# Topology Festival

May 11–13, 2018

## Abstracts of Talks (2016)

### Combinatorial Hodge Theory of Manifolds, Geometries, and Minkowski Weights

We discuss applications of Hodge theory, a part of algebraic geometry, to problems in combinatorics. We moreover present situations in which these deep algebraic theorems themselves can be shown combinatorially and some of the many tools we developed in this context. (Joint work with June Huh and Eric Katz.)

### Phased Matroids and Matroids Over Hyperfields

A matroid is a combinatorial analog to a subspace of a vector space $F^n$, where $F$ is an arbitrary field, and an oriented matroid is a matroid with extra structure analogous to the case $F=\mathbb{R}$. This extra structure leads to surprisingly rich interaction between combinatorics, geometry, and topology, and so it is natural to look for a generalization of this corresponding to $F=\mathbb{C}$, or for other fields. In a 2012 paper Emanuele Delucchi and I laid the foundations for such a theory for $F=\mathbb{C}$, and in very recent work Matt Baker has put nearly all these types of theories into a beautiful common framework. This talk will trace the development of this line of thought and discuss some of the many new questions opened by recent work.

### Topological Combinatorics — an Introduction and Retrospective

The areas of topology and combinatorics have a long history of fruitful interactions. However, due to a number of influential results in the 1970s and 1980s, it makes sense to speak of topological combinatorics as an identifiable new area of mathematics since then. This will be a guided tour and a retrospective.

### Intersection Patterns of Finite Sets and of Convex Sets

Intersection patterns of finite sets as well as of convex sets can be studied with tools from equivariant topology. Under certain conditions one can directly relate the intersection patterns of finite sets to those of convex sets. This correspondence yields a common generalization of results from Tverberg-type theory and lower bounds for chromatic numbers of uniform intersection hypergraphs.

### Representation Stability and $S_n$-Module Structure in the Homology of the Partition Lattice

In recent years, important families of symmetric group representations have come to be better understood through the perspective of representation stability, a viewpoint introduced and developed by Thomas Church, Jordan Ellenberg, and Benson Farb, among others. A fundamental example of representation stability is the $S_n$-module structure for the $i$-th cohomology of the configuration space of $n$ distinct, labeled points in the plane, or more generally in a connected, orientable manifold. When the manifold is the plane, this $S_n$ module structure on $i$-th cohomology translates to the Whitney homology of the partition lattice, via an $S_n$-equivariant version of the Goresky-MacPherson formula, and it is also closely related to the rank-selected homology of the partition lattice. This talk will survey the combinatorial literature regarding the partition lattice and discuss what new things this can tell us about representation stability for configuration spaces, along with some other related new results regarding the partition lattice. This is joint work with Vic Reiner.

### Topology of Random Simplicial Complexes (introductory talk)

We will survey the rapidly developing field of stochastic topology, especially from a combinatorial point of view. In particular, we will discuss a number of results for the 2-dimensional random simplicial complex introduced by Linial and Meshulam. We will pay particularly attention to thresholds for various topological properties.

### A Bouquet of Spheres (research talk)

Why are so many simplicial complexes and posets studied in combinatorics homotopy equivalent to a wedge sum of spheres? We will discuss a probabilistic answer to this question, namely that for certain natural measure, this happens with high probability. This proves a conjecture from 2005. This talk is based on joint work in progress with Chris Fowler, Christopher Hoffman, and Greg Malen.

### Geometric Topology Meets Computational Complexity

Now that the geometrization conjecture has been proven, and the virtual Haken conjecture has been proven, what is left in 3-manifold topology? One remaining topic is the computational complexity of geometric topology problems. How difficult is it to distinguish the unknot? Or 3-manifolds from each other? The right approach to these questions is not just to consider quantitative complexity, i.e., how much work they take for a computer; but also qualitative complexity, whether there are efficient algorithms with one or another kind of help. I will discuss various results on this theme, such as that knottedness and unknottedness are both in NP; and I will discuss high-dimensional questions for context.

### The Triangulation Conjecture

The triangulation conjecture stated that any $n$-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology.

### Flexibility in High Dimensional Contact Geometry

A contact structure on a manifold is a sort of odd dimensional cousin of symplectic geometry, and has a number of connections to Hamiltonian dynamics, complex geometry, geometric topology, and quantum field theory. A natural question to ask is, which smooth manifolds admit contact structures? There are some simple homological obstructions. Recently, it was shown that these obstructions are the only ones, using $h$-principle type techniques. In this survey talk, we will introduce the dynamics of contact topology, and give a vague sketch of the tools going into the proof. We will then discuss a number of results connected to this, such as the classification of overtwisted contact manifolds, how to detect them, and new constructions of Liouville cobordisms.

### Obstructions to Nielsen Realization

Let $M$ be a manifold, and let $\textrm{Mod}(M)$ be its mapping class group. The Nielsen realization problem for diffeomorphisms asks, “Can $G < \textrm{Mod}(M)$ be lifted to the diffeomorphism group $\text{Diff}(M)$?” This group-theoretic question is related to the topology of fiber bundles with fiber $M$. In the case $M$ is a closed surface, the answer is “yes" for finite $G$ (by work of Kerckhoff) and “no" for $G=\textrm{Mod}(M)$ (by work of Morita). For most infinite $G < \textrm{Mod}(M)$, we have no idea. I will discuss some obstructions that can be used to show that certain groups don’t lift. Some of this work is joint with Nick Salter.

### Geometry vs. Topology: On 4-Polytopes and 3-Spheres

There have been massive efforts to understand the parameter spaces of convex polytopes — and great results such as the “$g$-Theorem” were achieved on the way. On the other hand, key questions are still open, already and in particular for the case of 4-dimensional polytopes/3-dimensional spheres. One crucial question is “fatness problem” for 4-dimensional polytopes, which is a key to the question whether we should expect the same answers for convex polytopes (which are discrete-geometric objects) and for cellular spheres (a topological model), at least asymptotically. I will argue that we should not, and present first results in this direction: The sets of $f$-vectors of 3-spheres and of 4-polytopes do not coincide. (Joint work with Philip Brinkmann)