AWM at JMM 2018

Specpial Session Abstracts

Contents

Friday, January 12, 2018, 8:00 a.m. – 10:50 a.m. (33B Upper Level, San Diego Convention Center)

AMS-AWM Special Session on Women in Symplectic and Contact Geometry and Topology, I

Polyfold Lab Report

Katrin Wehrheim*, Math Department, Evans Hall #3840, Berkeley, CA 94720

I will survey various results on applications and extensions of Hofer-Wysocki-Zehnder’s polyfold theory, such as fiber products of polyfold Fredholm sections’ equivariant transversality – existence and obstructions; equivariant fundamental class; Gromov-Witten axioms; two polyfold proofs of the Arnold conjecture. These are joint with or due to Peter Albers, Ben Filippenko, Joel Fish, Wolfgang Schmaltz, and Zhengyi Zhou.

A Smith inequality for fixed point Floer cohomology

Jingyu Zhao*, Brandeis University (jzhao0105@gmail.com)

We will describe an analogue of the classical Smith inequality for cyclic group of prime order p for fixed point Floer cohomology, which compares the ranks of the fixed point Floer cohomology of a symplectomorphism to its p-th iterations. The proof uses a construction of an equivariant p-th power map. This work in progress is based on the previous work by P. Seidel in the case of p=2.

Homological mirror symmetry for the genus 2 curve in an abelian variety and its SYZ mirror. Preliminary report

Catherine Kendall Asaro Cannizzo* (cannizzo@math.berkeley.edu)

Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs X and Y such that the complex geometry on X mirrors the symplectic geometry on Y . It allows one to deduce information about Y from known properties of X. Strominger-Yau-Zaslow (1996) described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich (1994) conjectured that a complex invariant on X, the bounded derived category of coherent sheaves, should be equivalent to a symplectic invariant of Y , the Fukaya category. This is known as homological mirror symmetry. In this project, we first use the construction of SYZ mirrors for hypersurfaces in abelian varieties following Abouzaid-Auroux-Katzarkov (2015), in order to obtain the manifolds X and Y . The complex manifold comes from the genus 2 curve as a hypersurface in its Jacobian torus, and we equip the SYZ mirror manifold with a symplectic form. We then describe progress made towards an embedding of the category on the complex side into the Fukaya category.

Rabinowitz Floer homology and mirror symmetry

Sara Venkatesh* (venkatesh@math.columbia.edu)

Rabinowitz Floer homology is a useful Floer theory for studying contact hypersurfaces in symplectic manifolds. Using intuition from mirror symmetry, I will illustrate how the Rabinowitz Floer homology of a bounding hypersurface detects elements of the Fukaya category living in the hypersurface. This result can be generalized to symplectic cohomology theories for cobordisms.

Symplectic topology of singularities

Roberta Guadagni* (rguadagni@math.utexas.edu)

While the study of singularities is a very developed field in many areas from algebraic geometry to PDEs, the symplectic point of view is still very much work in progress. Symplectic forms are naturally defined only on smooth spaces, therefore even giving the right definition of symplectic manifolds with singularities isn’t straightforward. After giving a (very brief) overview of the problem, I will focus on the symplectic geometry (and topology) of algebraic singularities embedded in a Kaehler ambient space. The simplest case in the symplectic Lefschetz singularity. For isolated singularities one can look at symplectic versions of the Milnor fibre (as done by A. Keating). Very little is known about the symplectic geometry of algebraic, non isolated singularities.

Friday, January 12, 2018, 1:00 p.m. – 5:50 p.m. (33B Upper Level, San Diego Convention Center)

AMS-AWM Special Session on Women in Symplectic and Contact Geometry and Topology, II

Annular Khovanov-Lee theory, representation theory, braids, and cobordisms

J. Elisenda Grigsby* (grigsbyj@bc.edu), Anthony M. Licata and Stephan M. Wehrli

I will survey the ways in which key representation-theoretic features of the annular Khovanov-Lee theory of braid closures give information about the surfaces they bound in the 4-ball as well as their dynamics when viewed as mapping classes of the punctured disk.

Satellite ruling polynomials and representations of the Chekanov-Eliashberg algebra. Preliminary report

Caitlin Leverson*, leverson@math.gatech.edu, and Dan Rutherford

Given a pattern braid β in J1(S1), to any Legendrian knot K in ℝ3 with the standard contact structure, we can associate the Legendrian satellite knot S(K). We will discuss the relationship between augmentations of the Chekanov-Eliashberg differential graded algebra of S(K) and certain representations of the Chekanov-Eliashberg differential graded algebra of K. For certain patterns, we can then relate a specialization of the ruling polynomial of S(K, β) to these representation numbers.

Immersed Lagrangian Fillings of Legendrian Submanifolds via Generating Families

Samantha Pezzimenti* (spezziment@brynmawr.edu), Bryn Mawr, PA

Given a Legendrian submanifold, we would like to know what geometric restrictions exist for its Lagrangian fillings. When the Legendrian admits a generating family (GF), there is a natural isomorphism between the GF-cohomology groups of the Legendrian and the cohomology groups of any GF-compatible embedded Lagrangian filling. I will show that a similar isomorphism exists for immersed GF-compatible Lagrangian fillings, which imposes restrictions on the minimum number and types of double points for any such filling. I will also give some constructions of immersed GF-compatible fillings.

Deformation of Singular Fibers of Genus 2 Fibrations and Small Exotic Symplectic 4-Manifolds

Anar Akhmedov and Sumeyra Sakalli* (sakal008@umn.edu)

In 1963, Kodaira classified all singular fibers in pencils of elliptic curves, and showed that in such a pencil, each fiber is either an elliptic curve or a rational curve with a node or a cusp, or a sum of rational curves of self-intersection -2. Later Namikawa and Ueno gave geometrical classification of all singular fibers in pencils of genus two curves. In their constructions they used algebro-geometric techniques. In this talk, I will give topological descriptions of certain singularity types in the Namikawa-Ueno’s list by presenting Lefschetz pencils of genus two curves in the Hirzebruch surfaces precisely. I will also discuss 2-nodal spherical deformation of certain singular fibers of genus two fibrations. Then by using them I will provide constructions of exotic, minimal, symplectic 4-manifolds homeomorphic but not diffeomorphic to CP2#6(− CP2), CP2#7(− CP2) and 3CP2#k(− CP2) for k = 16,…,19. This is a joint work with Anar Akhmedov.

Symplectic embeddings and toric geometry. Preliminary report

Daniel Cristofaro-Gardiner and Tara S Holm* (tara.holm@cornell.edu), Department of Mathematics, Cornell University, Ithaca, NY 14850, and Alessia Mandini and Ana Rita Pires

I will discuss my ongoing work with Cristofaro-Gardiner, Mandini and Pires. We study symplectic embeddings of four- dimensional eillipsoids into toric symplectic four-manifolds. Embedded contact homology capacities provide complete information about such embeddings. The embedding capacity function admits infinite staircases in several known settings (due to McDuff – Schlenck; Frenkel – Müller; and Cristofaro-Gardiner – Kleinman). We find three more, and conjecture that these are the only infinite staircases among toric surfaces.

Mean action of area-preserving diffeomorphisms of the annulus. Preliminary report

Morgan Weiler* (morgan.w@berkeley.edu), Dept. of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA 94720-3840

Given an area-preserving diffeomorphism ψ of a closed annulus which is a rotation near the boundary, we can define an “action function” from the annulus to the reals which captures the dynamics of ψ. We study this action function via a filtration on embedded contact homology introduced by Hutchings, which is applied after realizing ψ as the Poincaré return map of a global surface of section for the Reeb flow on a contact three-manifold.

Arboreal Skeleta

Laura Starkston* (lstarkst@stanford.edu)

We will talk about how to encode the symplectic geometry of a 2n-dimensional Weinstein manifold in an n-dimensional core subcomplex called the skeleton. The skeleton generally has complicated singularities, but we will discuss how we can work with representative skeleta with only a nice combinatorial and topological class of singularities aligning with Nadler’s arboreal singularities. Then we will discuss how to compare Weinstein manifolds like cotangent bundles using arboreal skeleta.

Saturday, January 13, 2018, 8:30 a.m.-12:00 p.m. (Room 8, Upper Level, San Diego Convention Center)

AWM Special Session on Noncommutative Algebra and Representation Theory, I

Color Lie rings and PBW deformations of skew group algebras

Siân Fryer* (sianfryer@math.ucsb.edu), Tina Kanstrup, Ellen Kirkman, Anne Shepler and Sarah Witherspoon

We examine color Lie rings arising from finite groups of diagonal matrices acting linearly on finite dimensional vector spaces, and show that (under certain conditions) their enveloping algebras are quantum Drinfeld orbifold algebras, i.e. PBW deformations of certain skew group algebras. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group can be realized as the universal enveloping algebra of a color Lie ring. Special cases of these results yield more familiar objects: for example, a Lie superalgebra is simply a color Lie ring with only two colors and base ring ?. This approach lends itself well to direct computation, and many concrete examples will be given.

Preprojective algebras

Gordana G Todorov* (g.todorov@northeastern.edu), Van Nguyen (nguyen@hood.edu) and Shijie Zhu (zhu.shi@husky.neu.edu)

In this talk I will discuss several descriptions of Preprojective Algebras associated to tree-type quivers. In the paper we combine results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, and we show that for any tree-type quiver, these different descriptions of preprojective algebra are all equivalent.

Preprojective algebras of tree-type quivers, (with Van Nguyen and Shijie Zhu), arXiv: 1612.01585.

Spline models for algebra and geometry

Elizabeth Drellich* (edrelli1@swarthmore.edu), 500 College Ave, Swarthmore, PA 19081

When engineers or computer graphics people use the word ”spline” they are talking about matching (polynomial) curves to smoothly model complicated surfaces. Dual to this idea is the concept of an ”algebraic spline.” Given a graph whose edges are labeled by ideals of a given ring R, an algebraic spline is a function f from the vertices to R such that if two vertices v and w are adjacent, then f (v) − f (x) is in the ideal labeling edge vw. Splines can be used to explicitly compute compute (equivariant) cohomology rings of algebraic varieties called GKM spaces. This talk will present recent work extending the spline model to new settings, both geometric and purely algebraic.

Detecting projectivity of modules and nilpotence in cohomology for finite dimensional (graded) Hopf algebras. Preliminary report

Julia Pevtsova* (julia@math.washington.edu), julia@math.washington.edu, and David J Benson, Srikanth B Iyengar and Henning Krause

For a finite group G, classical theorems of Quillen and Chouinard tell us how to detect whether a class in mod p cohomology is nilpotent or whether a module is projective: one has to restrict to elementary abelian subgroups of G. For connected finite group schemes, the detecting family consists of one-parameter subgroups as shown by Suslin, Friedlander, and Bendel. I’ll give an overview of these results and describe some new developments in this direction for for finite supergroup schemes.

slk friezes from Grassmannians. Preliminary report

Karin Baur, Eleonore Faber, Sira Gratz, Khrystyna Serhiyenko* (khrystyna.serhiyenko@berkeley.edu) and Gordana Todorov

Conway-Coxeter frieze is a lattice of shifted rows of positive integers satisfying the diamond rule: the determinant of any 2×2 matrix formed by the neighboring entires is 1. It is know that cluster-tilted algebras of type A are in bijections with such friezes. In particular, given an Aulsander-Reiten quiver of such algebra B we can apply the specialized Caldero- Chapoton map to every indecomposable B-module and obtain a frieze.
Morier-Genoud et al. studied generalized friezes called slk friezes, which are lattices of positive integers where the determinant of any k × k matrix is 1. In a similar manner, we investigate how slk friezes can be obtained from cluster categories C associated to the Grassmannian Gr(k, n). In particular, we determine a finite collection of objects in C that can be arranged to produce a frieze. If Gr(k, n) is of finite type we can view the corresponding frieze inside the Auslander-Reiten quiver of C such that 2 × 2 diamonds in a frieze arises from triangles.

Deformations of semisimple Hopf algebras

Adriana Meija Castano (sighana25@gmail.com), Florianopolis, Brazil, Susan Montgomery (smontgom@usc.edu), Los Angeles, CA, Sonia Natale (natale@famaf.unc.edu.ar), Cordoba, Argentina, Maria Vega (maria.vega@usma.edu), West Point, NY , and Chelsea Walton* (notlaw@temple.edu), Philadelphia, PA 19122

I will discuss the bi-Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension p3 and of dimension pq2 . This is joint work with Adriana Mejia Castano, Susan Montgomery, Sonia Natale, and Maria Vega. (Our project began at the BIRS WINART workshop in March 2016.)

Friday, January 12, 2018, 2:00 p.m. − 4:50 p.m. (Room 8, Upper Level, San Diego Convention Center)

AMS-AWM Special Session on Noncommutative Algebra and Representation Theory, II

A Weyl character type formula for cluster variables in type A

Vyjayanthi Chari* (chari@math.ucr.edu) and Matheus Brito

A prime representation of a Hopf algebra is defined to be one which cannot be written as a tensor product of two non–trivial representations. In 2009, Hernandez and Leclerc showed that a family of representations of the quantized enveloping algebra of an affine Lie algebra were precisely the cluster variables in a cluster algebra of type A.
In recent work, we have proved that these prime representations admit a BGG type resolution and deduce a Weyl character formula for these representations. In this talk we shall discuss the corresponding closed formula for a cluster variable in terms of the initial generators x1, · · · , xn, x′1, · · · , x′n of the cluster algebra.

Gerstenhaber structure of a class of special biserial algebras

Joanna Meinel and Van C. Nguyen* (nguyen@hood.edu), Department of Mathematics, Hood College, Frederick, MD 21701, and Bregje Pauwels, Maria Julia Redondo and Andrea Solotar

For any integer N ≥ 1, we consider a class of self-injective special biserial algebras AN given by quiver and relations over a field k. We study the Gerstenhaber structure of its Hochschild cohomology ring HH*(AN). This Hochschild cohomology ring is a finitely generated k-algebra, due to the results by Snashall and Taillefer. We employ their cohomology computations and Suárez-Álvarez’s approach to compute all Gerstenhaber brackets of HH*(AN). Furthermore, we study the Lie algebra structure of the degree-1 cohomology HH1(AN) as embedded into a direct sum of Virasoro algebras and provide a decomposition of HHn(AN) as a module over HH1(AN). This joint project was started at the WINART workshop at BIRS in April 2016.

On classification of modular categories by dimension

Julia Plavnik* (julia@math.tamu.edu), Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77840

The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories have also applications in different areas of mathematics like topological quantum field theory, representation theory, among others.
A complete classification of modular categories seems to be out of reach at the moment. Therefore a lot of efforts are done in advance in the classification of these categories under certain restrictions. Different directions have been considered: classification by rank, and by dimension, by dimension of the simple objects, among others.
In this talk, we will focus on the classification of modular categories by dimension. We will start by introducing some of the basic definitions and properties of these categories. We will also present different examples to understand better the structure and the notion of dimension in this setting.
The idea of the talk is to give a panorama of the current situation of the classification program for modular categories based on their dimension.

Explicit constructions of Segal-Sugawara vectors

Natasha Rozhkovskaya* (rozhkovs@math.ksu.edu)

The center of an affine vertex algebra at the critical level of a simple finite-dimensional Lie algebra is a commutative algebra. In this talk we will report on the explicit formulas for generators, the Segal-Sugawara vectors, in the case of exceptional Lie algebra of type G2 ( which is a joint work with A. I. Molev and E. Ragoucy).

Computing weight multiplicities

Pamela E. Harris* (peh2@williams.edu), 33 Stetson Ct, Bascom House Room 106C, Williamstown, MA 01267

Central to the study of the representation theory of Lie algebras is the computation of weight multiplicities, which are the dimensions of vector subspaces called weight spaces. The multiplicity of a weight can be computed using a well-known formula of Kostant that consists of an alternating sum over a finite group and involves a partition function. In this talk, we present some recent results related to questions regarding the number of terms contributing nontrivially to Kostant’s weight multiplicity formula along with some formulas to compute q-weight multiplicities for certain finite-dimensional Lie algebras. The work presented is in collaboration with a group of undergraduate research students at Williams College: Kevin Chang, Edward Lauber, Haley Lescinsky, Grace Mabie, Gabriel Ngwe, Cielo Perez, Aesha Siddiqui, and Anthony Simpson.

An elliptic Schur-Weyl construction of the rectangular representation of the DAHA

David Jordan and Monica Vazirani*, Department of Mathematics, One Shields Ave, Davis, CA 95616

Building on the work of Calaque-Enriquez-Etingof, Lyubashenko-Majid, and Arakawa-Suzuki, Jordan constructed a functor from quantum D-modules on general linear groups to representations of the double affine Hecke algebra (DAHA) in type A. When we input quantum functions on GL(N) the output is L(kN), the irreducible DAHA representation indexed by an N × k rectangle. For the specified parameters L(kN) is Y-semisimple, i.e. one can diagonalize the Dunkl operators. We give an explicit combinatorial description of this module via its Y-weight basis.
This is joint work with David Jordan.