If k is a field of characteristic zero, a theorem of Lurie and Pridham establishes an equivalence between formal moduli problems and differential graded Lie algebras over k. We generalise this equivalence in two different ways to arbitrary ground fields by using “partition Lie algebras”. These mysterious new gadgets are intimately related to the genuine equivariant topology of the partition complex, which allows us to access the operations acting on their homotopy groups (relying on earlier work of Dyer-Lashof, Priddy, Goerss, and Arone-B.). This is joint work with Mathew.

# Past Topology Seminar

In an article published in 2009, Dave Benson described, for a finite group $G$, the mod $p$ homology of the space $\Omega(BG^\wedge_p)$ --- the loop space of the $p$-completion of $BG$ --- in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we have tried to better understand Benson's result by generalizing it. We showed that when $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is its geometric realization, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a plus construction of $|\mathcal{C}|$ with respect to homology with coefficients in $R$, then $H_*(\Omega(|\mathcal{C}|^+_R);R)$ is the homology any chain complex of projective $R\mathcal{C}$-modules that satisfies certain conditions. Benson's theorem is then the special case where $\mathcal{C}$ is the category associated to a finite group $G$ and $R=F_p$, so that $p$-completion is a special case of the plus construction.

A proper simply connected one-ended metric space is call semi-stable if any two proper rays are properly homotopic. A finitely presented group is called semi-stable if the universal cover of its presentation 2-complex is semi-stable.

It is conjectured that every finitely presented group is semi-stable. We will examine the known results for the cases where the group in question is relatively hyperbolic or CAT(0).

Topological field theories give rise to a wealth of algebraic structures, extending

the E_n algebra expressing the "topological OPE of local operators". We may interpret these algebraic structures as defining (slightly noncommutative) algebraic varieties and stacks, called moduli stacks of vacua, and relations among them. I will discuss some examples of these structures coming from the geometric Langlands program and their applications. Based on joint work with Andy Neitzke and Sam Gunningham.

Answering a question of Milnor, Grigorchuk constructed in the early eighties the

first examples of groups of intermediate growth, that is, finitely generated

groups with growth strictly between polynomial and exponential.

In joint work with Laurent Bartholdi, we show that under a mild regularity assumption, any function greater than exp(n^a), where `a' is a solution of the equation

2^(3-3/x)+ 2^(2-2/x)+2^(1-1/x)=2,

is a growth function of some group. These are the first examples of groups

of intermediate growth where the asymptotic of the growth function is known.

Among applications of our results is the fact that any group of locally subexponential growth

can be embedded as a subgroup of some group of intermediate growth (some of these latter groups cannot be subgroups in Grigorchuk groups).

In a recent work with Tianyi Zheng, we provide near optimal lower bounds

for Grigorchuk torsion groups, including the first Grigorchuk group. Our argument is by a construction of random walks with non-trivial Poisson boundary, defined by

a measure with power law decay.

Recent tools make it possible to partition the space of rational Dehn

surgery slopes for a knot (or in some cases a link) in a 3-manifold into

domains over which the Heegaard Floer homology of the surgered manifolds

behaves continuously as a function of slope. I will describe some

techniques for determining the walls of discontinuity separating these

domains, along with efforts to interpret some aspects of this structure

in terms of the behaviour of co-oriented taut foliations. This talk

draws on a combination of independent work, previous joint work with

Jake Rasmussen, and work in progress with Rachel Roberts.

We present a state-sum construction of TFTs on r-spin surfaces which

uses a combinatorial model of r-spin structures. We give an example of

such a TFT which computes the Arf invariant for r even. We use the

combinatorial model and this TFT to calculate diffeomorphism classes of

r-spin surfaces with parametrized boundary.

The problem of "unbounded rank expanders" asks

whether we can endow a system of generators with a sequence of

special linear groups whose degrees tend to infinity over quotient rings

of Z such that the resulting Cayley graphs form an expander family.

Kassabov answered this question in the affirmative. Furthermore, the

completely satisfactory solution to this question was given by

Ershov and Jaikin--Zapirain (Invent. Math., 2010); they proved

Kazhdan's property (T) for elementary groups over non-commutative

rings. (T) is equivalent to the fixed point property with respect to

actions on Hilbert spaces by isometries.

We provide a new framework to "upgrade" relative fixed point

properties for small subgroups to the fixed point property for the

whole group. It is inspired by work of Shalom (ICM, 2006). Our

main criterion is stated only in terms of intrinsic group structure

(but *without* employing any form of bounded generation).

This, in particular, supplies a simpler (but not quantitative)

alternative proof of the aforementioned result of Ershov and

Jaikin--Zapirain.

If time permits, we will discuss other applications of our result.

I will discuss the quantum-field-theory origins of a classic result of Goresky-Kottwitz-MacPherson concerning the Koszul duality between the homology of G and the G-equivariant cohomology of a point. The physical narrative starts from an analysis of supersymmetric quantum mechanics with G symmetry, and leads naturally to a definition of the category of boundary conditions in two-dimensional topological gauge theory, which might be called the "G-equivariant Fukaya category of a point." This simple example illustrates a more general phenomenon (also appearing in C. Teleman's work in recent years) that pure gauge theory in d dimensions seems to control the structure of G-actions in (d-1)-dimensional QFT. This is part of joint work with C. Beem, D. Ben Zvi, M. Bullimore, and A. Neitzke.

My talk is based on joint work with Claire Debord (Univ. Auvergne).

We will explain why Lie groupoids are very naturally linked to Atiyah-Singer index theory.

In our approach -originating from ideas of Connes, various examples of Lie groupoids

- allow to generalize index problems,

- can be used to construct the index of pseudodifferential operators without using the pseudodifferential calculus,

- give rise to proofs of index theorems,

- can be used to construct the pseudodifferential calculus.