Topology Seminar

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

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 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.

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.

Math Department of Mathematics