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This series of talks will cover various notions of zeta functions associated to graphs.

Singular perturbation and the boundary layer phenomena appear very often in several applications today. Presence of a small parameter in the differential equation changes the behavior of the solution rapidly. Uniform meshes are inadequate for the convergence of numerical solution. The aim of the present talk is to consider the adaptive mesh generation for these problems based on moving mesh strategy. I shall start this talk with a small introduction on singular perturbation. The analytical and computational difficulties in the existed methods will be discussed. The concept of moving mesh strategy and its implementation will be explained. A nonlinear system of differential equations with delay will be considered to show the difference between a priori and a posteriori generated meshes after discussing my researches on this field. The parameter uniform a posteriori based error estimate for a system of reaction-diffusion problems and a priori based convergence analysis for a parabolic convection-diffusion problem will be presented with computational evidence.

We investigate the $L^p$ boundedness of the lacunary maximal function $A_rf$ associated to the spherical means on the Heisenberg group. By suitable adaptation of an approach of M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions, which lead to new unweighted and weighted estimates. In order to prove the result, several properties of the spherical means have to be accomplished, namely, the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator. This is a joint work with Hait, Bagchi and Roncal.

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I’m thinking of a person in the audience. How long will it take you to find whom, using only Yes/No questions? We consider this puzzle, which underlies search theory, with a twist: I’m choosing the person according to a known distribution, and your goal is to minimize the expected number of questions. How does the performance of your strategy depend on the type of question you’re allowed to ask? On the type of distribution I am allowed to choose? What happens if I can lie? Joint work with Yuval Dagan (MIT), Ariel Gabizon (Zcash), Daniel Kane(UCSD), Shay Moran (IAS).

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