The Riemann hypothesis asks about the location of zeros of the Riemann zeta function. More generally, one may consider the analogue of this hypothesis for L-functions. It is also of interest to study the distribution of non-zero values. We will discuss some old and recent results on this problem.

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.

This lecture is based on the paper "Generalized Hamming Weights for Linear Codes" by V.K. Wei. In this lecture I will define generalized Hamming weights and then discuss monotonicity of Hamming weights and duality theorem. Determination of complete weight hierarchy of binary Reed-Muller code will also be outlined.

This lecture is based on the paper "Generalized Hamming Weights for Linear Codes" by V.K. Wei. In this lecture I will define generalized Hamming weights and then discuss monotonicity of Hamming weights and duality theorem. Determination of complete weight hierarchy of binary Reed-Muller code will also be outlined.

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.

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.

http://www.math.iitb.ac.in/~seminar/ 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.

The next two talks in the Geometry-Topology seminar (on 04/03/16 and 11/03/16) will be on introduction to the basics of topological K-theory. Topological K-theory attaches to each topological space, a ring gotten using isomorphism classes of topological vector bundles. In the first lecture, I will prove some basic facts about complex vector bundles on compact, Hausdorff spaces and introduce the K-groups. The second lecture will be largely devoted to sketching a proof of Bott periodicity; a central theorem in this subject. Two classical (and related) applications of this theorem are in proving non-parallelizability of spheres other than those of dimension 1,3 and 7 and non-existence of finite-dimensional division algebras over R except in dimensions 1,2, 4 and 8.

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.