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.

Ramanujan's master theorem states that under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this power series. Ramanujan applied this theorem to compute several definite integrals and power series and this explains why it is referred as "Master Theorem". In this talk we shall try to explain its analogue for radial sections of line bundles over the real hyperbolic space. This a joint work (in progress) with Prof. Swagato K Ray.

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 this talk, we will have a look at three results, starting with the following. [Berlekamp-Welch] Given a univariate polynomial function over F_q with data corruption at t < q/2 points, we can recover the function completely if the degree of the function is sufficiently low. A generalization of the above is as follows, where instead of 'recovering' the function, we find all its 'close approximates'. [Madhu Sudan] Given data points (x_i,y_i), i \in [n], and parameters k and t, we can list all polynomials with degree at most k, which satisfy at least t data points. This result can further be generalized as follows. [Madhu Sudan] Given data points (x_i,y_i) with weights w_i, i \in [n], and parameters k and W, we can list all polynomials with degree at most k such that the sum of weights of data points satisfied by the polynomial is at least W. The last two results provide list-decoding of Reed-Solomon codes.

It is well known that a "good" large subset of the Euclidean space contains approximately as many lattice points as its volume. This need not hold for a general subset. On the other hand, a classical theorem of Siegel asserts that for any subset of positive measure, the "average" number of points (in an appropriate sense) of a general unimodular lattice contained in it, equals the measure of the set. In place of the average over the entire space of lattices one may also ask for analogous results for smaller subclasses. In a recent work with Jayadev Athreya, we explored this issue, with some modifications that place the problem in perspective, for the case of symplectic lattices, viz. lattices (in even-dimensional spaces) obtained from the standard lattice under symplectic transformations. In this talk I shall describe the overall asymptotics in this case, together with the historical background of the results and techniques involved.

We shall continue the discussion on the results of Marina Ratner on unipotent flows, and the techniques involved.

A tricolored-sum-free set in F_2^n is a collection of triples {(a_i,b_i,c_i)}_{I=1..m} such that a) for each I, a_i+b_i+c_i=0 b) If a_i+b_j+c_k = 0, then I=j=k. The notion of a tricoloured-sum-free set generalizes the notion of a capset to F_2^n. The basic question here is: How large can a tricolored-sum-free set be? We will see the following two (recent) results. i) Kleinberg's upper bound of 6\binom{n}{n/3} for a tricolored-sum-free set. This in conjunction with a previous result of his establishing a lower bound of \binom{n {n/3}2^{-\sqrt{16n/3}} gives almost asymptotically tight results. ii) Ben Green (in 2005) proved the following BOOLEAN REMOVAL LEMMA: Given \epsilon>0 there exists \delta depending only on epsilon such that the following holds: Write N=2^n. If X,Y,Z are subsets of F_2^n if by deleting \epsilon N elements from X,Y, Z altogether, one can eliminate all arithmetic triangles (triples (x,y,z) with \in X,y\in Y,z\in Z such that x+y+z=0) then there are at most \delta N^2 arithmetic triangles in (X,Y,Z). Green's proof establishes a bound for1/(\delta) which is a tower of 2s of length poly(1/\epsilon). We will look at a recent result of Fox and Lovasz (junior) who obtained an almost tight bound for this delta-epsilon dependence with \delta =O(\epsilon^{O(1)}).

We study the problem of discriminating Gaussian processes by analyzing the behavior of the underlying probability measures in an infinite-dimensional space. Motivated by singularity of a certain class of Gaussian measures, we first propose a data based transformation for the training data. For a J class classification problem, this transformation induces complete separation among the associated Gaussian processes. The misclassification probability of a component-wise classifier when applied on this transformed data asymptotically converges to zero. In finite samples, the empirical classifier is constructed and related theoretical properties are studied. This is a joint work with Juan A. Cuesta-Albertos.

We shall discuss a new method of computing (integral) homotopy groups of certain manifolds in terms of the homotopy groups of spheres. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we verify a conjecture of J. C. Moore. This is joint work with Somnath Basu.

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.