# Algebra Seminar

## Past sessions

### 19/04/2018, 11:30 — 12:30 — Room P3.10, Mathematics Building

Julien Ducoulombier, *ETH Zurich*

### Swiss Cheese operad and applications to embedding spaces

During this talk, I would like to give an overview of the (relative) delooping theorems as well as applications to spaces of long embeddings. In particular, we show that the space of long embeddings and the space of ($k$)-immersions from $\mathbb{R}^d$ to $\mathbb{R}^m$ are weakly equivalent to an explicit ($d+1$)-iterated loop space and an explicit ($d+1$)-iterated relative loop space, respectively. Both of them can be expressed in term of derived mapping spaces of coloured operads. Such a pair is a typical example of Swiss-Cheese algebra.

### 26/10/2017, 14:30 — 15:30 — Room P4.35, Mathematics Building

Geoffroy Horel, *Université Paris XIII*

### Mixed Hodge structure, Galois actions and formality

Given a dg-algebra or any algebraic structure in chain complexes, one may ask if it is quasi-isomorphic to its homology equipped with the zero differential. This property is called formality and has important consequences in algebraic topology. For example it forces the collapse of certain spectral sequences. In this talk I will explain how mixed Hodge structures can be used to prove formality when working with rational coefficients. I will also explain work in progress using Galois actions as a replacement for mixed Hodge structures in the case of torsion coefficients. This is joint work with Joana Cirici.

### 03/04/2017, 14:30 — 15:30 — Room P3.10, Mathematics Building

Ricardo Campos, *ETH Zurich*

### Configuration spaces of points and their homotopy type

Given a manifold $M$, one can study the configuration space of $n$ points on the manifold, which is the subspace of $M^n$ in which two points cannot be in the same position. The study of these spaces from a homotopical perspective is of interest in very distinct areas such as algebraic topology or quantum field theory. However, even if we started with a simple manifold $M$, despite the apparent simplicity such configuration spaces are remarkably complicated; even the homology of these spaces is reasonably unknown, let alone their (rational/real) homotopy type.

In this talk, I will give an introduction to the problem of understanding configuration spaces and present a combinatorial/algebraic model of these spaces using graph complexes. I will explain how these models allow us to answer fundamental questions about the dependence on the homotopy type of $M$. I will explain how these models give us new tools to address other problems such as understanding embedding spaces or computing factorization homology.

This is joint work with Thomas Willwacher based on arXiv:1604.02043.

### 20/01/2017, 16:00 — 17:00 — Room P3.10, Mathematics Building

Ieke Moerdijk, *University of Utrecht*

### Shuffles and Trees

The notion of "shuffle" of linear orders plays a central role in elementary topology. Motivated by tensor products of operads and of dendroidal sets, I will present a generalization to shuffles of trees. This combinatorial operation of shuffling trees can be understood by itself, and enjoys some intriguing properties. It raises several questions of a completely elementary nature which seem hard to answer.

### 12/12/2016, 15:00 — 16:00 — Room P4.35, Mathematics Building

Nasir Sohail, *Wilfrid Laurier University*

### Epimorphisms and amalgamation for ordered monoids

I shall introduce the amalgamation problem for partially ordered monoids (briefly pomonoids), and discuss its connection with dominions and epimorphisms. Examples will be given to show that amalgamation of monoids is subjected to severe restrictions if a (compatible) partial order is introduced on top of the binary operation. I shall then show that the introduction of order, however, does not affect the special amalgamation and (hence) epimorphisms in the sense that an epi in the 2-category of pomonoids is necessarily an epi in the (underlying) category of monoids.

### 30/09/2016, 10:30 — 11:30 — Room P4.35, Mathematics Building

Simon Henry, *Collège de France*

### On the homotopy hypothesis and new algebraic model for higher groupoids and homotopy types.

In his unpublished manuscript "Pursuing stacks" Grothendieck gave a definition of infinity groupoids and conjectured that the homotopy category of his infinity groupoids is equivalent to the homotopy category of spaces.

This conjecture (called the homotopy hypothesis) is still an open problem, and in fact there is a lot of expected basic results concerning his definition of infinity groupoids that are open problems. For these reasons, one prefers nowadays to use less problematic definitions of infinity groupoids, typically involving simplicial sets, as a starting point for higher category theory.

But Grothendieck's definition also has a lot of good properties not shared by the simplicial approaches: it is considerably closer to the intuitive notion of infinity category, it has a more general universal property, it is considerably simpler to extend to infinity categories, etc. And most recently realized, it can be defined within the framework of the Homotopy type theory program, while the definition of simplicial objects in this framework is considered to be one of the most important open problems of this program.

In this talk I will discuss a new sort of definitions of infinity groupoids that are inspired from Grothendieck's definition but that do not share any of its problems while retaining most of its advantages. We will also state a precise and simple looking technical conjecture which implies that Grothendieck definition is a special case of our framework, and hence also implies Grothendieck's homotopy hypothesis and most of the conjectures related to Grothendieck's definition.

### 05/07/2016, 15:30 — 16:30 — Room P3.10, Mathematics Building

Daniel Berwick-Evans, *University of Illinois at Urbana-Champaign*

### Elliptic cohomology, loop group representations, and 2-dimensional field theories

Elliptic cohomology, loop group representations, and 2-dimensional field theories have been linked since birth, though the precise nature of the relationship remains quite mysterious. I'll talk about some recent progress, wherein physics-inspired techniques over moduli spaces of (super) tori furnish analytic constructions of Euler classes in elliptic cohomology over the complex numbers. These classes have equivariant refinements (also constructed via field theory techniques) that can be identified with characters of positive energy representations of loop groups. This is joint work with Arnav Tripathy.

### 03/03/2016, 14:30 — 15:30 — Room P4.35, Mathematics Building

Geoffroy Horel, *MPI, Bonn*

### The operad of little disks, differential topology and Galois theory

The operad of little $n$-disks is a fundamental object in algebraic topology that was introduced as a way of recognizing $n$-fold loop spaces. I will recall its definition and then survey some recent work of Dwyer–Hess and Boavida–Weiss relating mapping spaces between the operads of little disks and spaces of knots and higher dimensional knotted objects. I will then describe a faithful action of the absolute Galois group of $\mathbb{Q}$ on the profinite completion of the operad of little $2$-disks.

### 21/01/2016, 15:00 — 16:00 — Room P3.10, Mathematics Building

Marcel de Jeu, *Leiden University*

### Positive representations

Many spaces in analysis are ordered (real) Banach spaces, or even Banach lattices, with groups acting as positive operators on them. One can even argue that such positive representations of groups are not less natural than unitary representations in Hilbert spaces, but contrary to the latter they have hardly been studied. The same holds for representations of ordered Banach algebras such that a positive element acts as a positive operator. Whereas there is an elaborate theory of $^\ast$-representations of $C^*$-algebras, hardly anything is known about positive representations of ordered Banach algebras, even though such representations are not rare at all.

We will sketch the gradually emerging field of “positive representations”, and mention some of the main problems (of which there are many) and results (of which there are still too few), jointly obtained with Ben de Pagter, Björn de Rijk, Sjoerd Dirksen, Xingni Jiang, Miek Messerschmidt, Dusan Radicanin, Mark Roelands, Jan Rozendaal, Frejanne Ruoff, and Marten Wortel.

The talk is meant as an advertisement for the topic and, more generally, for studying groups and Banach (lattice) algebras of operators on Banach lattices. The step from single operator theory on Hilbert spaces to groups and algebras of operators was taken in the first half of the 20th century, and now the field of Positivity could be ripe for a similar development.

Note the room change!

### 10/12/2015, 16:30 — 17:30 — Room P3.10, Mathematics Building

Edgar Costa, *Dartmouth College*

### Equidistributions in arithmetic geometry

Consider an algebraic variety defined by system of polynomial equations with integer coefficients. For each prime number $p$, we may reduce the system modulo $p$ to obtain an algebraic variety defined over the field of $p$ elements.

A standard problem in arithmetic geometry is to understand how the geometry of one of these varieties influences the geometry of the other.

One can take a statistical approach to this problem.

We will illustrate this with several examples, including: polynomials in one variable, algebraic curves and surfaces.

### 16/12/2014, 11:00 — 12:00 — Room P3.10, Mathematics Building

Iva Halacheva, *University of Toronto*

### Shift of Argument Algebras and the Cactus Group

For any semisimple Lie algebra $g$, there is a family of maximal commutative subalgebras of $U(g)$, the shift of argument algebras, parametrized by regular semisimple elements. They have simple spectrum, and the fundamental group of their moduli space is the pure cactus group. In type A, the resulting monodromy action agrees with the action of the pure cactus group on crystals defined using Schutzenberger involutions. We conjecture that this is also true in general. Skew-howe duality relates this result to an analogous one for the Gaudin model of commutative subalgebras in the $n$-th tensor power of $U(g)$.

### 17/07/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building

Anthony Blanc, *Université de Montpellier 2*

### Topological K-theory of complex non-commutative Spaces

It was known for some time by Bondal and Toën that an appropriate
notion of topological K-theory of dg-categories will furnish a
candidate for a rational structure on the periodic cyclic homology
of a smooth and proper dg-category. The main motivation comes from
the conjecture by Katzarkov-Kontsevich-Pantev that there exists a
pure non-commutative Hodge structure on the periodic homology of a
smooth and proper dg-algebra. I will present a meaningful
definition of topological K-theory of dg-categories over the
complex, using the topological Betti realization functor. This
definition is based on non-trivial results involving a
generalization of Deligne's proper cohomological descent. Finally I
will talk about the case of finite dimensional algebras.

### 17/07/2013, 15:00 — 16:00 — Room P3.10, Mathematics Building

Peter Trapa, *University of Utah*

### Unitary representations of reductive Lie groups

Unitary representations of Lie groups appear in many parts of
mathematics: in harmonic analysis (as generalizations of the sines
and cosines appearing in classical Fourier analysis); in number
theory (as spaces of modular and automorphic forms); in quantum
mechanics (as "quantizations" of classical mechanical systems); and
in many other places. They have been the subject of intense study
for decades, but their classification has only recently recently
emerged. Perhaps surprisingly, the classification has inspired
connections with interesting geometric objects (equivariant mixed
Hodge modules on flag varieties). These connections have made it
possible to extend the classification scheme to other related
settings. The purpose of this talk is to explain a little bit about
the history and motivation behind the study of unitary
representations and offer a few hints about the algebraic and
geometric ideas which enter into their study. This is based on a
recent preprint with Adams, van Leeuwen, and Vogan.

### 07/03/2013, 02:30 — 03:30 — Room P3.10, Mathematics Building

Bob Oliver, *Université Paris XIII*

###
Local equivalences between finite Lie groups

Fix a prime $p$. Two finite groups $G$ and $H$ will be called
$p$-locally equivalent if there is an isomorphism from a Sylow
$p$-subgroup $S$ of $G$ to a Sylow $p$-subgroup $T$ of $H$ which
preserves all conjugacy relations between elements and subgroups of
$S$ and $T$.

Martino and Priddy proved that if the $p$-completed classifying
spaces ${\mathrm{BG}}_{p}$ and ${\mathrm{BH}}_{p}$ are homotopy equivalent, then $G$ and $H$
are $p$-locally equivalent. They also conjectured the converse, a
result which has since been proven, but only by using the
classification theorem of finite simple groups.

Anyone who works much with finite groups of Lie type (such as
linear, symplectic, or orthogonal groups over finite fields)
notices that there are many cases of $p$-local equivalences between
them. For example, if $q$ and $q\prime $ are two prime powers such that
${q}^{2}-1$ and $(q\prime {)}^{2}-1$ have the same 2-adic valuation, then
${\mathrm{SL}}_{2}(q)$ and ${\mathrm{SL}}_{2}(q\prime )$ are 2-locally equivalent.

In joint work with Carles Broto and Jesper Møller, we proved,
among other results, the following very general theorem about such
$p$-local equivalences between finite Lie groups.

**Theorem**: Fix a prime $p$, a connected, reductive
group scheme $G$ over $Z$, and a pair of prime powers $q$ and $q\prime $
both prime to $p$. Then $G(q)$ and $G(q\prime )$ are $p$-locally
equivalent if $\stackrel{\u203e}{\u27e8q\u27e9}=\stackrel{\u203e}{\u27e8q\prime \u27e9}$ as closed subgroups of ${Z}_{p}^{\times}$.

Our proof of this theorem is topological: we show that the
$p$-completed classifying spaces have the same homotopy type, and
then apply the theorem of Martino and Priddy mentioned above. The
starting point is a theorem of Friedlander, which describes the
space $\mathrm{BG}(q{)}_{p}$ as a “homotopy fixed space” of a some self map
of $\mathrm{BG}(C{)}_{p}$ of a certain type (an “unstable Adams operation”).
This is combined with a theorem of Jackowski, McClure, and Oliver
that classifies more precisely the self maps of $\mathrm{BG}(C{)}_{p}$; and with
a result of Broto, Møller, and Oliver which says that under
certain hypotheses on a space $X$, the homotopy fixed space of a
self equivalence $f$ of $X$ depends (up to homotopy type) only on
the closed subgroup $\stackrel{\u203e}{\u27e8f\u27e9}$ in the group
$\mathrm{Out}(X)$ of all homotopy classes of self equivalences of $X$.

Currently, no other proof seems to be known of this purely
algebraic theorem.

### 08/01/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building

Luke Wolcott, *University of Western Ontario*

###
Bousfield lattices, quotients, ring maps, and non-Noetherian rings

Given an object $X$ in a compactly generated tensor triangulated
category $C$ (such as the derived category of a ring, or the stable
homotopy category), the Bousfield class of $X$ is the collection of
objects that tensor with $X$ to zero. The set of Bousfield classes
forms a lattice, called the Bousfield lattice $\mathrm{BL}(C)$. First, we
will look at examples of when a functor $F:C\to D$ induces a
lattice map $\mathrm{BL}(C)\to \mathrm{BL}(D)$, and will describe several lattice
quotients and lattice isomorphisms. Second, we will focus on
homological algebra; a ring map $f:R\to S$ induces, via extension
of scalars, a functor $D(R)\to D(S)$, and this induces a map on
Bousfield lattices. Third, we specialize to a specific map between
some interesting non-Noetherian rings.

#### See also

http://www.math.ist.utl.pt/~ggranja/Wolcott-IST.pdf

### 15/11/2012, 15:30 — 16:30 — Room P3.10, Mathematics Building

Enxin Wu, *University of Western Ontario*

###
Doing differential geometry on the irrational torus, a rush
introduction to diffeology.

The quotient group of the $2$-torus modulo a line of an irrational
slope is an interesting geometric object. However, as a quotient
topological space it is indiscrete. In noncommutative geometry, the
study of this geometric object (called the irrational torus) is
based on the study of a ${C}^{*}$-algebra related to it. In this talk,
we will introduce a geometric way (called diffeology) to study the
irrational torus. If time permits, I will talk about the usual
geometric information of the irrational torus from the
diffeological point of view: the tangent bundle, de Rham
cohomology, smooth homotopy groups, etc.

### 17/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building

Marco Robalo, *Université Montpellier 2*

###
Noncommutative Motives: A universal characterization of the motivic
stable homotopy theory of schemes.

In this talk I will explain a new approach to the theory of
noncommutative motives based in the construction of a motivic
stable homotopy theory for the noncommutative schemes of
Kontsevich.

In the first part of the talk, we review the original motivic
homotopy theory of schemes as constructed by
Voevodsky-Morel-Jardine and explain its recent universal
characterization. The fundamental step in this characterization is
to understand the nature of the construction of symmetric spectrum
objects in a model category M. I will try to sketch the idea in
detail.

In the second part we explain the construction of a new motivic
theory for the noncommutative schemes which mimics the classical
one for schemes. Every scheme gives rise to a noncommutative one
and because of the universal property described in the first part,
this assignment gives birth to a canonical monoidal comparison map
between the commutative and the new noncommutative motivic
theories. This work is ongoing and it is part of my PhD thesis
under the direction of B. Toën in the Université de
Montpellier.

The talk will require (at all times) the language of
$\mathrm{\infty}$-categories, for which I will provide a small introduction
in the beginning.

### References

- M. Kontsevich,
Noncommutative motives
- J. Lurie,
Higher Algebra
- M. Robalo,
Non-commutative motives I: A Universal Characterization of the
Motivic Stable Homotopy Theory of Schemes
- G. Tabuada, A guided tour
through the Garden of Noncommutative Motives

### 12/07/2012, 16:30 — 17:30 — Room P4.35, Mathematics Building

Tibor Beke, *University of Massachusetts*

###
The sign pattern theorem and Brouwer's fixed point theorem

This work grew out of my attempt at concocting a proof of the
Brouwer fixed point theorem that is suitable for a first course in
topology. It should not involve algebraic topology and special
tricks like the no-retraction theorem, and should make the
statement itself plausible. Already in dimension two, Brouwer's
fixed point theorem is quite surprising and (visually) not very
compelling --- a contrast to the one-dimensional case where the
statement is equivalent to the intermediate value theorem that is
visually "obvious". We present a proof the Brouwer fixed point
theorem as a higher-dimensional generalization of the intermediate
value theorem. The proof itself is purely combinatorial and reduces
to the "sign pattern theorem" about (higher dimensional) matrices
containing two types of symbols, + and -. This talk should be
suitable (and hopefully, ideal) for undergraduate students.

### 03/07/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building

Maria Vaz Pinto, *Instituto Superior Técnico*

### The regularity and the vanishing ideal of parameterized codes associated to even cycles

Let $X$ be an algebraic toric set in a projective space over a finite field $K$, parametrized by the $s$ edges of a graph $G$. We give an explicit combinatorial description of a set of generators of the vanishing ideal of $X$, $I(X)$, when $G$ is an even cycle or a connected bipartite graph with pairwise vertex disjoint even cycles. We also show a formula for the regularity of $S/I(X)$, where $S$ is a polynomial ring in $s$ variables over $K$.

### 15/12/2011, 15:00 — 16:00 — Room P3.10, Mathematics Building

Mike Paluch, *Instituto Superior Técnico*

### A simplicial guide to Voevodsky's motives

Let $V$ be a smooth complex projective variety. Since $V$, when endowed with the classical topology carries the homotopy type of a finite CW complex, it follows that the normalized chain complex $NZ\mathrm{Sin}(V)$ of the free simplicial abelian group of the singular complex of V can be viewed as an object in the derived category $D(\mathrm{Ab})$ of bounded complexes of finitely generated abelian groups. As stated by Beilinson and Vologodsky a basic objective of the theory of motives is to find a triangulated category $\mathrm{DM}$, whose formulation does not depend on the metric topology of the field of complex numbers, and functors from the category algebraic varieties to $\mathrm{DM}$ and from $\mathrm{DM}$ to $D(\mathrm{Ab})$ such that the composite functor carries $V$ to $NZ\mathrm{Sin}(V)$. In this talk I shall present a simplicial perspective of Voevodsky's theory of motives in terms of additive and simplicially additive Grothendieck topologies and a simplicial candidate for $\mathrm{DM}$.