Let X= (X,O_X) be a projective scheme over a field and with twisting sheaf O_X(1). Let F be a coherent sheaf of O_X-modules. The cohomology table of F is defined as the family h F:= (hi(X,F(n)))(i,n)?N0×Z. We give a survey on results about the set of cohomology tables hC={hF|(X,F)?C},for certain classes C of pairs (X,F). We particularly look at the following questions: (1) Under which conditions is the set hC finite, if C is the class of all pairs (X,F) for which F has a given dimension? (2) Under which conditions is the set hC finite, if C is the class of all pairs (X,F) in which X = P‘r is a given projective space and F is an algebraic vector bundle over X? (3) What can be said if X runs throught all smooth complex projective surfaces and F=OX is the structure sheaf of X? Our results are related to the theory of Hilbert functions, Hilbert polynomials and Hilbert schemes, but also to Castelnuov-Mumford regularity and to vanishing results for cohomology.

An affine domain A over a field k is called a sub-principle plane if it satisfies the following: 1. A=k[p,q] \subset k[u,v] where k[u,v] is a polynomial ring in two variables over k. 2. There is a polynomial g \in k[u,v] such that k[p,q,g] = k[x,y]. We will discuss the problem of identifying properties of p,q which ensure the condition of A being a sub-principle plane. The problem is clearly important in order to determine if the polynomial F(X,Y,Z) defining the kernel of the homomorphism k[X,Y,Z] \mapsto k[u,v] is an abstract plane. A detailed description of A is hoped to help with the solution of the three dimensional epimorphism Problem.

The classical version of Glivenko Cantelli thm asserts uniform convergence of the empirical cdf to the true cdf for iid real valued random variables. In this talk we extend that result to regenerative sequences, exchangeable sequences and stationary sequences all with possible delays. We discuss the extension to the vector case. This is based on joint work with Vivek Roy.

The drift of a real-valued random sequence at a particular time is equal to the conditional expected change in the sequence over the next time step, given the information known about the sequence up to the given time. If the drift is zero the sequence is known as a martingale. The actual change in the sequence is equal to the drift plus a conditional mean zero deviation. After each time step, a new drift can be calculated, and the random deviations from the drift add up over time. It is thus important to bound the cumulative effect of the deviations, to quantify whether the values of the sequence over a long period of time evolve according to the drift. This talk identifies an incomplete list of bounds implied by drift that have been used in many applications, including to analyze the performance of randomized algorithms for non-convex global optimization problems.

In this work, we investigate the extremal behaviour of left-stationary symmetric \alpha- stable random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls (in the Cayley graph) of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying quasi-invariant action of the free group but is different from what happens in the case of alpha stable random fields indexed by Zd. The presence of this new dichotomy is confirmed by the study of stable random fields generated by the canonical action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Sullivan. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a new class on point processes that we have termed as randomly thinned cluster Poisson processes. This limit too is very different from that in the case of a lattice. This talk is based on a joint work with Sourav Sarkar, who carried out a significant portion of the work in his master’s dissertation at Indian Statistical Institute.

Brauer-Thrall conjectures for representation theory of Artin algebra's was proved many years ago (in 1968). However the techniques invented by Auslander to prove this conjecture has found more applications than just proving the original conjectures. These techniques have been extended in commutative algebra to study Maximal Cohen-Macaulay modules over Cohen-Macaulay isolated singularities. I will also discuss a result of mine in this direction.

Many moduli spaces in algebraic geometry are constructed as quotients of algebraic varieties by a reductive group action using geometric invariant theory. In this talk we explain two such examples: moduli of coherent sheaves on a projective variety and moduli of quiver representations. In both cases, we introduce and compare two stratifications: a Harder-Narasimhan stratification associated to the notion of stability for the moduli problem and a stratification coming from the geometric invariant theory construction. In nice cases, these stratifications can be used to give recursive formulas for the Betti numbers of the moduli spaces.

We will prove that the second stable homotopy group of the Eilenberg Maclane space is completely determined by the Schur multiplier. Then we will discuss about the Schur multipliers of Noetherian groups. Time permitting, we will also discuss Noether's Rationality problem. All of the above will be shown as an application of a group theoretical construction.

http://www.math.iitb.ac.in/~seminar/colloquium/colloq-02-nov-16.pdf

In this talk we present the main problems and some recent results on Teichmüller spaces of surfaces with boundary.