# Topology Seminar

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

### Winter Quarter, 2019

- February 19,
**Jens Kjaer** (Notre Dame)

Unstable v_1-periodic Homotopy Groups through Goodwillie Calculus
**Abstract**: It is a classical result that the rational homotopy groups,

, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on

. The closest we can get to a similar statement for general homotopy groups is the Goodwillie spectral sequence, which computes the homotopy group of a space from its “spectral Lie algebra”. Unfortunately both input and differentials are hard to get at. We therefore simplify the homotopy groups by taking the unstable

-periodic homotopy groups,

(note

recovers rational homotopy groups). For h=1 we are able to compute the K-theory based

-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute

in a very different way from the original computation by Davis.

- February 26,
**Dan Margalit** (Georgia Tech)

Congruence subgroups of braid groups
**Abstract**: Using the Burau representation, we obtain certain subgroups of the braid groups called the level m congruence subgroups. The level 2 subgroup is the pure braid group. In this talk we will investigate the level 4 subgroup. To begin, we will give an infinite generating set. We will also consider the first cohomology group of the level 4 braid group and show that – just like the pure braid group – it exhibits a form of symmetry called representation stability. This talk represents joint work with Tara Brendle and Kevin Kordek.

- March 5,
**Craig Westerland** (University of Minnesota)

Quantum shuffle algebras and homology of braid groups
**Abstract**: The homology of braid groups has been known (by the work of Arnol’d and Cohen) for a long time. If we enlarge this problem by considering homology with coefficients in a nontrivial representation, there are numerous results establishing homological stability for various classes of coefficients, usually polynomial in some sense. In this talk, we will describe a method of computing the homology of braid groups with coefficients in exponential coefficient systems. Surprisingly (at least to me), the answer may be formulated in terms of the homological algebra of “quantum shuffle algebras” which have a deep connection to the theory of quantum groups.

This is joint work with Jordan Ellenberg and TriThang Tran. The results of these computations were used to establish the upper bound in the function field version of Malle’s conjecture on the distribution of Galois groups (which will be the subject of the colloquium on Monday 4 March).

- March 12,
**Kyle Ormsby** (Reed)

### Spring Quarter, 2019

### Fall Quarter, 2018

- October 2,
**Dev Sinha** (UO)

The mod-two cohomology of DX and QX
**Abstract**: The spaces of maps from an n-sphere to the nth suspension of a space form a directed system, whose limit is called

X or QX. This space is rightly viewed both as a basic mapping space to study and the underlying space for the “free derived abelian group on X.” In this talk I give a recent Hopf-ring based approach to the mod-two cohomology ring of QX, which was originally calculated by Dung. Our approach is through the finite divided powers of X, whose cohomology rings were previously only implicitly known. We present geometric representatives for the cohomology of DX which if it could be extended to QX (for X =

) would give such for the stable cohomology of mapping class groups.

- October 9,
**Eric Ramos** (UO)

Commutative Algebra in the Configuration spaces of Graphs
**Abstract**: Let G be a graph, thought of as a 1-dimensional simplicial complex. Then the n-stranded configuration space of G is the space F_n(G) = \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j\} / S_n, where S_n is the symmetric group on n letters. While one would hope to say something meaningful about the homology groups H_i(F_n(G)) of these spaces, it is known that they can be quite chaotic. Following recent trends in algebra, we therefore shift our focus to studying all of these groups simultaneously +_n H_i(F_n(G)), sacrificing knowledge about individual homology groups for statements which are more asymptotic in nature. In particular, it can be shown that +_n_i(F_n(G)) can be encoded as the additive group of some finitely generated graded module over an integral polynomial ring. Specializing to the case of trees, we compute the generating degree of this module, and show that it naturally decomposes as a direct sum of graded shifts of square-free monomial ideals. As an application we show that the homology groups of F_n(G), in the case of trees, are only dependent on the degree sequence of G. We conclude the talk by discussing what little is known in the general case, and provide a theorem describing aspects of the Hilbert polynomials of these modules.

- October 16,
**Clayton Shonkwiler** (Colorado State)

The Geometry of Topologically Constrained Random Walks
**Abstract**: Random walks in 3D are a standard simple model of polymers like proteins or DNA in solution. Despite being physically quite unrealistic (they have no thickness, no stiffness, and are not prevented from self-intersecting), such walks surprisingly exhibit the same scaling behavior as linear polymers forming open chains. From a geometric perspective, a random walk is just a choice of n independent random directions on the sphere of possible directions, which makes simulation simple and computations tractable.

However, modeling polymers with nontrivial topology presents additional challenges: the edges of the walk are no longer independent, so the joint distribution of edges is not a product distribution. Geometrically, this means the space of possible conformations is a high-dimensional manifold (or singular space) which is not the product of low-dimensional spaces. In this talk, I will describe how the differential and symplectic geometry of these spaces can be exploited to give sampling algorithms, to compute expectations of geometric quantities like radius of gyration, and to address questions like: what is the probability that a loop random walk is knotted?

- October 23,
**Haofei Fan** (UCLA)

Unoriented cobordism maps on link Floer homology
**Abstract**: We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a disoriented link cobordism. Furthermore, we will discuss some potential applications on the involutive upsilon invariants and unoriented four-ball genus.

- November 6,
**Marco Golla** (University of Nantes)

**Special Time:** 9am in 210 Deady.

Smooth concordance bounds on wrapping numbers
**Abstract**: Suppose that K is a knot in the solid torus T. A natural invariant of K is the wrapping number of K, i.e. the minimal number intersections of K with a meridional disc of T, up to isotopy. I will talk about an analogue 4-dimensional question for knots up to concordance (i.e., essentially, up to an annulus in T x [0,1]). The main tool will come from (twisted) correction terms in Heegaard Floer homology. This is joint work with Daniele Celoria.

- November 13,
**Charles Katerba** (Montana State)

Searching for closed essential surfaces in knot complements with character varieties
**Abstract**: Culler-Shalen theory uses a 3-manifold’s (P)SL(2,C) character variety to construct essential surfaces in the manifold. It has been a fundamental tool over the last 35 years in low-dimensional topology. Much of its success is due to a solid understanding of the essential surfaces with boundary that can be constructed with the theory. It turns out, however, that not every surface with boundary is detected. Moreover, one can also construct closed essential surfaces within this framework. In this talk, we will discuss a module-theoretic perspective on Culler-Shalen theory and apply this perspective to show that there are knot complements in

which contain closed essential surfaces, none of which are detected by Culler-Shalen theory. As a corollary, we will construct an infinite family of closed hyperbolic Haken 3-manifolds whose representations into PSL(2, C) have a special number-theoretic property.

- November 20,
**Claudius Zibrowius** (U. British Columbia)

Khovanov homology and the Fukaya category of the 4-punctured sphere
**Abstract**: About 15 years ago, Bar-Natan showed how to associate with a tangle a chain complex over a certain finite category such that the homotopy type of this chain complex is a tangle invariant generalising Khovanov homology. In this talk, I will describe two geometric interpretations of this chain complex for 4-ended tangles in terms of immersed curves on the tangle boundaries. The main algebraic tool for studying these immersed curves will be the category of peculiar modules, which I originally introduced in my PhD thesis in the context of a Heegaard Floer theory for 4-ended tangles. As an application of a general pairing theorem, I will describe a geometric interpretation of the kappa invariant of sutured tangles originally defined by Liam Watson.

This talk is about joint work in progress together with Liam Watson and Artem Kotelskiy, which was inspired by recent work of Hedden, Herald, Hogancamp and Kirk.

- November 27, Course organization meeting

Topology Seminar

Previous years: 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