In the next couple of lectures, we will see some applications of the so called Polynomial Method to problems in Combinatorics. We will focus on the following three applications: 1. Joints Problem: For a set L of lines in R^3, a point p in R^3 is said to be a joint in L if there are at least three non-coplanar lines in L which pass through p. We will discuss a result of Guth and Katz who showed an upper bound on the maximal number of joints in an arrangement of N lines. 2. Lower bounds on the size of Kakeya sets over finite fields: For a finite field F, a Kakeya set is a subset of F^n that contains a line in every direction. We will discuss a result of Dvir showing a lower bound of C_n*q^n on the size of any Kakeya set over F^n, where C_n only depends on n and F is a finite field of size q. 3. Upper bounds on the size of 3-AP free sets over finite fields: We will then move on to discuss a recent result of Ellenberg and Gijswijt who showed that if F is a finite field with three elements, and S is a subset of of F^n such that S does not that does not contain three elements in an arithmetic progression, then |S| is upper bounded by c^n for a constant c < 3.

Hilbert in 1888 studied the inclusion Pn,2d ⊇ Σn,2d, where Pn,2d and Σn,2d are respectively the cones of positive semidefinite forms and sum of squares forms of degree 2d in n variables. He proved that: “Pn,2d = Σn,2d if and only if n = 2,d = 1, or (n,2d) = (3,4)”. In order to establish that Σn,2d Pn,2d for all the remaining pairs, he demonstrated that Σ3,6 P3,6, Σ4,4 P4,4, thus reducing the problem to these two basic cases. In 1976, Choi and Lam considered the same inclusion for symmetric forms and claimed that Hilbert’s characterisation above remains unchanged. They demonstrated that establishing the strict inclusion reduces to show it just for the basic cases (3,6),(n,4)n≥4. In this talk, we will explain the algebraic geometric ideas behind these reductions and how we extended these methods to investigate the above inclusion for even symmetric forms. We will present our leading tool a “Degree Jumping Principle”, an analogue of reduction to basic cases and construction of explicit counterexamples for the basic pairs. As a complete analogue of Hilbert’s theorem for even symmetric forms, we establish that “an even symmetric n-ary 2d-ic psd form is sos if and only if n = 2 or d = 1 or (n,2d) = (n,4)n≥3 or (n,2d) = (3,8)”. This is a joint work with S. Kuhlmann and B. Reznick.

I will introduce the subject of algebraic vector bundles on projective varieties in this talk. Vector bundles are used in commutative algebra in several contexts. For example, they provide geometric interpretation of tight closure of an ideal. They were used to show that tight closure does not commute with localization. A subtle notion of the semistability of vector bundles plays an important role in this subject. I will try to explain the relevance of this notion and discuss some examples. This talk will be a prequel to a talk by Prof. Nitin Nitsure on October 14.

Abstract. Sums of squares representations of polynomials is of fundamental importance in real algebraic geometry and goes back to the 1888 seminal paper of Hilbert. The main theorem in his paper is a full characterisation of all pairs (n,d) for which every nonnegative polynomial of a fixed degree d in a given number of variables n is a sum of squares of polynomials. Ninety years later, Choi and Lam asserted that this characterisation remains unchanged for symmetric forms. In this talk first some key observations and problems related to Hilbert’s theorem will be discussed. We then complete the above assertion of Choi-Lam. Along the way, we shall also discuss briefly how test sets for positivity of symmetric polynomials play an important role in establishing this assertion.

The purpose of hypocoercivity is to obtain rates for solutions of non-purely diffusive equations, in asymptotic regimes. This is a very useful technique for kinetic equations. After reviewing some easy results based on hypo-ellipticity, the lecture will focus on linear kinetic equations without regularizing effects and the L2 hypocoercivity method. Some motivations will be introduced, with a toy model. The core of the lecture will be a theoretical result based on a joint work with C. Mouhot and C. Schmeiser. Initially intended for systems with compactness or confinement in position space and simple local equilibira, the method has been extended to various local equilibria in velocities and non-compact situations in positions. It is also flexible enough to include non-local transport terms associated with Poisson coupling. Some recent results rely on various, deep functional inequalities. An application to the linearized Vlasov-Poisson-Fokker-Planck system will also be briefly presented.

The talk is based on my joint work with Prof Peter Beelen. In this talk, we recall the notion of Majority logic decoding for binary codes introduced by Reed in the late 60s. We also recall some basic notions of coding theory, construction of Grassmann codes and its basic parameters. Lines in Grassmannians are closely related to parity checks of Grassmann codes. We exploit this property of Grassmann codes together with several other properties of point-line geometry of Grassmannian to construct certain kind of parity checks for these codes. In the end, we use these parity checks and the idea of Reed's majority logic decoding to correct certain errors for Grassmann codes.

Several nonparametric ageing classes have been established in literature based on the various reliability characteristics. The talk will cover certain probabilistic issues such as, reliability bound, moment bound, closure under the formation of weak limits, characterisation theorem, etc., which have been hitherto unknown in the literature, for the New Better than Average Failure Rate (NBAFR) class of life distributions. We further explore the validity of these results in the context of a more general ageing class related to NBAFR family. In the inferential part of my presentation, I will discuss a problem of testing exponentiality against an alternative which is dened based on the Laplace-Stieltjes transform, namely the so-called L-class. The asymptotic distributions of our scale-invariant test statistics are derived and consistency of the test established. General expressions of the local approximate Bahadur eciencies for the test statistics are obtained and evaluated for typical alternatives. The performance of the test is assessed by means of a simulation study and through application to some real life data sets.

Let R be a normal noetherian domain with a finite group of automorphisms G. Samuel's result says that the kernel of the natural homomorphism Cl(R^G)\to Cl(R) is a subgroup of H^1(G,U), where U is the group of units in R. If R is divisorially unramified over R^G then the kernel is isomorphic to H^1(G,U). This enables us to find the divisor class groups of rings of invariants of finite group actions on polynomial or power series rings. If time permits two general constructions of cyclic unramified coverings of normal varieties will be discussed.

Two elements of a group G are said to be z-equivalent if their centralizers are conjugate within G. The z-equivalence is a weaker relation than the conjugacy relation. Let G be an algebraic group defined over a field k. Steinberg, proved that when G is a reductive group and k is an algebraically closed field, G(k) has finitely many z-classes. This result is generalised to more general base field k which are of type (F). In this talk, we discuss the results on this problem.

Let M be a finitely generated seminormal submonoid of the free monoid \mathbb Z_+^n and let k be a field. Then Anderson conjectured that all finitely generated projective modules over the monoid algebra k[M] is free. He proved this in case n=2. Gubeladze proved this for all n using the geometry of polytopes. In a series of 3 lectures, we will outline a proof of this theorem.