The use of differential equations (equations involving functions and their derivatives) to model physical phenomenon traces back to Sir Isaac Newton and his Philosophiæ Naturalis Principia Mathematica. Their applicability is seen in economics, engineering, physics and many other areas.

My graduate research focused on studying properties of the data-to-solution map associated to the initial value problem that models the motion of an inviscid and incompressible fluid. The first derivation of such system of partial differential equations is attributed to Leonhard Euler. Modern derivations use Newton’s second law of motion, and the laws of conservation of mass and energy. In particular, my work was motivated by the following observations:

The question of global (in time) existence, uniqueness and regularity of solutions of the Euler equations in three dimensional space is a major open problem in analysis and it is closely related to the Clay Millennium Prize Problem for the Navier-Stokes equations, where the fluid is assumed viscous.

At the beginning of the 20th century Jacques Hadamard noticed that accumulation of small errors in the initial data of mathematical models of physical phenomena could produce large error in the solutions. He then proposed that besides the existence and uniqueness of solutions, such models should have an extra property: the solutions should depend continuously on the initial data. 

My doctoral dissertation is available here:

Regularity properties of the solution map of the incompressible Euler equations.

A more detailed and .pdf formated version can be found here: arXiv.

Many STEM majors take single variable integral calculus. A central topic there is that of approximating quantities, more generally functions. The simplest functions one learns about are polynomials and the notion of accuracy quickly translates to number of terms; thus one spends some time studying infinite sums and their convergence.  In that vein, Gauss theorem comes in to fill in some of the gaps in the theory of convergence of summations exactly when other tests have shown to be "inconclusiveness". 

During my second vist to OSU, the summer before graduate school,  I decided to spend some weeks with my colleague and friend Samuel Perez Ayala studying under Dr. Aurel Stan. We worked on providing with a new mathematical proof to this theorem based on observation made by Stan prior to the summer program, that combined, a generalization of the Gel'Fand's Trick, infinite products and many applications of Young's Inequality. Besides the required summer school, we also attended (sometimes) the graduate immersion program at OSU and we studied baby Rudin together, there for the first time, on our own.

Naaria can complete a job in 1 hour and Degran in 1.5 hours. How much time will they spent together finishing one job?

This was the question that motivated this particular exploration project. The inequality between the arithmetic and harmonic means of n positive numbers, can be interpreted as the fact that the time necessary for n workers, each working at a constant speed (productivity), to complete a job working together, is less than or equal to 1/(n^2) of the sum of the individual times required by the workers laboring alone to complete that job. We show first that if we consider n workers, who are becoming tired in time, and whose productivity decreases exponentially in time at a rate that depends only on the difficulty of the job performed, but not on the workers, then the above inequality still holds. The proof relies on Jensen inequality. Finally, we extend this result to the case in which we have n workers, becoming tired continuously in time, and whose order of productivity remains the same in time, that means, if worker 1 productivity is greater than or equal to worker 2 productivity at time 0, then worker 2 productivity never exceeds worker 1 productivity.

I was lucky to find in Dieff Vital and Raemeon Cowan, two very accomplished friends who also wanted to learn from Dr. Aurel Stan while in the summer school SAMMS at Ohio State University during the summer of 2014.

During the summer of 2013 I had my first undergraduate research experience at MSRI-UP. The program was lead at the time by Dra. Ivelisse Rubio and the research was directed by Dra. Rosa Orellana. At the time, Dr. Alexander Diaz was a graduate student at Notre Dame and he was the research mentor and guide assigned to my team who also included Rita Zevallos and Francis Castro.

During two weeks we were exposed to various questions and techniques in enumerative combinatorics and for 4 weeks we tackled our question of counting signed permutations that admitted a peak set. That is, permutations for which we knew the exact positions of values larger than their neighbors. For details, see the arxiv link above. To this day, I still believe this experience changed my appreciation and understanding of what it means to do mathematics.