# Topology Seminar 2020

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

### Winter Quarter, 2020

- January 14,
**Joan Licata** (Australia National University)

**Special Time:**9am in 104 Deady

Foliated Open Books—or, How I Learned to Stop Worrying About How to Glue Open Books
**Abstract**: In this talk, I’ll discuss recent joint work with Vera Vertesi to develop a new version of an open book decomposition for contact three-manifolds. Our foliated open books join a zoo of existing types of open books, so I’ll focus on the senses in which ours is a natural construction which provides a user-friendly approach to cutting and gluing.

- 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)

Fibered, homotopy-ribbon knots and the Poincaré Conjecture
**Abstract**: The last unsolved remnant of the century-old Poincaré Conjecture posits that every smooth 4-manifold with the homotopy type of the 4-sphere is diffeomorphic to the 4-sphere. A homotopy 4-sphere that is built without 1-handles can be encoded as a $n$-component link with an integral Dehn surgery to

. I’ll describe a program to prove that such spheres are smoothly standard in the case that $n=2$ and one component of the link is fibered, which has been carried out in joint work with Alex Zupan in the case that the fibered knot is a generalized square knot. I’ll discuss how this relates to the problem of classifying ribbon disks for a fibered knot, and, time permitting, I’ll outline how the theory of trisections connects this work to the Andrews-Curtis Conjecture and the Generalized Property R Conjecture.

- February 11,
** Maggie Miller** (Princeton)

Light bulbs in 4-manifolds
**Abstract**: In 2017, Gabai proved the light bulb theorem, showing that if

and

are 2-spheres homotopically embedded in a 4-manifold with a common dual, then with some condition on 2-torsion in

one can conclude that

and

are smoothly isotopic. Schwartz later showed that this 2-torsion condition is necessary, and Schneiderman and Teichner then obstructed the isotopy whenever this condition fails. I showed that when

does not have a dual, we may still conclude the spheres are smoothly concordant.

I will talk about these various definitions and theorems as well as new joint work with Michael Klug generalizing the result on concordance to the situation where has an immersed dual (and may have none), which is a common condition in 4-dimensional topology.

- February 18,
**Allison Miller** (Rice)

Satellite operations and knot genera
**Abstract**: The satellite construction, which associates to a pattern knot P in a solid torus and a companion knot K in the 3-sphere the so-called satellite knot P(K), features prominently in knot theory and low-dimensional topology. Besides the intuition that P(K) is “more complicated” than either P or K, one can attempt to quantify how the complexity of a knot changes under the satellite operation. In this talk, I’ll discuss how several notions of complexity based on the minimal genus of an embedded surface whose boundary is the given knot and which satisfies various additional constraints change under satelliting. In the case of the classical Seifert genus of a knot, Schubert gives an exact formula. In the 4-dimensional context the situation is more complicated, and depends on whether we work in the smooth or topological category: the smooth category is asymptotically similar to the classical setting, but our main results show that the topological category is much weirder. This talk is based on joint work with Peter Feller and Juanita Pinzón-Caicedo.

- February 25,
**Rostislav Akhmechet** (University of Virginia)

Stable homotopy refinement of quantum annular homology
**Abstract**: We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra, and Wehrli. Using an equivariant version of the Burnside category approach of Lawson-Lipshitz-Sarkar, we associate to an annular link

a naive equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of

. This is joint work with Slava Krushkal and Michael Willis.

- March 10,
**Umut Varolgunes** (Stanford)

Seifert form of chain type invertible singularities
**Abstract**: Consider complex quasi-homogeneous polynomials with only an isolated singularity whose monomials are linked to each other in the manner of a chain:

. In this talk, I will explain how to compute the Seifert form of such a polynomial using specially chosen Morsifications. The computation relies on an explicit understanding of certain vanishing cycles as matching cycles. If time permits, I will also explain some attempts to categorify this computation.

### 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

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