# Topology Seminar

This seminar is held on Tuesdays at 3pm in 210 Deady.

### Fall Quarter, 2019

- October 1, Organizational Meeting
- October 8,
**Eric Ramos** (UO)

Spanning subspace configurations and representation stability
**Abstract**: Let V_1, V_2, V_3,… be a sequence of Q-vector spaces where V_n carries an action of S_n for each n. Representation stability is one notion of when the sequence {V_n} has a limit. In this talk, we will introduce the tools of representation stability as they apply to the homology groups of the variety X_{n,k} of spanning configurations. That is, the variety of n-tuples (L_1,…,L_n) of lines in C^k that satisfy L_1 + … + L_n=C^k as vector spaces. Using representation stability, we will study stability phenomena for the homology groups H_q(X_{n,k}) as the parameter (n,k) grows. This is joint work with Brendan Pawlowski and Brendon Rhoades.

- October 15,
**Boris Botvinnik** (UO)

Spin manifolds with fibered singularities: geometry and topology
- October 22,
**Nikolai Saveliev** (U. Miami)

On the monopole Lefschetz number
**Abstract**: Given an involution on a rational homology 3-sphere Y making it into a branched cover of S^3, we prove a formula for the Lefschetz number of the map induced by this involution on the reduced monopole homology of Y. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg-Witten invariants of 4-manifolds. It has various applications in 4-dimensional topology, gauge theory, knot theory, and contact geometry, and a recent extension to higher order diffeomorphisms. This is a joint project with Jianfeng Lin and Daniel Ruberman.

- October 29,
**Clover May** (UCLA)

Decomposing C2-equivariant spectra
**Abstract**: Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood, even for G=C2, the cyclic group of order two. The structure theorem for RO(C2)-graded cohomology with Z/2 coefficients substantially simplifies computations. The algebraic structure theorem says the cohomology of any finite C2-CW complex decomposes as a direct sum of two basic pieces: cohomologies of representation spheres and cohomologies of spheres with the antipodal action. This decomposition lifts to a splitting at the spectrum level. In joint work with Dan Dugger and Christy Hazel we extend this result to a classification of compact modules over the genuine equivariant Eilenberg-MacLane spectrum HZ/2.

- November 5,
**Yoosik Kim** (Boston U.)

T-equivariant disc potentials of Fano toric manifolds
**Abstract**: In this talk, I explain how to construct an equivariant SYZ mirror using an equivariant Lagrangian Floer theory on the Morse model. In the case of (semi-) Fano toric manifolds, the mirrors recover Givental’s equivariant mirrors, which can compute the equivariant quantum cohomology. This talk is based on joint work with Siu-Cheong Lau and Xiao Zheng.

- November 5,
**Tadayuki Watanabe** (Shimane U.)

**Special Time:** 11am in 210 Deady Hall

Some exotic nontrivial elements of the rational homotopy groups of Diff(S^4)
**Abstract**: Kontsevich’s characteristic classes for framed smooth homology sphere bundles were defined by Kontsevich as a higher dimensional analogue of Chern-Simons perturbation theory in 3-dimension, developed by himself. In this talk, I will present an application of Kontsevich’s characteristic class to a disproof of the 4-dimensional Smale conjecture, which says that the group of self-diffeomorphisms of the 4-sphere has the same homotopy type as the orthogonal group O(5).

- November 12,
**Heather Lee** (U. Washington)

Global homological mirror symmetry for genus two curves
**Abstract**: Mirror symmetry is a duality between symplectic and complex geometries, and the homological mirror symmetry (HMS) conjecture was formulated by Kontsevich to capture this phenomenon by relating two triangulated categories. There is a mirror map that is a local isomorphism between the moduli space of complex structures on one manifold and the moduli space of complexified Kahler structures on the mirror manifold. In this talk, we will identify this mirror map globally in the context of HMS for the example of genus two complex curves.

In her recent thesis, Cannizzo proved HMS with the B-model being a version of the bounded derived category of coherent sheaves on a family of genus two curves. The A-model is the Fukaya-Seidel category on the Landau-Ginzburg mirror (Y, v_0), where Y is a toric Kahler manifold of real dimension 6 and v_0: Y –> C is a symplectic fibration. Cannizzo’s thesis is for a particular family of complex parameters on the genus two curve. We upgrade her result to include all complex parameters and the corresponding Kahler parameters of the mirror, and we identify the mirror map globally which will exhibit interesting wall crossing phenomena.

This is a joint project with Haniya Azam, Catherine Cannizzo, and Chiu-Chu Melissa Liu.

- November 13,
**Tom Hockenhull** (Glasgow)

**Special Time:** 12pm in 206 Deady Hall

Koszul duality and Knot Floer homology
**Abstract**: ‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of ‘(bordered) Heegaard Floer homology’ by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality which links older constructions in Heegaard Floer homology with the bordered Heegaard Floer homology of three-manifolds with torus boundary. I shan’t assume any existing knowledge of Koszul duality or any form of Heegaard Floer homology.

- December 3, Course Organization Meeting

### Winter Quarter, 2020

- January 21,
**Allen Boozer** (UCLA)

Computer Bounds for Kronheimer-Mrowka Foam Evaluation
**Abstract**: Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. One outgrowth of their approach is the definition of a functor J^flat from the category of webs and foams to the category of integer-graded vector spaces over the field of two elements. Of particular interest is the relationship between the dimension of J^flat(K) for a web K and the number of Tait colorings Tait(K) of K. I describe a computer program that strongly constrains the possibilities for the dimension and graded dimension of J^flat(K) for a given web K, in many cases determining these quantities uniquely.

- January 28,
**Jeffrey Meier** (Western Washington University)
- February 11,
** Maggie Miller** (Princeton)
- February 18,
**Allison Miller** (Rice)

### Spring Quarter, 2020

- April 28,
**Sherry Gong** (UCLA)

Topology Seminar

Previous years: 2019 2018 2017 2016 2015 2013 2012 F11 S10 W10 F09 F08 S08 W08 F07 S07 W07 F06 F05-S06 F04-S05 F03-S04 F02-S03 F00-S01 F99-S00 F98-S99 F97-S98