Seminário de Álgebra 

Cohomology with $\mathbb{Z}/p$-coefficients is represented by a stable object, an Eilenberg-MacLane spectrum $H\mathbb{Z}/p$. Classically, since $\mathbb{Z}/p$ is a field, any module over $H\mathbb{Z}/p$ splits as a wedge of suspensions of $H\mathbb{Z}/p$ itself. Equivariantly, cohomology and the module theory of $G$-equivariant Eilenberg-MacLane spectra are much more complicated.

For the cyclic group $G=C_p$ and the constant Mackey functor $\underline{\mathbb{Z}}/p$, there are infinitely many indecomposable $H\underline{\mathbb{Z}}/p$-modules. Previous work together with Dugger and Hazel classified all indecomposable $H\underline{\mathbb{Z}}/2$-modules for the group $G=C_2$. The isomorphism classes of indecomposables fit into just three families. By contrast, we show for $G=C_p$ with $p$ an odd prime, the classification of indecomposable $H\underline{\mathbb{Z}}/p$-modules is wild. This is joint work in progress with Grevstad.

The negative answer to Hilbert's 10th problem tells us that determining whether or not an algebraic variety should carry any rational points or not is impossibly hard (literally!). The same problem even for curves is difficult: For elliptic curves, this is the subject of the celebrated Birch and Swinnerton-Dyer conjecture. I will survey recent results on this problem, and explain briefly an explicit p-adic analytic construction of rational points of infinite order on elliptic curves of analytic rank one (settling a conjecture of Perrin-Riou). These final bits of my mostly expository talk will be a report on joint works with Rob Pollack & Shu Sasaki, and with Denis Benois.

Quandles are algebraic structures that play a prominent role in knot theory and also form a class of set-theoretic solutions of the Yang-Baxter equation. After a brief introduction to quandles, I will focus on recent developments and open problems, including the Hayashi conjecture and the isomorphism problem for principal quandles.

Given a locally compact group G, the decomposition of the space of square integrable functions on G into irreducible unitary representations of G (“irreps”) is one of the basic desires in harmonic analysis. Not all irreps appear in such a decomposition; those which do are called tempered. The decomposition has a discrete as well as a continuous parts; the irreps which appear in the discrete part are called square integrable, and are much simpler analytically than general tempered irreps. Loosely speaking, tempered irreps can be thought of as “on the verge” of being square integrable. Although this intuition is rather classical, we discuss a new possible formal interpretation of it. This is joint work with D. Kazhdan.

In this expository lecture, we will explain how results from universal algebra can be used to construct free objects in various algebraic categories, such as the free unital vector lattice algebra over a vector space. Starting from these free objects of an algebraic nature, we then proceed to construct normed objects, such as unital Banach lattice algebras, that solve universal problems in analytic categories. These, in turn, can be used to construct free objects in larger analytic categories, such as the free complete locally convex-solid topological vector lattice algebra over a Banach space.

This is joint work with Walt van Amstel.

This talk describes an extension of Ravenel-Wilson Hopf ring techniques to $C_2$-equivariant homotopy theory. Our main application and motivation for introducing these methods is a computation of the $RO(C_2)$-graded homology of $C_2$-equivariant Eilenberg-MacLane spaces. The result we obtain for $C_2$-equivariant Eilenberg-MacLane spaces associated to the constant Mackey functor $\underline{\mathbb{F}}_2$ gives a $C_2$-equivariant analogue of the classical computation due to Serre at the prime 2. We also investigate a twisted bar spectral sequence computing the homology of these equivariant Eilenberg-MacLane spaces.

Classical knot theory is the study of smooth embeddings of circles in the 3-sphere up to the ambient isotopy. One of the fundamental problems in this field is the classification of knots, for which one needs invariants. The fundamental group of a knot complement space is a well-known invariant, but there are examples where it fails to distinguish knots. Around the 1980s, Matveev and Joyce introduced an almost complete knot invariant using quandles, known as knot quandles.

In the talk, I will introduce the notion of residual finiteness of quandles and prove that all link quandles are residually finite. Using the preceding result, we will see that the word problem is solvable for link quandles. I will talk about the orderability properties of link quandles. Since all link groups are left-orderable, it is reasonable to speculate that link quandles are left (right)- orderable. In contrast, we will see that the orderability of link quandles behave quite differently than that of the corresponding link groups.

We characterize intermediate subalgebras between a C*-algebra $A$ and the crossed product $A\rtimes_\alpha G$ or $\ell^1(G,A,\alpha)$, for actions $\alpha$ of discrete groups $G$ on a C*-algebras $A$.

I will describe how one can associate a “classifying space” to an infinity (alias homotopy) sheaf on the category of manifolds. This is based on joint work with D. Berwick-Evans and D. Pavlov (arXiv:1912.10544.), and is a homotopical strenghtening of a theorem of Madsen and Weiss. Examples abound.

In 1952, Dirac proved that any graph on $n$ vertices with minimum degree $n/2$ contains a Hamiltonian cycle, i.e. a cycle which passes through every vertex of the graph exactly once. We prove a resilience version of Dirac’s Theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\epsilon \gt 0$, a.a.s. the following holds: let $G_0$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2 + \epsilon)d$. Then, $G_0$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved. This is joint work with Padraig Condon, Alberto Espuny Díaz, Daniela Kuhn and Deryk Osthus.

In this talk, we are going to present a user-friendly approach to derived geometry. One of our goals is to convince the audience that the notions of derived manifold / scheme / space / stack are just as natural as their classical counterparts. After having introduced the basic techniques, we will apply them to study certain moduli spaces of geometrical origin. Derived geometry also has recently found applications in arithmetics, which we will try to explain in the last part of the talk.

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.

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.

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.

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.

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.

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.

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.

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.

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.