Click on the title to download a PDF version of the beamer file. 
Research Talks:
Mellin Transform Techniques for the Mixed Problem in Two Dimensions 
In this talk I will discuss the boundary value problem with mixed Dirichlet and Neumann boundary conditions for the Laplacian and the Lame system in infinite sectors in two dimensions.

Using a potential theory approach the problem is reduced to inverting a singular integral operator (SIO) naturally associated with the problem on appropriate function spaces. Mellin transform techniques are then employed in the study of the spectrum of the aforementioned SIO.
Inveting Double Layers on Lebesgue Spaces on the Boundary of Lipschitz Domains
Double layers arise naturally in connection with boundary value problems (BVPs) for second order elliptic operators with datum in Lebesque spaces on the boundary of the domain in question. In fact the solvability of the Dirichlet and Neumann problems hinges on the ability of inverting an operator of the type (1/2)I+T on Lp where T is of double layer type.

The first part of the talk will be focused on the key tool for inverting such operators on L2, namely Rellich type identities/estimates. Concretely, we shall show the equivalency of the L2 norms for the tangential gradient and the normal derivative of a harmonic function in a Lipschitz domain, whose gradient has a square integrable non-tangential maximal function. In the context of Lipschitz domains, Rellich estimates have been used first by G. Verchota in his Ph.D. thesis to treat BVPs for the Laplacian.

In the second part, I will present a two-dimensional mechanism inspired by the work of M. Riesz which allows us to establish invertibility on Lp for each p bigger than or equal to 2.

Spectral Properties of Hardy Kernel Operators and Application to Second Order Elliptic Boundary Value Problems 
In this talk I will present an old result of David Boyd from the 70's regarding the Lp spectrum of Mellin convolution type singular integral operators, for p between 1 and infinty, and then I will discuss its relevance to establishing well-posedness results for second order elliptic boundary value problems in polygonal domains in two dimensions. 

Jump Formulas for Tempered Distributions
In this talk I will present and prove a useful jump formula in the class of tempered distributions which prefigures similar phenomena at the level of singular integral operators. This is following closely the presentation in the book, "Distributions, Partial Differential Equations, and Harmonic Analysis", by Dorina Mitrea.

Undergraduate Talks:
On the Proof of the Friendship Theorem
In this talk we will discuss the proof of the Friendship Theorem. The origin of the Theorem or who gave it the human touch is still unknown.The proof presented is due to Paul
Erdos,Alfred Renyi and Vera Sos, this was first and most accomplished proof given for the problem,But several other proofs exist. The proof uses a combination of combinatorics, and linear algebra.
I will after show, how this problem leads to the study of Kotzig’s Conjecture. This lecture is appropriate for any undergraduate student in mathematics.

On the Partial Fraction of the Cotangent Function, and its Application to Riemann's Zeta Function
​In this talk we will discuss the partial fraction expansion for the cotangent function, result originally included by Euler in his introduction to Analysin Infinitorum from 1748. This is done using a very clever and simple argument by Gustav Herglotz, known as the “Herglotz trick”.We shall also discuss how, a few years later, Euler was able to use this partial fraction expansion to find the values of the Riemann’s Zeta function evaluated at positive integers. This lecture is appropriate for any undergraduate student in mathematics.

On Polya's Theorem for Polynomials with one Complex Variables
​In this talk I will present and proof a theorem of Polya on Polynomials. The proof is due to George Polya and goes back to the early 1900. The proof is done with some nice tricks and using Chebyshev's Theorem. The proof presented in this talk follows closely the proof in the book "Proofs from The Book" by Martin Aigner and Gunter Ziegler. This lecture is appropriate for any undergraduate student is mathematics.