Matrix integrals play an important role in random matrix theory, 2d quantum gravity, and the topology of moduli spaces of Riemann surfaces. In this first lecture we shall look at a perturbative expansion for matrix integrals via Feynman graphical methods.

In this talk we will discuss uniqueness results for inductive types, with particular focus on identity types.

In the last talk we introduced Equivariant Chow groups and proved that they are well defined using a certain double fibration argument. In this talk we will present the functoriality of the flat pull backs and proper pushforwards of Equivariant cycles. We will review Grothendieck's axiomatic construction of Chern classes in the classical context and use it to define 'Equivariant Chern classes'. If time permits we will compute some examples of Equivariant Chow groups and a result which allows us to compute such groups just using the Equivariant Chow groups of the maximal torus.

The classical notion of convexity of subsets of the Euclidean space metamorphoses into several distinct types of (geodesic) convexity in the more general Riemannian setting , this phenomenon, along with its geometric implications, being the main topic of this talk.

The circle is uniquely characterized by the property that among all simple closed plane curves of given length L, the circle of circumference L encloses maximum area. In this talk, we will recount some geometric generalizations and applications of this inequality. We shall also consider various geometric and analytic inequalities closely connected to the isoperimetric inequality, such as Brunn-Minkowski's, Wirtinger's, Poincaré's and Sobolev's inequalities.

I will present a potential approach to construction of universal fibrations in the categories of presheaves of infinity-groupoids.