The vertices of the Tamari lattice are binary trees with an order relation defined by a rotation on binary trees. Such a lattice can be realized geometrically as a convex polyhedron, called the associahedron. I will begin with a survey of this active subject and then will give some extensions of the Tamari lattice in relation with the so-called rational Catalan combinatorics and higher diagonal coinvariants (joint work with L.-F. Pr�ville-Ratelle).

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 shall consider a collocation based quadrature method for obtaining stable approximations for the solution of ill-posed integral equations.

Unlike integer factorization, a reducible holomorphic eta quotient may not factorize uniquely as a product of irreducible holomorphic eta quotients. But whenever such an eta quotient is reducible, the occurrence of a certain type of factor could be observed: We conjecture that if a holomorphic eta quotient f of level M is reducible, then f has a factor of level M. In particular, it implies that rescalings and Atkin-Lehner involutions of irreducible holomorphic eta quotients are irreducible. We prove a number of results towards this conjecture: For example, we show that a reducible holomorphic eta quotient of level M always factorizes nontrivially at some level N which is a multiple of M such that rad(N) = rad(M) and moreover, N is bounded from above by an explicit function of M. This implies a new and much faster algorithm to check the irreducibility of holomorphic eta quotients. In particular, we show that our conjecture holds if M is a prime power. We also show that the level of any factor of a holomorphic eta quotient f of level M and weight k is bounded w.r.t. M and k. Further, we show that there are only finitely many irreducible holomorphic eta quotients of a given level and provide a bound on the weights of such eta quotients. Finally, we give an example of an infinite family of irreducible holomorphic eta quotients of prime power levels.

We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated fourth order Liouville theorem to the problem and possible link with the De Giorgi's conjecture. This is an attempt to prove the McKenna-Walter conjecture which is open for the last two decades.

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.

In this lecture we will first revisit Hodge theory from the point of view of the Heat equation. In particular, we will see how to prove the Hodge theorem by assuming the existence of the Heat Kernel. We will then look at the asymptotic expansion of the Heat Kernel and see how it leads to the signature theorem. In particular we will see how both the Hodge theorem and the signature theorem are special cases of the general statement "Analytical Index = Topological Index" (which is basically the statement of the Atiyah Singer Index Theorem).

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 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.