In a finite projective 3-space considered as an incidence geometry, an ovoid is the analogue of the sphere in Euclidean 3-space, introduced independently by Segre and Tits. Elliptic quadrics are generic examples of ovoids. A projectively nonequivalent family of ovoids were constructed by Tits which is closely related to the Suzuki groups and Moufang sets. These are the only known families of ovoids. Apart from being objects of intrinsic interest, they are fundamentally related to some important combinatorial structures like inversive planes, generalised quadrangles, permutation polynomials, group divisible designs, etc. Their classification and understanding their distribution in the projective 3-space are the fundamental problems regarding them. However, several first questions about them are yet to be settled. To name a few: the structure of the intersection of any two of them, packing the projective 3- space by ovoids, the number of such objects, up to projective equivalence. As an introduction to this topic, we discuss some interesting results and mention some open problems. I will make an effort to keep the talk elementary.

Mumford forms are sections of a certain line bundle defined over the moduli space of smooth algebraic curves of genus g>0. In this talk we discuss the relationship of Mumford forms with a certain Bosonic measure coming from String theory, and their estimates.

In this talk I will explain some results on the extensions of mod-p characters of affine pro-p Iwahori–Hecke algebras. As a preliminary application we compute the degree one extensions of smooth representations of SL(2,Q_p). These calculations also reveal interesting phenomenon on Iwahori subgroup cohomology of smooth representations. If time permits I will explain how these extensions can be related to local Galois representations.

We shall present a simple proof due to Vijay Kodiyalam. This proof makes use of the fact that transform of a complete ideal in a two-dimensional regular local ring R in a quadratic transform of R is again complete. It also uses a structure theorem, due to Abhyankar, of two-dimensional regular local rings birationally dominating R.

The Riemann-Roch theorem is fundamental to algebraic geometry. In 2006, Baker and Norine discovered an analogue of the Riemann-Roch theorem for graphs. This theorem is not a mere analogue but has concrete relations with its algebro-geometric counterpart. Since its conception this topic has been explored in different directions, two significant directions are i. Connections to topics in discrete geometry and commutative algebra ii. As a tool to studying linear series on algebraic curves. We will provide a glimpse of these developments. Topics in commutative algebra such as Alexander duality and minimal free resolutions will make an appearance. This talk is based on my dissertation and joint work with i. Bernd Sturmfels, ii. Frank-Olaf Schreyer and John Wilmes and iii. an ongoing work with Alex Fink.

Zariski's first paper in algebra written in 1938 proved among many other results that product of complete ideals is complete in the polynomial ring $K[X,Y]$ where $K$ is an algebraically closed field of characteristic zero. This was generalised to two-dimensional regular local rings in Appendix 5 of Zariski-Samuel's classic "Commutative Algebra". We will present a new proof of this theorem using a formula of Hoskin-Deligne about co-length of a zero-dimensional complete ideal in a two-dimensional regular local ring in terms of quadratic transforms of $R$ birationally dominating $R.$

In the first half of the talk, we shall recall Koszul filtation and Grobner flag. Let R be a standard graded algebra. If R has a Koszul filtation, then R is Koszul. If R has a Grobner flag, then R is G-quadratic. I will mention an important result of Conca, Rossi, and Valla: Let R be a quadratic Gorenstein algebra with Hilbert series 1 + nz + nz^2 + n^3. Then for n=3 and n=4, R is Koszul. In the second half of the talk, we shall focus on class of strongly Koszul algebras. If time permits, I will prove that Koszul algebras are preserved under various classical constructions, in particular, under taking tensor products, Segre products, fibre products and Veronese subrings.

The Robinson-Schensted-Knuth (RSK) correspondence is a bijection from the set of matrices with non-negative integer entries onto the set of pairs of semistandard Young tableaux (SSYT) of the same shape. SSYT can be expressed as integral Gelfand-Tsetlin patterns. We will show how Viennot's light-and-shadows algorithm for computing the RSK correspondence can be extended from matrices with non-negative integer entries to matrices with non-negative real entries, giving rise to real Gelfand-Tsetlin patterns. This real version of the RSK correspondence is piecewise-linear. Indeed, interesting combinatorial problems count lattice points in polyhedra, and interesting bijections are induced by volume-preserving piecewise-linear maps.

We will give some basic definitions and take a few examples of motives. The reference is A. J. Scholl's article (1991, Seattle).