Algebraic Topology, Geometry and Physics Seminar

27 May 2014: Alessandro Zampini; Quantumness and classicality, 2; This is the second part of last week's talk.

20 May 2014: Alessandro Zampini; Quantumness and classicality, 1; In this talk I shall describe how the geometry of C^*-algebras and Jordan algebras can be useful to investigate whether a physical system has a quantum vs. a classical description.

13 May 2014: Gennaro Di Brino; Foundations of Homotopical Algebraic Geometry, 6; Starting with a brief review of what was done in the last talk and making more precise some of the notions we introduced last week, today we will finally deal with the definition of a stack in the sense of Toen and Vezzosi. We will then give an idea of the reason why they can be used as the building blocks for derived algebraic geometry.

29 Apr 2014: Gennaro Di Brino; Foundations of Homotopical Algebraic Geometry, 5; Simplicial Presheaves and Model Sites: The purpose of the fifth talk of our series will be to deal with simplicial presheaves on a model category and to define model topologies. We will then introduce pre-stacks and stacks on model sites.

8 Apr 2014: Gennaro Di Brino; Foundations of Homotopical Algebraic Geometry, 4; Towards homotopy theory for model categories: In our fourth meeting, we will deal with the small object argument and finish talking about the example of the category of nonnegatively graded chain complexes. We will then end with a brief sketch of the definition of the homotopy category of a model category.

1 Apr 2014: Gennaro Di Brino; Foundations of Homotopical Algebraic Geometry, 3; In our third talk, we will introduce model categories, following Quillen. After presenting the basics of model category theory, we will focus mostly on the example provided by the category of non-negatively graded chain complexes of modules over an associative ring with unit.

18 Mar 2014: Gennaro Di Brino; Foundations of Homotopical Algebraic Geometry, 2; Motivated by the representability criterion we stated last time, we will deal with Grothendieck topologies and sheaves (see last part of last week's abstract). Time permitting, we will then start talking about model categories.

11 Mar 2014: Gennaro Di Brino; Foundations of Homotopical Algebraic Geometry, 1; In this series of talks we will use the various formulations of the category-theoretical Yoneda lemma to explore Homotopical Algebraic Geometry (HAG 1 and 2), the foundational work of Toen and Vezzosi.
The classical setting: We plan to start from a standard representability proof for a (contravariant) functor from the category of schemes to that of sets. We will see how the original lemma of Yoneda comes into play in that setting and we will use the classical representability criterion as the starting point for the introduction of Grothendieck topologies on a category, and sheaves on Grothendieck sites.

25 Feb 2014: Benoît Jubin; Super Harish-Chandra pairs and graded Lie algebras; I will briefly recall the notions of super Lie groups and their morphisms, the construction of the super Lie algebra of a super Lie group, and super Lie group actions. Then, I will introduce super Harish-Chandra pairs and their morphisms. I will construct a functor from the category of super Lie groups to that of super Harish-Chandra pairs, and a quasi-inverse of that functor, proving the equivalence of these two categories with a constructive proof. These constructions are due to Kostant, Koszul, Vishnyakova, Carmeli–Fioresi, among others. Time permitting, I will say how this can be adapted to the $\mathbb{Z}$-graded case.

17 Dec 2013: Alessandro Zampini; The Hochschild–Kostant–Rosenberg theorem; The aim of the talk is to describe how the differential calculus on a topological space can be given a homological interpretation, and then to show how this allows (Connes' approach) to extend such a notion to non-commutative spaces.

10 Dec 2013: Benoît Jubin and Norbert Poncin organized an international workshop on Higher Lie Theory from 9 to 11 December 2013.

26 Nov 2013: Jian Qiu; Integration of Lie infinity algebras (Henriques' method); In contrast to the integration method due to Getzler, who designed an iteration procedure to solve the Maurer–Cartan (MC) equation directly, Henriques placed a filtration on the Lie infinity algebra, that allows him to solve the MC equation in successive approximations. His method dispenses from the nilpotency restriction of Getzler, yet on the other hand the filtration is not valid when the Lie algebra has terms in the positive grading.

12 Nov 2013: Benoît Jubin; DG Lie groups and DG Lie algebras, I; After a review of graded manifolds, I will introduce graded Lie groups and show how to construct their tangent graded Lie algebras, using left-invariant vector fields. Then, I will recall the notions of homological vector field and dg manifold, and I will introduce dg Lie groups and show how to associate to each of them a dg Lie algebra. In the process, I will survey the theory of graded Hopf algebras, and, as a warm-up, I will show how multiplicative vector fields on Lie groups give rise to differentials on their Lie algebras (via the Van Est isomorphism). I will hint at the inverse constructions (the “integration problem”) of all these steps.

Algebraic Topology, Geometry and Physics