We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

The main conjecture of Iwasawa theory (formulated by R. Greenberg) predicts a close relationship between certain algebraic objects (Selmer groups) and certain analytic objects (p-adic L-functions). We will review the statements of the main conjecture in various contexts. At the end, we will see the algebraic analog of certain p-adic factorization formulae obtained by Gross and Dasgupta.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field F is called exceptional APN, if it is also APN on infinitely many extensions of F. In this article we consider the most studied case of F=F_2n. A conjecture of Janwa-Wilson and McGuire-Janwa-Wilson (1993/1996), settled in 2011, was that the only monomial exceptional APN functions are the monomials x_n, where n= 2k+ 1 or n= 22k?2k+ 1 (the Gold or the Kasami exponents, respectively). Aubry, McGuire and Rodier conjectured that the only exceptional APN function is one of the monomials just described. One of our results is that all functions of the form f(x) =x^2^k+1+h(x) (for any odd degree h(x) \neq 2^l+ 1) with (k, l) = 1), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture. One ingredient in deriving this result is the proof we present of our earlier conjecture on the relatively primeness of exceptional multivariate polynomials in the Gold case. Up until now, the main tool used by most researchers in the study of exceptional APN functions, has been the method of Janwa, McGuire and Wilson to prove the absolute irreducibility of multivariate polynomials. The algorithmic approach is based on intersection multiplicity theory and Bezout’s theorem, and computations initiated in Janwa and Wilson. Our techniques of establishing absolute irreducibility rely on repeated hyperplane intersections, linear transformations, reductions,and the known APN monomial functions. We apply the estimates of Weil, Bombieri, Deligne, Lang-Weil, Ghorpade-Lachaud on rational points on varieties over finite fields to demonstrate exceptional properties. The absolute irreducible hypersurfaces are related to hyper-plane sections of Fermat varieties, and are of independent interest. We will discuss applications of our results in the construction of algebraic geometric codes, cryptography, combinatorics, finite geometry, sequence design, and Ramanujan graphs.

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.

I will talk about my results on Modica type estimates for parabolic reaction diffusion equationsobtained in a series of joint work with Prof. Nicola Garofalo. I will describe the connection ofthese estimates to a geometric conjecture of De Giorgi. I will also show how one can obtaincertain rigidity/symmetry type results as a consequence of these estimates.

Voevodsky's conjecture states that numerical and smash equivalence coincide for algebraic cycles. I shall explain the conjecture in more detail and talk about some of the examples for which this conjecture is known.

The Moser-Trudinger embedding has been generalized by Adimurthi and Sandeep to a weighted version. We prove that the supremum is attained, generalizing a well-known result by Flucher, who has proved the case \beta= 0

We will start by briefly recalling some of the tools from analytic number theory which go into the study of L-functions. This will include the summation formula, the trace formula (Petersson/Kuznetsov) and the circle method. The main focus will be the subconvexity problem (and its applications). We will briefly recall the ideas of Weyl and Burgess, and then in some detail cover the amplification technique as developed by Duke, Friedlander and Iwaniec. We will also discuss the works of Michel and his collaborators. After discussing the scopes and shortfalls of the amplification technique, we will move towards more current techniques. The ultimate goal will be to discuss the status of GL(3) subconvexity.