The cohomology of $SL_n(\mathbb{Z})$ and its finite index subgroups are important in many areas of mathematics ranging from number theory to manifold topology. The low degree cohomology stabilizes and is well understood. In this lecture series, I will focus on the more mysterious high dimensional cohomology. Using Borel-Serre duality, this can be computed by studying Steinberg modules which are defined via Tits buildings.

I will describe how the homotopy groups of spheres may be generated using suspensions, Whitehead products and higher order operations starting from the identity map and the inclusion of wedge summands. This is joint work with David Blanc and Debasis Sen.

Let $QI(R)$ ($QI(R_+)$) denote the group of quasi-isometries of the real line ($[0,\infty)$ resp.). We can split $QI(R)$ as ($QI(R_+) \times QI(R_-)$) $\rtimes \langle t\rangle$ and $t$ has order $2$. We will introduce an invariant for the elements of $QI(R_+)$ and split it further into smaller units. We will then show that a quotient of $QI(R_+)$ gives an example of left orderable group which is not locally indicable.

The theory of orderable groups in group theory started with seminal works of Dedekind and Holder. In recent years, the possibility of ordering interesting groups (Thompson's group, braid groups, knot groups) have attracted the interest of people coming from different fields. In this talk we will discuss algebraic properties of left orderable and bi-orderable groups, sufficient criteria to test orderability, space of orderings and orderability of some well-known groups. This theory has also a natural dynamical aspect, which will be discussed too. In this regard, locally indicable groups will be discussed.

First I review the study of cut locus, and the thread construction of ellipsoid. Next, I will talk about several topics of so-called intuitive geometry, for example, reversing polyhedra.

Model categories provide a general abstract framework in which to do homotopy theory. This will be a purely introductory, expository talk in which we will introduce the definition of a model category and run through the axioms at a leisurely pace giving examples and motivation. After covering the minimal definition we will briefly discuss cofibrantly generated model structures, simplically enriched model structures, and left Bousfield localization. Time permitting, the goal will be to describe a useful simplicial model category structure on simplicial presheaves on Cartesian spaces, a large family of objects that encompasses smooth manifolds and stacks. The basic concepts from category theory that we need will be briefly reintroduced as necessary.

A diffeological space consists of a set $X$ together with a collection $D$ of set functions $U \to X$, where $U$ is a Euclidean space, that satisfies three simple axioms. In this talk, we will describe how this simple definition provides a new, powerful framework for differential geometry. Namely, every finite dimensional smooth manifold is a diffeological space, as are many infinite dimensional ones, orbifolds, and many other objects of interest in differential geometry. Further, the category of diffeological spaces is much better behaved than the category of finite dimensional smooth manifolds, in a way that we will make precise. Despite the fact that diffeological spaces are much more general than manifolds, many classical constructions in differential geometry still make sense for them, such as tangent spaces, differential forms, homotopy theory, and fiber bundles. However, recent results show that many of the cherished and basic theorems of smooth manifold theory fail for general diffeological spaces, but this failure opens up worlds of interesting possibilities. We will review two such results. One being the difference between the internal and external tangent space of a diffeological space, and the obstruction between Cech cohomology and deRham cohomology. If time permits, I will discuss the recent work of my preprint “Diffeological Principal Bundles and Principal Infinity Bundles.”