Relative categories are maybe the most naive model for abstract homotopy theory (just categories with a subcategory of "weak equivalences"). Barwick and Kan showed that the category of relative categories has a model structure, Quillen equivalent to the Joyal model structure on simplicial set, which has infinity-categories as fibrant objects. We will show that model categories define fibrant relative categories and also discuss other aspects of the homotopy theory of relative categories.

We shall give several examples of Yang-Mills Theories.

We introduce Voevodsky's Univalence Axiom and see some of its consequences in type theory. We also start studying various definition of type-theoretic equivalences.

Let $X$ be a topological space with an action of a topological group $G$. We want to relate to $X$ an algebraic object that reflects both the topology and the action of the group. The first candidate is the cohomology ring $H^*(X/G)$: however, if the action is not free, the space $X/G$ may have some pathology. The Borel construction allows to replace $X$ by a topological space $X'$ which is homotopically equivalent to $X'$ and the action of $G$ on $X'$ is free.

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

The purpose of this talk will be to illustrate how various abstract techniques from algebraic geometry (i.e. cohomology, Riemann Roch) can be used to study algebraic surfaces. Algebraic surfaces are smooth projective varieties over \mathbb{C} of dimension 2. After reviewing the basics of linear systems and divisors on surfaces, we will study morphisms determined by linear systems on the Hirzebruch surfaces, a particularly nice class of algebraic surfaces. Depending on time, other applications and results will be described, such as the relation of Hirzebruch surfaces to the Enriques-Kodaira classification or the classication of degree n-1 nondegenerate surfaces in P^{n}.

For an algebraic variety X over an arbitrary field F, a classical question asks whether X has a K-point for a given field extension K/F. When X is a generalized Severi-Brauer variety, we may extend this question to ask not only about K-points, but about K-forms of any closed Schubert subvariety. In this talk, we consider an algebraic form of this question concerning data from the underlying central simple algebra of X. We then discuss applications to the Grothendieck group and Chow group of X, generalizing a result of N. Karpenko for usual Severi-Brauer varieties. This is part of ongoing work with D. Krashen and N. Lemire.