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Seminars During Academic Year 2018-19

Fall 2018

Wednesday September 5, 2018 in RH104
Jonathan Campbell (Vanderbilt University), Fixed Point Theory and the Cyclotomic Trace

Fixed point theory has been the motivation for many of the most celebrated results of 20th century mathematics: the Lefschetz fixed point theorem, the Atiyah-Singer index theorem, and the development of etale cohomology. In this talk I'll describe work, joint with Kate Ponto, that relates classical fixed point theory to algebraic K-theory, topological Hochschild homology and the cyclotomic trace. The relationship seems to clarify both domains, and readily suggests generalizations, via machinery of Lindenstrauss-McCarthy, that relate to dynamical zeta functions. The link turns on careful considerations of the bicategorical structure of THH. Prerequisites: knowledge of the stable homotopy category, and an appetite for category theory; I will strive for the talk to be otherwise self-contained.

Wednesday September 19, 2018
G. Cazassus (IU), Equivariant Floer theory via cotangent bundles
Wednesday September 26, 2018 in RH 104
Min Hoon Kim (KIAS), A family of freely slice good boundary links.

Title: A family of freely slice good boundary links.

Abstract: The still open topological surgery conjecture for 4-manifolds is equivalent to the statement that all good boundary links are freely slice. In this talk, I will show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links. This is joint work with Jae Choon Cha and Mark Powell.

Wednesday October 10, 2018
Juanita Pinzon-Caicedo (NCSU): Satellites of infinite rank in the smooth concordance group

Many winding number zero satellite operations induce an infinite rank function on the smooth concordance group

Wednesday October 17, 2018 in RH 104
Stephan Stolz (Notre Dame), "Invertible topological field theories are SKK manifold invariants"

Topological field theories in the sense of Atiyah-Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A field theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant. Here these letters stand for schneiden=cut, kleben=glue and kontrolliert=controlled, meaning that E(M) does not change when modifying the manifold by cutting and gluing along hypersurfaces in a controlled way. The main result of this joint work with Matthias Kreck and Peter Teichner is that the map described above gives a bijection between topological field theories and SKK manifold invariants.

Wednesday October 24, 2018 in RH 104
Siddhi Krishna (BC), Taut Foliations, Positive 3-Braids, and the L-Space Conjecture

The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll discuss how to build taut foliations for manifolds obtained by surgery on positive 3-braid closures. No background in Heegaard-Floer or foliation theories will be assumed.

Wednesday October 31, 2018 in RH 104
Lisa Piccirillo (Texas), The Conway knot is not slice

Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. I’ll give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the Kirby problem list. I will also use this construction to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

Wednesday November 7, 2018
Enrico Toffoli, Rho invariants for 3-manifolds with toral boundary
Tuesday November 13, 2018 in RH 104
Junghwan Park (Georgia Tech), Rational cobordisms and integral homology

We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.

Wednesday November 14, 2018 in RH 104
Nathan Dowlin, A spectral sequence from Khovanov homology to knot Floer homology

Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.

Spring 2019

Tuesday January 29, 2019 in RH 104
Lee Kennard (Syracuse)
Wednesday February 13, 2019 in RH 104
Birgit Richter (Hamburg), "Splitting results for higher THH"

For a commutative ring (spectrum) R and a (pointed) finite simplicial set X one can build the tensor product X \otimes R. Important special cases are (topological) Hochschild homology for X=S^1 and torus homology for X=S^1 x...xS^1. In joint work with Bobkova, Hoening, Lindenstrauss, Poirier and Zakharevich we investigate how we can play off the algebraic variable R against the space variable and use this to show several splitting results.

Tuesday February 19, 2019 in RH 104
Dror Bar-Natan (University of Toronto), Computation without Representation

A major part of "quantum topology" (you don't have to know what's that) is the definition and computation of various knot invariants by carrying out computations in quantum groups (you don't have to know what are these). Traditionally these computations are carried out "in a representation", but this is very slow: one has to use tensor powers of these representations, and the dimensions of powers grow exponentially fast.

In my talk I will describe a direct-participation method for carrying out these computations without having to choose a representation and explain why in many ways the results are better and faster. The two key points we use are a technique for composing infinite-order "perturbed Gaussian" differential operators, and the little-known fact that every semi-simple Lie algebra can be approximated by solvable Lie algebras, where computations are easier.

This is joint work with Roland van der Veen and continues work by Rozansky and Overbay.

Wednesday February 20, 2019
Claudius Zibrowius (University of British Columbia), "Khovanov homology and the Fukaya category of the 3-punctured disc"

This talk will focus on a classification result for complexes over a certain quiver algebra and its consequences for Khovanov homology of 4-ended tangles. In particular, I will introduce a family of immersed curve invariants for pointed 4-ended tangles, whose intersection theory computes reduced Khovanov homology. This is joint work in progress with Artem Kotelskiy and Liam Watson, which was inspired by recent work of Matthew Hedden, Christopher Herald, Matthew Hogancamp and Paul Kirk.

Wednesday March 20, 2019 in RH 104
Maggie Miller (Princeton), Concordance of Light Bulbs

I will describe a concordance version of Gabai's 4-dimensional light bulb theorem. The light bulb theorem says that two 2-spheres $R$ and $R'$ that are homotopically embedded in a 4-manifold and mutually intersect another 2-sphere $G$ (with trivial normal bundle) in one point are isotopic, modulo a condition on 2-torsion in the fundamental group of the 4-manifold. I'll show that if we relax the condition on $R'$ so that $R'$ intersects $G$ in many points, then we may still conclude that $R$ and $R'$ are concordant (modulo the same 2-torsion obstruction). The proof is constructive, and will entail explicitly constructing a 3-manifold handle by handle in a movie of 5-dimensional space.

Wednesday March 27, 2019
Nate Bottman, The symplectic (A-infinity,2)-category

title: The symplectic (A-infinity,2)-category

abstract: I will describe ongoing work, partly joint with Katrin Wehrheim and Shachar Carmeli, which aims to implement a complete version of functoriality for the Fukaya category by constructing Symp, an (A-infinity,2)-category whose objects are symplectic manifolds. Symp associates operations to chains in 2-associahedra, which are configuration spaces of marked vertical lines in the plane. There are three aspects to this project: algebra (define notions of 2-associahedra and (A-infinity,2)-categories), analysis (regularize moduli spaces of pseudoholomorphic quilts involving the strip-shrinking degeneration), and computational (how can we make computations in Symp?). I will touch on each of these aspects, focusing on the algebraic one, and, time permitting, the computational one.

Wednesday April 3, 2019 in RH 104
Arunima Ray (MPI, Bonn), "The 4-dimensional sphere embedding theorem"

The disc embedding theorem for simply connected 4-manifolds was proved by Freedman in 1982 and forms the basis for his proofs of the h-cobordism theorem, the 4-dimensional Poincaré conjecture, 4-dimensional surgery, and the classification of simply connected 4-manifolds, all in the topological category. The disc embedding theorem for more general manifolds is proved in the book of Freedman and Quinn. However, the geometrically transverse spheres claimed in the outcome of the theorem are not constructed. We close this gap by constructing the desired transverse spheres. We also outline where transverse spheres appear in surgery and the classification of 4-manifolds and give a general 4-dimensional sphere embedding theorem. This is a joint project with Mark Powell and Peter Teichner.

Wednesday April 10, 2019
Vardan Oganesyan (Stony Brook), Monotone Lagrangian submanifolds and toric topology

Let N be the total space of a bundle over some k-dimensional torus with fibre Z, where Z is a connected sum of sphere products. It turns out that N can be embedded into C^n as a monotone Lagrangian submanifold. Also, it is possible to construct embeddings of N with different minimal Maslov number and get submanifolds which are not Lagrangian isotopic. In addition, we can show that some of our embeddings are smoothly isotopic but not isotopic through Lagrangians.

Wednesday May 1, 2019 in RH 104
Ben Knudsen (Harvard), Higher enveloping algebras and configuration spaces of manifolds

We provide Lie algebras with enveloping algebras over the little n-disks operads. These algebras enjoy a combination of good formal properties and computability, the latter afforded by a Poincare–Birkhoff–Witt type result. The main application pairs this theory with the theory of factorization homology in a study of configuration spaces, leading to a number of computations and qualitative results.

Tuesday June 4, 2019 in RH 104
Allison Moore (University of California, Davis), Obstructions to band surgery

Band surgery is a topological operation that transforms a knot into a new knot or link. When the operation respects orientations on the links involved, it is called coherent band surgery, otherwise it is called non-coherent. We will classify all band surgery operations from the trefoil knot to the T(2, n) torus knots and links, and prove some related obstructions to non-coherent band surgery. This is done by analyzing the behavior of the Heegaard Floer d-invariants under integral surgery along knots in the branched double cover of the links involved in the banding. We’ll also mention some newer results on the structure of the band-Gordian graph.

Tuesday July 16, 2019
Taehee Kim (Konkukk University, S. Korea), "Polynomial splittings of knot concordance invariants"

In this talk we will address the following question: if two nonslice knots have coprime Alexander polynomials, then are they not concordant? I will talk about the results on this question regarding various concordance invariants in both smooth and topological settings. Parts of this talk are based on joint work with Min Hoon Kim and Se-Goo Kim.

IU/PU/IUPUI Joint Topology Seminar

Fall 2018: Saturday October 13, 2018
IU/PU/IUPUI Joint Topology Seminar: Paul VanKoughnett (Purdue) and Ernie Fontes (Ohio State)