted Alexander polynomials of hyperbolic links and a conjecture of Dunfield, Friedl and Jackson*- In this talk we will discuss the twisted Alexander polynomial associated to the holonomy representation of a hyperbolic knot. Dunfield, Friedl and Jackson conjecture that it determines the genus and fiberedness of a hyperbolic knot. We will survey recent results on the conjecture and explain its generalization to hyperbolic links. This is a joint work with Anh Tran.

- Yasushi Nagai (Keio):
*Relations between abstract patterns and the corresponding dynamical systems*- Objects such as tilings, Delone sets, almost periodic functions and measures have proved to be interesting in connection with quasicrystals. In this talk we start from an investigation of relations between Delone sets and their corresponding dynamical systems. Next we give a general framework to discuss tilings, Delone sets, functions and measures in a systematic way and generalize the results on relations between Delone sets and their dynamical systems.

- Shintaro Nishikawa (Penn State):
*Talk: K-amenability and the Baum-Connes Conjecture for groups acting on trees*- I will describe K-amenability and the Baum-Connes Conjecture for groups acting on trees. I will mention that this work has been generalized in two directions by seeing trees either as Euclidean buildings or as CAT(0)-cubical complexes. Basic reference of this talk will be the work done by P. Julg and A. Valette in 1984.

*Poster: K-amenability and the Baum-Connes Conjecture for a-T-menable groups*- I will describe K-amenability and the Baum-Connes Conjecture for groups properly and isometrically acting on Hilbert spaces (so called a-T-menable groups).

- Jun Nonaka (Waseda U. Senior HS):
*The growth rates of ideal Coxeter polyhedra in hyperbolic 3-space*- Kellerhals and Perren conjectured that the growth rates of the reflection groups given by compact hyperbolic Coxeter polyhedra are always Perron numbers. We proved that the conjecture holds in the context of ideal Coxeter polyhedra in hyperbolic 3-space. We also obtained the bounds from below the growth rates of composite ideal Coxeter polyhedra by the growth rates of its ideal Coxeter polyhedral constituents. This talk is based on a joint work with Ruth Kellerhals.

- Genki Omori (Tokyo Tech):
*A small normal generating set for the handlebody subgroup of the Torelli group*- We consider the handlebody subgroup of the Torelli group, i.e. the intersection of the handlebody group and the Torelli group of an orientable surface. The handlebody subgroup of the Torelli group is related to integral homology 3-spheres through the Heegaard splittings. In this talk, we give a small normal generating set for the handlebody subgroup of the Torelli group.

- Nobuhiko Otoba (Regensburg):
*Truncation of the Yamabe invariant*- The Yamabe invariant, also known as the sigma constant, of a closed smooth manifold is the real number that measures its curvature properties. For example, it is positive if and only if the manifold admits a metric of positive scalar curvature. Ammann, Dahl, and Humbert showed that the Yamabe invariant, as the map from the diffeomorphism classes, factors through the bordism group up to truncation. We discuss whether it further factors through the connective real K-group. This talk is based on joint work in progress with Professors Bernd Ammann (Regensburg) and Michael Joachim (Muenster).

- Emma Previato (BU):
*Special functions in physics: A 21st-century version of Riemann's and Klein's constructions*- Elliptic functions were recently generalized to all Riemann surfaces; many problems remain open, both theoretical and applied. Implementations in: geodesic motion; statistical mechanics; non-linear wave equations; and random-matrix theory will be illustrated.

- Steve Rosenberg (BU):
*Random holonomy and algebraic structures*- I'll discuss a toy model of a 0+1 dimensional TQFT that replaces formal path integrals with rigorous Brownian motion constructions of random holonomy. In the trivial case, this reproduces the standard tensor algebra/Fock space of a vector space. For nontrivial cases, we get deformations of the tensor algebra.

- Mao Shinoda (Keio):
*Talk: Non-generic properties of optimizing measures for continuous functions*- The main purpose of the ergodic optimization is to describe invariant measures which optimize the space average of a "performance" function. A general belief is that, for sufficiently regular generic functions the uniqueness and the low-complexity of optimizing measures hold. We prove, on the other hand, there exits a dense subset of continuous functions which have uncountably many ergodic optimizing measures. The main idea of our proof is the application of the Bishop Phelps theorem to the context of optimizing measures.

*Poster: Introduction to ergodic optimization*- In this poster, I will describe the background of ergodic optimization, which aims to single out invariant probability measures optimizing the space average of a "performance" function. By introducing some dynamical systems and performance functions, I will discuss the relation between the space average of functions and behavior of dynamical systems.

- Matt Szczesny (BU):
*Hall algebras of coherent sheaves on schemes over F_1*- The Hall algebra of a category encodes the structure of extensions between objects. Classically, this construction was applied to finitary abelian categories, such as the category of representations of a quiver or coherent sheaves on a smooth projective curve over a finite field, and produced parts of classical and affine quantum groups. I will discuss the Hall algebra construction in the context of non-additive context, where it produces a variety of combinatorial Hopf algebras such as the Connes-Kreimer Hopf algebras of trees and graphs. When applied to the category of coherent sheaves on schemes over F_1, the resulting Hopf algebras hint at rich representation-theoretic structures coming from the geometry of higher-dimensional varieties.

- Airi Takeuchi (Keio):
*Poster: K-theory of locally compact spaces*- In the topological K-theory, K-groups of compact spaces are constructed from vector bundles over these spaces, and K-groups of locally compact spaces are defined via one-point compactification. In this poster, I will describe K-groups of Euclidean spaces and spheres with the periodicity of K-theory. I will also compute K-groups of real and complex projective spaces by using six-term exact sequences.

- Asahi Tsuchida (Hokkaido):
*Talk: A theory of solvability of generalized Hamiltonian systems and a study on abnormal extremals of rank two distributions*- There is a natural symplectic structure on the tangent bundle of a symplectic manifold. A generalized Hamiltonian system is a particular type of a Lagrangian submanifold of the tangent bundle and is regarded as a generalization of a first order differential equation. In this talk, we consider existence of the solutions and smoothness of the solutions of generalized Hamiltonian systems. We also consider an application of them to study abnormal curves on sub-Riemannian geometry.

*Poster: A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank*- We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorphism by using the Hamilton vector fields.

- Jackson Walters (BU):
*Vertex algebras associated to toroidal algebras*- In the literature there have been vertex algebras associated to toroidal Lie algebras, a certain N-variable generalization of affine Kac-Moody algebras. I will describe a project aimed at recovering these vertex algebras geometrically using factorization algebras.

- Weiwei Wu (Georgia):
*Dehn twists on Lagrangian spherical space forms*- Seidel's famous theorem describes a mapping cone relation induced by Lagrangian Dehn twists along spheres in Fukaya categories. Recently, this was generalized to Lagrangian submanifolds diffeomorphic to projective spaces in my earlier joint work with Cheuk-Yu Mak using Lagrangian cobordisms. In this talk, we further generalize the cone relation to Lagrangians diffeomorphic to finite quotients of spheres and projective spaces by techniques of SFT. Algebraically, these Dehn twists are all twists along certain spherical functors. This is an ongoing joint work with Cheuk-Yu Mak.

- Xiao Zheng (BU):
*SYZ Mirror Symmetry of Hypertoric Varieties*- Hypertoric varieties are hyperkahler analogs of toric varieties, obtained as hyperkahler quotients of the quaternion spaces by hamiltonian torus actions. Examples include T*P^n and the crepant resolution of A_n singularities. In this talk, I will outline a construction of SYZ mirrors of hypertoric varieties.