The cohomology and homotopy groups of moduli spaces of smooth manifolds are some of the most basic invariants of manifold bundles: the former contain all the characteristic classes and the latter classify smooth bundles over spheres. Complete calculations of these groups are challenging, even for the simplest compact manifolds. It is then desirable to know, at least, some qualitative information, for example whether these groups are (degreewise) finitely generated.

In this talk, I will discuss a method to attack this question which leads to the following theorem: if \(M\) is a closed smooth manifold of even \(\text{dimension }\gt 5\) with finite fundamental group, then the cohomology and higher homotopy groups of \({\rm BDiff}(M)\) are finitely generated abelian groups.

This is joint work with M. Krannich and A. Kupers.

The space \(U(n)/N(n)\), where \(N(n)\) is the normalizer of the maximal torus in \(U(n)\), arises naturally in many areas of mathematics and can be identified with the unordered flag manifolds. In this talk, we will introduce the concept of the \(n\)-fold extended symmetric power of a space \(X\), and describe its cohomology as a Hopf ring. We will also demonstrate homological stability for the spaces \(\left\{U(n)/N(n)\right\}\), and describe the stable cohomology ring of \(U(n)/N(n)\). If there is sufficient time, we may also see some of the low-dimensional computations that have been done in the case where \(n = 3, 4\). This is joint work with Lorenzo Guerra.

In this talk, I'll compute the equivariant cohomology algebras of locally k-standard T-manifolds. Then, we discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism.

In this talk, I'll introduce the category of locally k-standard T-manifolds which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment-angle manifolds. They are smooth manifolds with well-behaved actions of tori. Then, I'll compute the fundamental groups and equivariant cohomology rings of these new manifolds.

One can naturally associate to any pointed topological space a topological monoid of continuous loops that start and end at base point with multiplication given by concatenation of loops. Every element in this topological monoid has an inverse up to homotopy, so it is “almost” a topological group. This passage does not loose any homotopical information and it is often useful to recast the homotopy theory of spaces in terms of topological monoids/groups. In algebraic topology one studies spaces by means of algebraic invariants. For instance, one can replace a topological space by a chain complex with additional structure, more precisely a coalgebra structure - through the classical singular chains construction.

In this series of talks, I will describe how to model algebraically the passage from a pointed space to its topological monoid of based loops as a natural construction -called the “cobar functor” and originally due to F. Adams- that takes a differential graded (dg) coalgebra and produces a dg algebra. When applied to the singular chains on a pointed space, the cobar functor produces a model for the chains on the based loop space. A slightly new perspective on this construction will allow us to generalize many existing results in the literature from simply connected spaces to spaces with arbitrary fundamental groups. We will discuss several applications and consequences of this new perspective such as obtaining algebraic models for non-simply connected spaces (extending the work of Sullivan, Quillen, Goerss, Mandell, and others) and models for the free loop space functor suitable for studying string topology in the non-simply connected setting.

I will discuss what cyclic cohomology and various string topology algebraic operations on it are. I will then discuss how these operations are related to counting the number of possibilities of gluing the sides of a \(2n\)-gon to obtain a surface of genus \(g\).

I will discuss how the answer to certain integrals on the space of Hermitian matrices is related to counting the number of possibilities of gluing the sides of a \(2n\)-gon to obtain a surface of genus \(g\).

The celebrated Otal-Croke marked length spectrum rigidity theorem recovers the geometry of a closed negatively curved surface from the lengths of closed geodesics on the surface. As an intermediate step of the proof, the dynamics of the (Anosov) geodesic flow is recovered from the periods. We generalize this dynamics rigidity result to the setting of volume preserving \(3\)-dimensional Anosov flows. In turn, it leads to a more general “weighted” marked length spectrum rigidity for negatively curved surfaces. This is joint work with Federico Rodriguez Hertz.

Annular knots can be thought of as knots in a solid torus. Recently, the study of annular knots using invariants from annular Khovanov homology has inspired interesting results. I will talk about a different approach to define invariants of annular knots using combinatorial knot Floer homology. I will also discuss its relationship with band rank and contact topological invariants of Legendrian knots.