Extended CV
This is something between a CV and a diary, intended as a place to keep a record of my mathematical activities. A more formal, summarized version of this information–with the irrelevant bits removed–can be found in my CV.
Education
I'm a fourth year Ph.D. student working with Dino Lorenzini.
I got my M.S. degree from Georgia Tech. My advisor there was Matt Baker. I took courses in real, complex, functional, and harmonic analysis; basic algebra, algebraic geometry, and algebraic number theory. I didn't write a thesis, but instead passed the doctoral qualifying exams.
I studied at the USB for six months in 2005, supervised by Pedro Berrizbeitia (pedrob@usb.ve). I took courses on primality testing, measure theory, and Fourier analysis.
In addition to taking courses at the USB, I studied at the IVIC with Carlos Di Prisco (cdiprisc@ivic.ve) and learned a lot of Ramsey theory.
Here I got my B.S. degree. Besides passing the standard courses in algebra, topology, geometry, and analysis, I took topics courses on chaos theory, numerical analysis, and mathematical logic.
Research
My research interests are in the field of arithmetic geometry. My dissertation work has to do with modular curves and torsion of elliptic curves defined over number fields of low degree.
As a student at UGA I have participated in several VIGRE research groups:A VIGRE research group led by Robert Rumely. My main contribution, joint with John Doyle, was to design and implement an algorithm for computing all numbers of bounded height in a given number field. Details may be seen here. We then used this work to compute preperiodic points of many quadratic polynomial maps, in a joint paper with Xander Faber.
A VIGRE research group led by Elham Izadi. We learned about Green's conjecture on the Clifford index of a curve in its canonical embedding. Though the conjecture is stated for smooth curves, it is known to be true for generic irreducible nodal curves. I studied and computed with a very particular example of reducible nodal curves (namely, binary curves) to see whether the conjecture seems to hold for these.
A VIGRE research group led by Angela Gibney. We attempted to disprove (and later to prove!) special cases of the Fulton conjecture.
A VIGRE research group led by Neil Lyall. We studied several instances of the interaction between combinatorics and Fourier analysis: the Szemeredi-Trotter incidence theorem, Besicovitch/Kakeya sets, sum/product estimates, and the Erdös distance problem.
These are a couple of independent research projects in which I ended up proving something that was already known, and having a lot of fun in the process:
A project supervised by Elham Izadi. I figured out exactly at what point the Hilbert function and Hilbert polynomial of a projective variety start agreeing. In the process I found a nice formula for the Hilbert function of any complete intersection, which turned out to be very useful for speeding up computations.
A problem I worked on at the suggestion of Matt Baker. The theorem was already known, but I was not able to find a complete proof of it anywhere, so I spent a couple of months constructing my own.
Teaching Experience
This was the second time I taught Calculus I for science and engineering majors, and I organized the course quite differently to the way it was the first time. There were optional individual presentations on topics like history of calculus, strange examples or counterexamples, proofs of relevant theorems, short programming projects, and others. I also asked that every student participate in one of several group projects that were meant to have them work together on something challenging that used what they learned in class for a practical purpose. It was fun to teach the course this way, because there was a lot of interaction among students, and between myself and the students, both in class and outside.
I taught Calculus I to a group of 40 science and engineering majors. Having never taught this course before, I did not try any new ideas. The course followed a fairly traditional style based on the syllabi of previous instructors. It was somewhat difficult to keep all the students interested, because many of them were already familiar with everything we were doing in class, while others had never seen it. The result of this was that sometimes the more advanced students were not learning much from my lectures. If I were to teach this course again I would make sure that I have a way to engage these students and keep them interested.
I taught Calculus I to a group of about 40 students. I set out to teach the best calculus course possible, and tried a few new ideas for organizing the class and explaining the material. The ideas were largely successful, and I very much enjoyed teaching the class because students really seemed to be progressing. There were, however, a few skills which I did not manage to fully convey to a portion of the class. Discussing these issues with more experienced instructors gave me some ideas to try in the future.
I taught Precalculus to a group of about 40 students. It was a new and certainly challenging experience having to teach concepts at such a low level. My duty was mainly to plan and deliver good lectures; all the grading was done by computer since the students took homework, quizzes, and tests by using the online system WebAssign. I attended two weekly meetings run by Lisa Townsley where we discussed all the details involved in teaching this course for the first time.
I taught Calculus II for the first time. I had been a TA for this class twice, but I still learned a lot from being the instructor. This was my first time teaching a summer course, and it required careful planning of time.
This was my first time as the instructor of a course. I taught Calculus III to a group of about 60 students, divided into two sections for recitation. My responsibilities were to give two lectures each week, write quizzes and tests, assign and grade computer projects, and coordinate the TA's. Quite a rewarding experience.
I ran biweekly problem sessions for the courses Calculus II, Finite Mathematics, and Survey of Calculus. My duties also included grading quizzes and exams, and I would often give review sessions before tests, at the students' request.
I ran weekly problem sessions for the courses Calculus I, Calculus II, Analysis I, General Topology, Geometry II, and p-adic Numbers.
As a supplement to my teaching, I have participated in three teaching seminars:
This was by far the most useful teaching seminar I've been in. It was an intensive three-week course filled with many kinds of activities: class discussions, model lectures, math history talks, a movie about teaching, presentations by math professors on particular aspects of teaching, talks by university staff about academic honesty and student disabilities, and practical sessions on technology in the classroom. Each student wrote a model statement of teaching philosophy, which was a difficult but highly educational task. We read and discussed the book "How To Teach Mathematics", by Steven Krantz, and we read an outline of the book "What the Best College Teachers Do", by Ken Bain. It was the discussions of these books that I found very educational. The seminar was organized by Robert Rumely.
In this seminar led by Malcolm Adams, we had class discussions about several subjects that may arise when teaching: How to prepare a syllabus, how to deal with dishonest or disrespectful students, the importance of having a teaching mentor, being mindful of board work, etc. We also gave mock lectures and offered constructive criticism on others' teaching style.
In this seminar led by Rena Brakebill, we had weekly meetings with all current TA's to discuss issues that came up in the courses we were teaching at the time. The instructor also videotaped a lecture given by each student, and privately suggested ways of improving the teaching style.
Talks
This was a 15-minute summary of a year's work together with John Doyle and Xander Faber on an algorithm for computing the set of rational preperiodic points of a given quadratic polynomial over a specified number field. I also gave a conjecturally complete list of preperiodic graphs arising over quadratic number fields.
I presented a fast algorithm which John Doyle and I designed for computing all numbers of bounded height in a given number field.
In celebration of BSD's 50th anniversary, I gave this survey talk aimed at undergraduate students. I explained the statement of the BSD conjecture for elliptic curves, its generalization by Tate, and some connections to my own research.
I discussed Qing Liu's algorithm for computing the reduction type of a curve of genus 2. A first step in his algorithm is to compute the stable reduction of the curve, and it was this step that I explained in detail.
I discussed the transition that occured in algebraic geometry from the study of projective varieties to the study of schemes. Most of the talk was focused on motivating the concept of a scheme with many examples. I also gave some idea of the uses of sheaf cohomology for the study of schemes.
I presented an important theorem of Roth in the field of diophantine approximation. I recounted the history of the theorem, its generalization in the Schmidt subspace theorem, and applications of these to studying integral points on varieties.
I presented the results of a computational project I undertook to figure out exactly when the Hilbert function and Hilbert polynomial of a variety start agreeing.
In this very long talk aimed mostly at undergraduate students, I described the fields of algebraic number theory, algebraic geometry, and commutative algebra; and how they interact in arithmetic geometry.
Awards
A competitive university-level award given to graduate students showing particularly strong potential to produce an excellent thesis. The award allows students to focus on their dissertation for one year with no teaching duties.
NSF VIGRE Fellowship (2008, 2010)
Awarded to outstanding Hispanic graduate students at Georgia Tech.
The University awards full scholarships for academic excellence to all students who maintain a GPA of at least 9 (out of 10).
Conferences Attended
A two-week summer school organized by Colorado State University and taking place at the Universidad de Costa Rica. It consisted of two short courses: Maximal Curves and Applications to Coding Theory, by Rachel Pries and Beth Malmskog, and Elliptic Curves and Applications to Public Key Cryptography, by Jeff Achter and Cassie Williams.
Four week-long courses on various subjects. I attended the course on deterministic primality tests, by Pedro Berrizbeitia.
A rather large number of talks on various aspects of logic, including the history and philosophy of logic, model theory, and algebraic logic. There were also a few intensive courses running the week before the conference. I sat in on Itay Neeman's course, "Introduction to Determinacy".