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.

In 1979, Lovasz gave a beautiful simple *randomized* algorithm to detect if a given bipartite graph has a perfect matching, which involves computing the determinant of the Tutte matrix of the graph at a randomly chosen point. The algorithm has the additional advantage of being easily parallelizable. The problem of coming up with a deterministic version of this algorithm remained open for over 35 years until it was solved in 2015 by Rohit Gurjar, Stephen Fenner and Thomas Thierauf. We will see this algorithm.

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

A question of Stillman asks: Is there a bound for the projective dimension of every homogeneous ideal generated in a polynomial ring, which depends solely on the knowledge of the degrees of the generators, and not on the number of variables? In joint work with A. Banerjee, we show that for every fixed degree sequence, the correctness of a conjectural bound can be proved by an implementable algorithm.

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 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.

We will see a recent result of Thomas Rothvoss, that gives a constructive proof of a classical theorem in combinatorial discrepancy theory due to Spencer. Roughly, the problem is to colour the elements of a finite universe U red and blue so that each of a collection C of subsets has as small a discrepancy between the number of red and blue elements in it. Spencer showed that when |U| = |C| = n, the discrepancy can be made as small as O(\sqrt{n}). For a long time, it remained an open problem to give a constructive proof of this theorem, until a solution was provided by Nikhil Bansal in 2010. An alternate proof of Spencer's theorem was provided by a result of Lovett and Meka in 2012, and this result is generalized in the work of Rothvoss from 2014, which I will describe.

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