: Normalized Matching Property (NMP) is a simple and natural generalization of the famous Hall's marriage condition for bipartite graphs, to the setting when the sizes of the two vertex classes are distinct. It is a well-studied notion in the context of graded posets and several well-known ones are known to have it (for instance the boolean lattice or the poset of subspaces of a finite dimensional vector space). However, in this talk, we will consider NMP with a 'random twist': if for every possible edge in a bipartite graph, we toss a coin in order to decide if we keep it or not, how biased must the coin be to expect to have NMP in the graph with high probability? We shall arrive at a sharp threshold for this event. Next, what can we say about explicit graphs that are known to behave 'random-like'? One of the earliest notions of a pseudorandom graph was given by Thomason in the 80s. We shall prove an 'almost' vertex decomposition theorem: every Thomason pseudorandom bipartite graph admits - except for a negligible portion of its vertex set - a partition of its vertex set into trees that have NMP and which arise organically through the Euclidean GCD algorithm.

In this talk, we shall define Milnor $K$-groups and Chow groups. We study various properties of these and also theorems relating both groups. In particular, we talk about Bloch's formula and Totaro's map. Towards the end, I shall state my results in this direction, which are joint work with Prof A. Krishna.

In this talk, we will discuss some well known regularity issues concerning equations of the form $u_t - div |\nabla u|^{p-2} \nabla u = 0$ for $12$) and the singular case ($p<2$) separately. Moreover in several instances, the estimates are not even stable as $p\rightarrow 2$. In this talk, I shall discuss two regularity estimates and give an overview on how to obtain uniform $L^{\infty}$ and $C^{0,1}$ estimates in the full range $\frac{2N}{N+2}

R.G. Swan and J. Towber showed that if (a 2 , b, c) is a unimodular row over any commutative ring R then it can be completed to an invertible matrix over R. This was strikingly generalised by A.A. Suslin who showed that if (a r! 0 , a1, . . . , ar) is a unimodular row over R then it can be com- pleted to an invertible matrix. As a consequence A.A. Suslin proceeds to conclude that if 1 r! ∈ R, then a unimodular row v(X) ∈ Umr+1(R[X]) of degree one, with v(0) = (1, 0, . . . , 0), is completable to an invertible matrix. Then he asked (Sr(R)): Let R be a local ring such that r! ∈ GL1(R), and let p = (f0(X), . . . , fr(X)) ∈ Umr+1(R[X]) with p(0) = e1(= (1, 0, . . . , 0)). Is it possible to embed the row p in an invertible matrix? Due to Suslin, one knows answer to this question when r = d + 1, without the assumption r! ∈ GL1(R). In 1988, Ravi Rao answered this question in the case when r = d. In this talk we will discuss about the Suslin’s question Sr(R) when r = d − 1. We will also discuss about two important ingredients; “homotopy and commutativity principle” and “absence of torsion in Umd+1(R[X]) Ed+1(R[X]) ”, to answer Suslin’s question in the case when r = d − 1, where d is the dimension of the ring.

: Motivated by the theory of compact surfaces, Yamabe wanted to show that on a given compact Riemannian manifold of any dimension there always exists a (conformal) metric with constant scalar curvature. It turns out that solving the Yamabe problem amounts to solving a nonlinear elliptic partial differential equation (PDE). The solution of the Yamabe problem by Trudinger, Aubin and Schoen highlighted the local and global nature of the problem and the unexpected role of the positive mass theorem of general relativity. In the first part of my talk, I will survey the Yamabe problem and the related issues of the compactness of solutions. In the second part of the talk, I will discuss the higher-order or polyharmonic version of the Yamabe problem: "Given a compact Riemannian manifold (M, g), does there exists a metric conformal to g with constant Q-curvature?" The behaviour of Q-curvature under conformal changes of the metric is governed by certain conformally covariant powers of the Laplacian. The problem of prescribing the Q-curvature in a conformal class then amounts to solving a nonlinear elliptic PDE involving the powers of Laplacian called the GJMS operator. In general the explicit form of this GJMS operator is unknown. This together with a lack of maximum principle makes the problem difficult to tackle. I will present some of my results in this direction and mention some recent progress.

We discuss the classification of tuples of commuting matrices over a finite field, up to simultaneous conjugation.

The purpose of these two lectures is to provide the proof of Cohen’s structure theorem for complete local rings (which Cohen proved in his PhD thesis 1942, Johns Hopkins University under the guidance of Oscar Zariski). In these lecture we deal with the equicharacteristic case. We give a modern and concise treatment by using the notion of formal smoothness which was introduced by Grothendieck in 1964 in EGA Chapter IV. It is closely connected with the differentials and throws new light to the theory of regular local rings and used in proving Cohen’s structure theorem of complete local rings.

: The purpose of these two lectures is to provide the proof of Cohen’s structure theorem for complete local rings (which Cohen proved in his PhD thesis 1942, Johns Hopkins University under the guidance of Oscar Zariski). In these lecture we deal with the equicharacteristic case. We give a modern and concise treatment by using the notion of formal smoothness which was introduced by Grothendieck in 1964 in EGA Chapter IV. It is closely connected with the differentials and throws new light to the theory of regular local rings and used in proving Cohen’s structure theorem of complete local rings.

In this talk, we will discuss an important object in number theory : L-functions. A well-known example is the Riemann zeta function. We will focus on the arithmetic properties of the special values of L-functions. These have very interesting applications to congruences between modular forms. We will give a gentle introduction to these concepts highlighting several examples and important results in the literature. We will present recent joint research with Abhishek Saha and Ralf Schmidt regarding special L-values and congruences of Siegel modular forms.

We will consider mainly the following situation. Let R,S be complete normal local domains over an alg. closed field k of char. 0 such that S is integral over R. Our aim is to describe three ideals in S; I_N, I_D, I_K (Noether, Dedekind, Kahler differents resp.) each of which capture the ramified prime ideals in S over R. In general these three ideals are not equal. An important special case when all are equal is when S is flat over R. The case when there is a finite group G of k-automorphisms of S such that R is the ring of invariants is already very interesting. Then many nice results are proved. These include works of Auslander-Buchsbaum, Chevalley-Shephard-Todd, Balwant Singh, L. Avramov, P. Roberts, P. Griffith, P. Samuel,.... I will try to discuss all these results. I believe that these results and ideas involved in them will be very valuable to students and faculty both.