UCLA Topology Seminar
2015/16

Unless otherwise noted, the seminar meets on Wednesdays and/or Fridays, 4-5pm in MS 5148. Please contact Mike Hill, Ko Honda or Ciprian Manolescu for more information.

Past seminars: 2014/15.


Winter 2016

Click on titles for abstracts.

Date Speaker Affiliation Title
Fri January 08 Ian Zemke UCLA Graph cobordism and basepoint moving maps in Heegaard Floer homology
Fri January 15 Kristen Hendricks UCLA A flexible construction of equivariant Floer cohomology
Fri January 29 Mike Menke UCLA An introduction to the four-color theorem
Fri February 05 Sangjin Lee UCLA Sutured manifolds and decompositions
Fri February 12 Jacob Rooney UCLA Instanton homology
Fri February 19 Haofei Fan UCLA Sutured instanton homology
Fri February 26 Mike Miller UCLA Khovanov homology is an unknot-detector, after Kronheimer-Mrowka
Fri March 04 Matt Stoffregen UCLA The Kronheimer-Mrowka approach to the four-color theorem: I
Fri March 11 Jianfeng Lin UCLA The Kronheimer-Mrowka approach to the four-color theorem: II

Fall 2015:

Date Speaker Affiliation Title
Wed September 30 Mike Miller UCLA Brown Representability
Wed October 7 Haofei Fan UCLA Spectra and examples following Adams
Wed October 14 Mike Menke UCLA Spectral sequences in stable homotopy
Fri October 16 Liam Watson Univ. Glasgow Taut foliations on graph manifolds
Mon October 19, 4:30-5:30pm, LATop (at USC) Jeff Danciger UT Austin Convex projective structures on non-hyperbolic three-manifolds
Mon October 19, 5:30-6:30pm, LATop (at USC) Faramarz Vafaee Caltech L-spaces and rationally fibered knots
Wed October 21 Jacob Rooney UCLA Atiyah-Segal Completion Theorem
Fri October 23 (3-4pm) Anastasiia Tsvietkova UC Davis Hyperbolic structures from link diagrams
Fri October 23 (4-5pm) Danielle O'Donnol Indiana University Spatial graph Floer Homology
Wed October 28 Kevin Carlson UCLA Adams' "Prerequisites for Carlsson's...": equivariant stabilization and the transfer
Thu October 29, 4:50-5:40pm, Colloquium Denis Auroux UC Berkeley Lagrangian tori, wall-crossing and mirror symmetry
Wed November 4 (3-4pm)
Igor Belegradek
Georgia Tech
Spaces of nonnegatively curved metrics
Wed November 4 (4-5pm) Matt Stoffregen
UCLA
Carlsson's Proof: statements and outline
Mon November 16, 4-5pm, LATop (at Caltech) Mike Hill UCLA A higher-height lift of Rohlin's Theorem: on \eta^3
Mon November 16, 5-6pm, LATop (at Caltech) Joshua Greene Boston College Definite surfaces and alternating links
Wed November 18 Diana Hubbard Boston College An annular refinement of the transverse element in Khovanov homology
Fri November 20 Akram Alishahi Columbia Heegaard Floer homology for tangles and cobordisms between them
Mon November 30, 4-5pm, LATop (at UCLA, MS 6229) Ailsa Keating Columbia Higher-dimensional Dehn twists and symplectic mapping class groups
Mon November 30, 5-6pm, LATop (at UCLA, MS 6229) Hiro Lee Tanaka Harvard Factorization homology and topological field theories
Wed December 2 (3-4pm) Ben Cooper University of Iowa Formal contact categories
Wed December 2 (4-5pm) Jianfeng Lin UCLA Deeper study of Carlsson's proof



Abstracts:

Watson: An L-space is a rational homology sphere with simplest possible Heegaard Floer homology. Ozsvath and Szabo have shown that if a closed, connected, orientable three-manifold has a coorientable taut foliation then it is not an L-space. I will explain how to prove the converse to this statement when restricting to graph manifolds. Combined with work of Boyer and Clay, this leads to an equivalence between graph manifold L-spaces and graph manifolds with non-left-orderable fundamental group. This is joint work with J. Hanselman, J. Rasmussen, and S. Rasmussen.

Tsvietkova: Thurston suggested a method for computing the hyperbolic structure of a 3-manifold that involves triangulating the manifold, and the method was later inmplemented in the program SnapPea by Jeff Weeks. The talk will briefly introduce an alternative method for computing the structure of a hyperbolic link, based on ideal polygons bounding the regions of a diagram rather than a trianguation. We will discuss the ongoing program that uses a blend of the alternative method with various techniques in order to relate diagrammatic properties of hyperbolic links to their intrinsic geometry.

O'Donnol: A spatial graph is an embedding, $f$, of a graph $G$ into $S^3$. For each transverse disk spatial graph, $f(G),$ we define a combinatorial invariant $HFG^-(f(G))$ which is a bi-graded module over a polynomial ring. The gradings live in $\mathbb{Z}$ and $H_1(S^3\smallsetminus f(G))$. This invariant is a generalization of combinatorial link Floer homology defined by Manolescu, Ozsvath, Sarkar (MOS) for links in $S^3$. To do this, we have generalized grid diagrams and grid moves. Following MOS, our invariant is the homology of a chain complex that counts certain rectangles in the grid. Although the chain complex depends on the choice of grid, the homology depends only on the embedding. Unlike many homology theories, our theory is not the categorification of an existing polynomial invariant. Thus taking the generalized Euler characteristic gives another new invariant, an Alexander polynomial for balanced spatial graphs. This is joint work with Shelly Harvey (Rice University).

Belegradek: I will explain how to determine homeomorphism type of the space of complete nonnegatively curved metrics on the plane and the 2-sphere in the smooth and in the Holder topologies.

Hubbard: In 2006, Plamenevskaya proved that the homology class of a certain distinguished element in Khovanov homology is an invariant of transverse links. In this talk I will define an annular refinement of this element, kappa, and I will show that while kappa is not an invariant of transverse links, it is a conjugacy class invariant of braids. We will see applications of kappa relating to transverse links, braid destabilization, and the word problem in the braid group. This work is joint with Adam Saltz.

Alishahi: Heegaard Floer homology was generalized for non-closed 3-manifolds with certain boundary decoration called sutured manifolds, by Juhasz, Eftekhary and I. Sutured manifolds can be described as a generalization of oriented tangles. We use this description to define a notion of cobordism between sutured manifolds. Associated with cobordisms between sutured manifolds we will describe the definition of an invariant homomorphism between Heegaard Floer homologies of the corresponding sutured manifolds. These maps generalize cobordism maps associated to 4-dimensional cobordisms between closed 3-manifolds and define cobordism maps for decorated cobordisms between pointed knots. This is a joint work with Eaman Eftekhary.

Cooper: To each oriented surface S we associate a differential graded category. The homotopy category is a triangulated category which satisfies properties akin to those of the contact categories studied by K. Honda. These categories are related to the algebraic contact categories of Y. Tian and the bordered sutured categories of R. Zarev.

Hendricks: In the past few years, equivariant Floer cohomology has been used to construct many spectral sequences between Floer-type invariants of three-manifolds and knots. We will give an alternative formulation of equivariant Lagrangian Floer cohomology, which can be used to show several of these spectral sequences are invariants of their topological input data and/or explicitly computable, and can also be applied to define new equivariant versions of several other Floer-type invariants. As an application, we construct a new knot concordance invariant which appears to be distinct from other concordance invariants from Floer theory. This is joint work with R. Lipshitz and S. Sarkar.