# Algebra Seminar

## Past sessions

### Positive representations of algebras of continuous functions

It is well known from linear algebra that a family of mutually commuting normal operators on a finite dimensional complex inner product space can be simultaneously diagonalised. Strongly related to this is the fact that a representation of a commutative C*-algebra on a Hilbert space is generated by a so-called spectral measure, taking its values in the orthogonal projections. A result by Ruoff and the lecturer asserts that a similar phenomenon occurs for positive representations of algebras of continuous functions on a substantial class of Banach lattices.

Recently, it has become clear that these two facts for Hilbert spaces and Banach lattices can be understood from one underlying general theorem for positive homomorphisms of algebras of continuous functions into partially ordered algebras. This result can be proved by purely order-theoretic methods.

In this lecture, we shall explain this theorem and how it relates to the two special cases mentioned above. We shall also sketch the theory of measure and integration in partially ordered vector spaces that is necessary to formulate and establish it.

This is joint work with Xingni Jiang.

### Non-commutative Boolean algebras

In this talk, I shall explain how the classical theory of Stone duality may be generalized to a non-commutative setting. This theory has connections with étale groupoids, quantales, groups and inverse semigroups.

Some of the work was joint with Alina Vdovina. I shall assume no prior exposure to this theory.

### A primer on the Section Conjecture — a bridge between arithmetic and homotopy

In 1983, Grothendieck wrote a letter to Faltings in which he formulated a conjecture for hyperbolic curves over fields which are finitely generated over the rationals. Remaining open to date, it carries the study of rational points on an algebraic variety to the realm of profinite groups. Assuming only a working knowledge of basic Algebraic Geometry, we formulate and motivate the Section Conjecture and outline some modern attempts to tackle it.

### Amenable algebras: algebraic and analytical perspectives

In this talk, we investigate the amenability of algebras from algebraic and analytical viewpoints.

We also consider its relationship with the

1. semi-simplicity of operator algebras and
2. crossed product Banach algebras associated with a class of $C^\ast$-dynamical systems.

### Dynamical systems for arithmetic schemes - the higher dimensional case

Extending the colloquium lecture, which essentially deals with $\operatorname{spec} \mathbb{Z}$ we discuss the general case of our construction of dynamical systems for arithmetic schemes. Functoriality and the relation to rational Witt vectors and Fontaine's $p$-adic period ring $A_\inf$ will also be explained if time permits.

### Equidimensional algebraic cycles and current transforms

In this talk we show how equidimensional algebraic correspondences between complex algebraic varieties can be used to construct pull-backs and transforms of a class of currents representable by integration. As a main application we exhibit explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth quasi-projective complex algebraic varieties to Deligne-Beilinson with integral coefficients.

We exhibit a few examples and indicate how this can be applied to Voevodsky’s motivic complexes. This is joint work with Pedro dos Santos and Robert Hardt.

### Cohomology of braids, graph complexes, and configuration space integrals

I will explain how three integration techniques for producing cohomology classes — Chen integrals for loop spaces, Bott-Taubes integrals for knots and links, and Kontsevich integrals for configuration spaces — come together in the computation of the cohomology of spaces of braids. The relationship between various integrals is encoded by certain graph complexes. I will also talk about the generalizations to other spaces of maps into configuration spaces (of which braids are an example). This will lead to connections to spaces of link maps and, from there, to other topics such as rope length, manifold calculus of functors, and a conjecture of Koschorke, all of which I will touch upon briefly. This is joint work with Rafal Komendarczyk and Robin Koytcheff.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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.

### 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!

### 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.

### 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 $U(g)$.

### 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.

### 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.

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Current organizer: Gustavo Granja