In 1986, Margulis, using methods from dynamics, proved A. Oppenheim's 1929 conjecture that for every indefinite irrational quadratic form in at least three variables, the values it takes at integer lattice points form a dense subset of the real line. Subsequently, Eskin-Margulis-Mozes proved an associated counting result, giving polynomial asymptotics for the number of lattice points of norm at most $T$ which get mapped to a fixed interval. In joint work with Margulis, we give an effective version if this result for almost every quadratic form.

classification of smooth structures on the complex projective space

We study some properties of Fatou and Julia sets for a family of holomorphic endomorphisms of C^k, k >1. In particular, the family we consider is a semigroup generated by various classes of holomorphic endomorphisms of C^k. Here we generalize a result from the dynamics of the iterates single holomorphic function in C^k, which was proved by Fornaess-Sibony(1998). Also we define recurrent Fatou components in this setup and give a classification result.

Let $(Q, q)$ be a inner product space over a commutative ring $R$, and consider the Dickson-Siegel-Eichler-Roy's subgroup of the orthogonal group $O_R(Q \perp H(R)^n)$, $n \geq 1$. We show that it is a normal subgroup of $O_R(Q \perp H(R)^n)$, for all $n$, except when $n = 2$. This is a joint work with A.A. Ambily.

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.

Let E be an elliptic curve, over a number field K, without complex multiplication. A theorem of Serre says that the image of the Galois representation of the absolute Galois group of K on the l-adic Tate module T_l(E) has open image in GL_2(Z_l). We shall give an outline of Serre's original proof of this theorem, emphasizing the ideas involved.

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.

We will give a construction to state a conjecture of Gan, Gross and Prasad about generic representations in L-Packets. We will use this construction to give a new proof of the classification of the Knapp-Stein R group associated to a unitary unramified character of a torus. Finally we prove the conjecture in a special case.

We use the dynamics on SL(3,R)/SL(3,Z) to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.