We will define principal bundles, connections and curvature. With the basics defined, we will construct the Chern-Weil homomorphism. Let E be a principal G-bundle on M with a connection D. Let F be the curvature of D, and g=Lie(G). The Chern-Weil homomorphism associates to each Ad-invariant polynomial on g, a well defined cohomology class in the de Rham cohomology H_{dR}^*(M). Let P be an Ad-invariant homogeneous polynomial of degree k on g. The Chern-Weil image of P is given by the closed 2k-form P(F^{2k}). Its class in H^{2k}_{dR}(M) does not depend on the choice of the connection. This class is functorial. We will conclude with a few examples.

We shall indicate a solution to a problem posed by Euclid: Prove or disprove the parallel postulate assuming only the first 4 axioms of Euclidean geometry. The solution leads to a new geometry called hyperbolic geometry. We shall only assume some familiarity with calculus.

At random points of the integer lattice we place boxes of random sizes. The question we ask is under what conditions the entire space is covered. Also if instead of the entire lattice, our underlying space is a subset of it, then what can we say about the coverage properties.

Differential geometric approach to the field of statistics gave rise to a branch of mathematics called the information geometry. Information geometry began as a study of the geometric structures possessed by a statistical model of probability distributions. A statistical model equipped with a Riemannian metric together with a pair of dual affine connections is called a statistical manifold. Information geometry typically deals with the study of various geometric structures on a statistical manifold. In this talk I present a brief description of the information geometric framework for the statistical estimation problem. First I describe two important class of geometric structures on a statistical manifold, the alpha-geometry and the ( F, G)-geometry. Then the role of the (F, G)-geometry in the study of dually flat structures of the deformed exponential family is discussed. Also I describe the geometric framework for the mismatched estimation problem in an exponential family. Finally I present some of the open research problems in the area of estimation in a deformed exponential family

The regularity lemma (due to E. Szemeredi) states that for a given $\epsilon>0$ every graph can be partitioned into $k< M(\epsilon)$ equitable parts {V_1,V_2,..,V_k} such that except for at most $\epsilon k^2$ of these pairs, the rest are $\epsilon$-regular. The proof of the regularity lemma obtains a $k$ which grows as a tower of 2's of height $\Omega(1/\epsilon^5)$. Gowers however showed that this type of tower growth is unavoidable, and simpler proofs were later given by Conlon and Fox. We shall look at another one which is considerably simpler, due to Shapira and Moschovitz.

Within crystallization theory, two interesting PL invariants for d manifolds have been introduced and studied, namely gem-complexity and regular genus. In this talk, we prove that, for any closed connected PL 4-manifold M, its gem-complexity k(M) and its regular genus G(M) satisfy: k(M) >= 3\Chi(M)+10m-6 and G(M)>=2\Chi(M)+5m-4, where rk(Pi_1(M))=m. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of PL 4-manifolds and a large class of PL 4-manifolds satisfy these bounds. We also provide upper bounds for regular genus and gem-complexity of manifold bundles over circle.

The order bound is a general method to obtain a lower bound for the minimum distance of an evaluation code. It is a very good bound in case the code is defined using Goppa's construction of codes from curves. In my talk I will outline the main ideas behind the order bound and make them more explicit in the case of one-point algebraic-geometry codes

A theorem of Bayer and Stillman asserts that if I is an ideal in a polynomial ring S over a field (in finitely many variables), then the projective dimension and regularlity of S/I are equal to those of S/Gin(I), where Gin(I) is the generic initial ideal of I in the reverse lexicographic order. In this series of talks, we will discuss the necessary background material, and prove the above theorem.

The goal of this talk will be to prove Thom’s theorem that the oriented cobordism ring when tensored with rational numbers is a polynomial algebra over Q generated by the cobordism classes of complex projective spaces.