In this talk I will prove that the i-th homotopy groups of a sphere S^n are finite when i is greater than n, except in one particular case, using the Serre spectral sequence. In the first half of the talk I will give the background material needed to understand the proof.

We will talk about a result of M. Katz and W. Messing, which says the following. From the Riemann hypothesis and the hard Lefschetz theorem in l-adic cohomology, the corresponding facts for any Weil cohomology follow.

An important class of problems in mathematics deals with the reconstruction of a structure from the eigenvalues of an associated matrix. The most famous such problem is: Can one hear the shape of a drum? Here we deal with the question: Which graphs are determined by the spectrum (eigenvalues) of its adjacency matrix? More in particular we ask ourselves whether this is the case for almost all graphs. There is no consensus on what the answer should be, although there is a growing number of experts that expect it to be affirmative. In this talk we will present several results related to this question. This includes constructions of cospectral graphs and characterizations of graphs by their spectrum. Some of these results support an affirmative answer, some support the contrary. It will be explained why the speaker believes that it is true.

A sunflower with p petals is a family of sets A_1,...,A_p such that the intersections of all pairs of distinct sets are the same. A famous conjecture in combinatorics, called the Sunflower conjecture, asserts a bound on the maximum size of any family of k-sets that does not contain a p-sunflower. We review some recent work by Ellenberg-Gijswijt and Naslund-Swain that proves a weak variant of this conjecture due to Erdos and Szemeredi.

We will prove a purely numerical criterion due to M. Artin to test the rationality of a surface singularity. In practice this is the criterion which is used when a rational surface singularity is being considered.

Tropical algebraic geometry is in the interface of algebraic and polyhedral geometry with applications to both these topics. We start with a gentle introduction to tropical algebraic geometry. We then focus on the tropical lifting problem and discuss recent progress. Tropical analogues of graph curves play an important role in this study.

Dyer's outer automorphism of PGL(2,Z) induces an involution of the real line, which behaves very much like a kind of modular function. It has some striking properties: it preserves the set of quadratic irrationals sending them to each other in a non-trivial way and commutes with the Galois action on this set. It restricts to an highly non-trivial involution of the set unit of norm +1 of quadratic number fields. It conjugates the Gauss continued fraction map to the so-called Fibonacci map. It preserves harmonic pairs of numbers inducing a duality of Beatty partitions of N. It induces a subtle symmetry of Lebesgue's measure on the unit interval. On the other hand, it has jump discontinuities at rationals though its derivative exists almost everywhere and vanishes almost everywhere. In the talk, I plan to show how this involution arises from a special automorphism of the infinite trivalent tree

We demonstrate that for discrete Markov dependent rvs, the normal approximation can be effectively replaced by compound Poisson approximation..In case of three state Markov chain, the effect of symmetry will be estimated.

Koszul algebras are the algebras over which the resolution of the residue class field is given entirely by linear matrices. This series of talks will be a survey on results obtained about Koszul algebras since they were introduced by Priddy in 1970. In the first talk, We shall see lots of examples of Koszul algebras, and discuss several characterizations of Koszul algebras.

Reed-Muller codes are among the most elementary and most studied codes. Less studied, but equally elementary are their projective counterparts, the protective Reed-Muller codes. Many open questions remain about these codes. Mathematically, a very interesting question is the determination of the generalized Hamming weights. The determination of these weights is equivalent to the determination of the maximum number of common solutions to certain system of polynomial equations. In this talk, I will give an overview of recent work and developments on the theory of generalized Hamming weights of projective Reed-Muller codes. This work was carried out together with Mrinmoy Datta and Sudhir Ghorpade.