In this talk, we will try to understand Topology of a quotient of the complex Stiefel manifold, which we call $m$-projective Stiefel manifold. More precisely, we consider quotients of complex Stiefel manifolds by finite cyclic groups whose action is induced by the scalar multiplication on the corresponding complex vector space. We will obtain a description of their tangent bundles, compute their mod $p$ cohomology and obtain estimates for their span (with respect to their standard differentiable structure). We will discuss the Pontrjagin and Stiefel-Whitney classes of these manifolds and give applications to their stable parallelizability. This is part of joint work with P. Sankaran.

In this talk I will construct manifolds and orbifolds with quasitoric boundary. I'll show that these manifolds and orbifolds with boundary has a stable complex structure. These induce explicit (orbifold) complex cobordism relations among quasitoric manifolds and orbifolds. In particular, we show that a quasitoric orbifold is complex cobordant to some copies of fake weighted projective spaces. The famous problem of Hirzebruch is that which complex cobordism classes contain connected nonsingular algebraic varieties? I'll give some sufficient conditions to show when a complex cobordism class may contain an almost complex quasitoric manifold. Andrew Wilfong give some necessary condition of this problem up to dimension 8.

Time-like metric spaces are topological spaces equipped with an order relation < and a distance function, where the distance from x and y is defined only when x < y. The notion was introduced by H. Busemann, and the motivation comes from the theory of relativity in physics. In this talk, I will describe the basics of this theory and give some examples.

In this seminar, we will study the structure of linear algebraic groups over complex numbers and over real numbers. The first lecture will be elementary.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

One of the most useful analogies in mathematics is the fundamental group functor (also known as the Galois Correspondence) which sends a topological space to its fundamental group while at the same time sending continuous maps between spaces to corresponding homomorphisms of groups in such a way that compositions of maps are preserved. A an obvious question one might ask is whether the fundamental group functor is "onto", that is: (1) is every group the fundamental group of a space and (2) every homomorphism the image of a continuous map between corresponding spaces? The easy answer to (1) is "yes" and the nonobvious answer to (2) is "it depends on the spaces". We'll introduce the harmonic archipelago as the shining example of a space with a strange fundamental group, define an archipelago of groups as a group theoretic product operation and finally describe how such products are (almost) all isomorphic to the fundamental group of the harmonic archipelago. We will study examples showing that there are group homomorphisms that cannot be induced by continuous maps on certain spaces and how the fundamental group of the harmonic archipelago factors through all such "discontinuous homomorphisms", how none of the examples is constructible (or even understandable in any reasonable way) and how one might detect spaces whose fundamental group allows them to be the codomain of such weird homomorphisms (the conjecture is that they contain the rational numbers or torsion). We'll talk a bit about the notion of cotorsion groups from classical Abelian group theory and how that notion can be generalized to non-Abelian groups by requiring certain types of systems of equations have solutions and then mention how countable groups which have solutions to such systems are always images of the fundamental group of an archipelago. In the end we're lead to an example of a countable group which we can prove is either the rational numbers or gives a counterexample to a nearly 50 year old conjecture in number theory: the Kurepa conjecture. So there is a little topology, a little homotopy theory, some group theory, a pinch of logic and a wisp of number theory in the talk. This is a distillation of work I've published recently with Hojka and Meilstrup (Proc AMS) and work that is still being written up with Hojka and Herfort.

Starting with a positive definite kernel $K$ defined on a bounded open connected subset $\Omega$ of $\mathbb C^d,$ we give several canonical constructions for producing new positive definite kernels on $\Omega,$ possibly taking values in $Hom(E)$ for some normed linear space $E$ of dimension $d.$ Specifically, this includes the curvature defined as the $d\times d$ matrix of real analytic functions $$\big ( \!\! \big ( \tfrac{\partial}{\partial_i \bar{\partial}_j} \log K \big ) \!\!\big ).$$ These kernels define an inner product on a submodule (over the polynomial ring) functions holomorphic on $\Omega.$ The completion is a Hilbert space on which the polynomials act by point-wise multiplication making it into a "Hilbert module". We will discuss hereditary properties, sub and quotient of these Hilbert modules.

Paul Erd\H{o}s and L\'{a}szl\'{o} Lov\'{a}sz proved that, for any positive integer $k$, up to isomorphism there are only finitely many maximal intersecting families of $k-$sets. So they posed the problem of determining or estimating the largest number $\N{k}$ of the points in such a family. They proved by means of an example that $N(k)\geq2k-2+\frac{1}{2}\binom{2k-2}{k-1}$. In 1985, Zsolt Tuza proved that the upper bound of $N(k)$ is best possible up to $2\binom{2k-2}{k-1}$. In this talk, we discuss the recent development of the upper bound on $N(k)$.

The classical form of Weyl's law is concerned with counting the eigenvalues of the Laplacian operator on a bounded domain. Atle Selberg generalized it in the 50s to show the existence of non-holomorphic modular forms. I will explain the Weyl's law with examples and how it counts automorphic forms, which are objects of number-theoretic interest.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.