The complexity of determining whether a graph has an induced odd cycle of length at least 5 (odd hole) is unknown. In this talk, I will describe a polynomial-time algorithm to do this if the input graph does not contain the bull (a particular 5-vertex graph that turns out to be important in the theory of induced subgraphs) as an induced subgraph. This is joint work with Maria Chudnovsky.

In this series of two talks, we will begin by discussing some Nullstellensatz-like results when the base field is finite, and outline the proofs. Next, we will discuss a combinatorial approach to determining or estimating the number of common zeros of a system of multivariate polynomials with coefficients in a finite field. Here we will outline a remarkable result of Heijnen and Pellikaan about the maximum number of zeros that a given number of linearly independent multivariate polynomials of a given degree can have over a finite field. A projective analogue of this result about multivariate homogeneous polynomials has been open for quite some time, although there has been considerable progress in the last two decades, and especially in the last few years. We will outline some results and conjectures here, including a recent joint work with Peter Beelen and Mrinmoy Datta.

In this second talk of the series I will continue the discussion on symplectic manifolds. I will introduce the moment map and use it to construct quotients of symplectic manifolds under certain actions of lie groups.

In this second talk of the series I will continue the discussion on symplectic manifolds. I will introduce the moment map and use it to construct quotients of symplectic manifolds under certain actions of lie groups.

Graph complexes are simplicial complexes arising from graphs. In this talk we mainly focus on two types of complexes: Neighborhood complexes and Hom complexes. The topology of these complexes are closely related to the chromatic number of the underlying graphs. We give a brief survey of the research have been done with respect to them in recent years. We also discuss some open problems related to them.

In this series of two talks, we will begin by discussing some Nullstellensatz-like results when the base field is finite, and outline the proofs. Next, we will discuss a combinatorial approach to determining or estimating the number of common zeros of a system of multivariate polynomials with coefficients in a finite field. Here we will outline a remarkable result of Heijnen and Pellikaan about the maximum number of zeros that a given number of linearly independent multivariate polynomials of a given degree can have over a finite field. A projective analogue of this result about multivariate homogeneous polynomials has been open for quite some time, although there has been considerable progress in the last two decades, and especially in the last few years. We will outline some results and conjectures here, including a recent joint work with Peter Beelen and Mrinmoy Datta.

In this series of two talks, we will begin by discussing some Nullstellensatz-like results when the base field is finite, and outline the proofs. Next, we will discuss a combinatorial approach to determining or estimating the number of common zeros of a system of multivariate polynomials with coefficients in a finite field. Here we will outline a remarkable result of Heijnen and Pellikaan about the maximum number of zeros that a given number of linearly independent multivariate polynomials of a given degree can have over a finite field. A projective analogue of this result about multivariate homogeneous polynomials has been open for quite some time, although there has been considerable progress in the last two decades, and especially in the last few years. We will outline some results and conjectures here, including a recent joint work with Peter Beelen and Mrinmoy Datta.

In this talk, I will discuss two applications of analysis and topology in constructing counterexamples to certain questions in commutative algebra. The talk will be fairly elementary and accessible to M.Sc students.