# Topology Seminar 2019

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

### Spring Quarter, 2019

- April 9, No seminar – Moursund Lecture (Amie Wilkinson)
- April 16,
**Juanita Pinzon Caicedo**(NC State)

Operations of Infinite Rank in Concordance**Abstract**: Oriented knots are said to be concordant if they cobound an embedded cylinder in the interval times the 3-sphere. This defines an equivalence relation under which the set of knots becomes an abelian group with the connected sum operation. The importance of this group lies in its strong connection with the study of 4-manifolds. Indeed, many questions pertaining to 4–manifolds with small topology (like the 4–sphere) can be addressed in terms of concordance. A powerful tool for studying the algebraic structure of this group comes from satellite operations or the process of tying a given knot P along another knot K to produce a third knot P(K). In the talk I will describe how to use SO(3) gauge theory to provide a general criterion sufficient for the image of a satellite operation to generate an infinite rank subgroup of the smooth concordance group. This is joint work with Matt Hedden. - April 23,
**Eric Ramos**(UO)

Categories of Graphs and Contractions**Abstract**: For a (connected, finite) graph G, we define its genus to be the quantity g := E – V + 1, where E is the number of edges of G and V is the number of vertices. While it is not the case that graph homomorphisms preserve this invariant, it is the case that contractions between graphs do. In this talk we will consider the category of all genus g graphs and contractions. More specifically, we consider integral representations of the opposite category, i.e. functors from the opposite category to Abelian groups. Using the combinatorics of graph minors, we will show that representations of this kind satisfy a Noetherian property. As applications of this technical result, we show that configuration spaces of graphs as well as Kazhdan–Lusztig polynomials of graphical matroids must satisfy certain strong finiteness conditions. This is joint work with Nick Proudfoot. - April 30,
**Brenda Johnson**(Union College)

Functor Precalculus**Abstract**: Functor calculi have been developed in a variety of forms and contexts in algebra and topology. Each of these calculi comes equipped with its own definition of polynomial or degree n functor. Such definitions are often formulated in terms of the behavior of the functor on certain types of cubical diagrams. Using the discrete calculus developed with Kristine Bauer and Randy McCarthy as a starting point, I will describe a category-theoretic framework, which we call a precalculus, that provides a means by which notions of degree for functors can be defined via cubical diagrams. I will also discuss how such precalculi can be used to produce functor calculi. This is work in progress with Kathryn Hess. - May 7, No seminar – Niven Lecture (Alison Etheridge)
- May 14,
**Philip Engel**(University of Georgia)

Penrose tilings and Hurwitz theory of leaf spaces**Abstract**: A group acting on an elliptic curve must have order N = 1, 2, 3, 4, or 6. We call the quotient an elliptic orbifold. Certain branched covers of the order N elliptic orbifold are in bijection with tiled surfaces, and form a lattice in the moduli space of N-ic differentials on Riemann surfaces. The enumerative theory of these branched covers suggests a phantom “elliptic orbifold” for all integers N. I will discuss work-in-progress with Peter Smillie proposing a definition for the Hurwitz theory of this non-existent object, and attempts to relate it to quasi-crystals in the moduli space of quintic differentials and the enumeration of Penrose-tiled Riemann surfaces. - May 28,
**Allison Moore**(UC Davis)

Heegaard Floer d-invariants and integral surgery**Abstract**: Heegaard Floer homology is an extensive package of invariants associated to a closed, oriented three-manifold equipped with a spin-c structure. One particularly useful piece of this package is the d-invariant, which is defined as the maximal grading of a non-torsion class in the Heegaard Floer module. Such d-invariants are in general difficult to compute. We will discuss how to leverage the “mapping cone” formula of Heegaard Floer homology in order to describe the d-invariants of integral surgeries along certain knots and links. We’ll also discuss some applications of the d-invariants in the context of lens space surgeries and band surgery along knots. Parts of this work are joint with E. Gorsky and B. Liu. Other parts are joint with T. Lidman and M. Vazquez.

### 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)

The structure of the homotopy groups of the motivic sphere spectrum**Abstract**: The homotopy groups of the motivic sphere spectrum are the structural constants governing the universe of (cellular) stable motivic homotopy theory. I will motivate and construct this bi-graded system of groups over a general base field, and then delve into their structural aspects, focusing on vanishing lines and behavior in the eta-periodic range, where the motivic Hopf map acts as an isomorphism. I will conclude by discussing a slice spectral sequence approach to the homotopy groups of the eta-periodic motivic sphere (joint with Oliver Röndigs) which provides complete information in case the base field has odd characteristic or cohomological dimension at most 1. This recovers a theorem of Andrews-Miller over C and suggests a “Witt-theoretic image of J pattern” in general.

### 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

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