### Seminars

#### By Year

##### Venue: LH-1, Mathematics Department

The talk will focus on congruences modulo a prime $p$ of arithmetic invariants that are associated to the Iwasawa theory of Galois representations arising from elliptic curves. These congruences fit in the framework of some deep conjectures in Iwasawa theory which relate arithmetic and analytic invariants.

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##### Venue: Lecture Hall - I, Dept. of Mathematics

In this talk, I will discuss threshold estimation for a regression function in some different settings. The threshold can either be a change–point, i.e. a point of jump discontinuity in an otherwise smooth curve, or the first time that a regression function crosses a certain level. Both problems have numerous applications in a variety of spheres, like biology (pharmacology, dose-response experiments) and engineering. Our goal is to estimate thresholds of this type given a fixed budget of points to sample from, but with the flexibility that batch sampling can be done in several stages, so that adaptive strategies are possible. Our strategy is to use multistage zoom-in procedures to estimate the threshold: an initial fraction of the sample is invested top come up with a first guess, an adequate neighborhood of the first guess is chosen, more points are sampled from this neighborhood and the initial estimate id updated. The procedure continues thus, ending in a finite number of stages. Such zoom-in procedures result in accelerated convergence rates over any one–stage method. Approximations to relative efficiencies are computed and optimal allocation strategies, as well as recipes for construction of confidence sets discussed.

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##### Venue: Lecture Hall - I, Dept. of Mathematics

Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear.We also discuss the case of arbitrary input sequence.

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##### Venue: Lecture Hall - I, Dept. of Mathematics

This is an expository talk whose aim will be to give an introduction to SLE (Schramm-Loewner evolution), discovered by Oded Schramm in 2000 to describe many critical statistical mechanical systems.

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##### Venue: Lecture Hall - III, Dept. of Mathematics

I will give a brief overview of Wiles’ proof of Fermat’s Last Theorem, and explain the connection between modular forms and elliptic curves via Galois representations e.g. the Taniyama-Shimura conjecture. In the second half, I’ll explain some recent results on p-adic modular forms and deformations of Galois representations. If time permits, I’ll outline a future project on ranks of elliptic curves.

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