**John Vastola**

**Undergraduate Major:** Math and Physics

**Future Plans:** Ph.D. in Mathematical Physics

# John Vastola

##### John Vastola was born and raised in Oviedo, Florida. He is pursuing Bachelor's degrees in Mathematics and Physics. His commitment to helping others understand science has led him to work as a math tutor at the UCF Math Lab, to collaborate with art students through the National Science Foundation funded ICubed project on posters presenting research to the general public, and to present dramatic science demonstrations to elementary, middle, and high school students. As the President of the Math Club and Vice-President of the Society of Physics Students, he works to get his fellow undergraduates excited about science, engaged in science outreach, and to build a sense of community within the math and physics departments. His research interests include mathematical physics, elliptic curves, and novel areas like infinite sums and continued radicals. John plans to obtain his Ph.D. in Mathematical Physics and become a college professor.

#### Systematically Evaluating Sums Using Integral Transforms, with Applications to Statistical and Quantum Physics

##### Conducted at the University of Central Florida through the McNair Scholars Program Summer Research Fellowship

**Awards: ** Best Poster Presentation at the 2015 Kennesaw Mountain Undergraduate Mathematics Conference.

** Mentor: **Dr. Costas Efthimiou, Assistant Professor – Department of Physics, University of Central Florida.

** Abstract: **
Evaluating sums analytically is a problem that is easy to pose and to give approximate solutions to, but that is difficult
to exactly solve in general. Many of the results that are known are byproducts of Fourier analysis, which
requires guessing that a series corresponds to a function; for obvious reasons, this is undesirable. A method of evaluating
sums using integral transforms is proposed which can reproduce many results obtained using other techniques. In particular,
representing polynomials as Laplace transforms gives some nontrivial exact results.
Some applications of the method are demonstrated, and extensions using integral representations of frequently appearing
functions are suggested. One useful representation of the gamma function is supplied, and used to provide both well-known
and more obscure results. Interestingly, the application of this integral representation to evaluating sums
suggests the introduction of a novel integral transform, which itself can be
used to evaluate sums. Some of thetransform's properties are given, and its usefulness
in other areas (like solving differential equations) is touched upon. Some physical problems involving
the partition functions of statistical mechanics, and some infinite sums appearing in quantum mechanics, are considered.

#### On the Boundedness or Unboundedness of the Ranks of Elliptic Curves

##### Conducted at the University of Central Florida as part of the ICubed National Science Foundation funded research project

** Mentor: **Dr. Costas Efthimiou, Assistant Professor – Department of Physics, University of Central Florida.

** Abstract: **
An elliptic curve is a nonsingular curve of genus one with at least one rational point. It is well known that a group law (with a simple geometric interpretation) can be constructed on a given elliptic curve; together with the group law, the set of rational points on an elliptic curve becomes an abelian group. In particular, by the Mordell-Weil Theorem, the set becomes a finitely generated abelian group. The minimal number of linearly independent nontorsion generators required—in addition to the finite number of torsion points—to generate a given point on a curve is called the rank of that curve. If elliptic curves are considered over the rational field, it is not known whether rank can be arbitrarily high. The highest known rank is at least 28, and it is thought that almost all elliptic curves have rank 0 or 1, making it difficult to improve on the current record. We will discuss the problem of finding curves of high rank in general, and consider some novel techniques used. We will also consider the opposite problem. In the function field case, rank was demonstrated to be unbounded by finding a family of curves in
which rank is unbounded. Honda's conjecture controversially asserts that this strategy will not work in the
number field case; it states that rank is bounded within a family of curves. We will consider numerical evidence in
support of Honda's conjecture, and discuss the implications for the Birch and Swinnerton-Dyer conjecture,
an outstanding open problem.

#### The Effect of Impurities on the Superconductivity of BSCCO-2212

##### Conducted at the University of Central Florida as part of the Honors in the Major undergraduate thesis program.

** Mentor: ** Richard Klemm, Ph.D., Associate Professor, Department of Physics, University of Central Florida

** Abstract: **
BSCCO is a high-temperature cuprate superconductor whose electronic structure is currently poorly understood. In particular, it is unclear whether its order parameter is consistent with s-wave or d-wave behavior. Leggett has suggested that its order parameter might take a certain form that is consistent with d-wave behavior. While some experiments on the surface of BSCCO seem to support this conclusion, other experiments have suggested that its order parameter is instead s-wave in the bulk. We present some quantum field theoretic calculations in the spirit of Abrikosov and Gorkov that suggest such an order parameter cannot be correct. We will demonstrate that having such an order parameter would mean that BSCCO's critical temperature would go to zero if it is sufficiently impure, contradicting experimental evidence otherwise. These calculations
lend support to the hypothesis that BSCCO is an s-wave rather than a d-wave superconductor

**Title:** TComputational Modeling of Short Baseline Neutrino Oscillation Experiments and a Search for the Existence of a Fourth Neutrino

##### Conducted at Vanderbilt University as part of the National Science Foundation-funded 2015 REU in Physics and Astronomy

** Mentor: ** David Ernst, Ph.D., Professor, Department of Physics and Astronomy, Vanderbilt University

** Abstract: **
Neutrino oscillations—the fact that one flavor of neutrino can later be measured as another flavor with some probability—lie outside the purview of
the Standard Model of particle physics, and thus represent an exciting frontier that is currently poorly understood. Short baseline reactor experiments conducted in the 1980's and 1990's to better understand them were consistent with no oscillations, contradicting contemporary higher precision experiments by Daya Bay and T2K, among others; this suggests that their analyses were flawed. Mention et. al. assert that a 3.5% increase in the flux of neutrinos in these experiments due to revised antineutrino spectra of U-235, Pu-239, Pu-241, and U-238 can partially account for this discrepancy. It is also believed that a three, or perhaps four neutrino model will better fit experimental data. To test these hypotheses, we have constructed a computational model of six short baseline experiments and reproduced their results to within reasonable accuracy.
We hope to make the aforementioned adjustments to our model and analyze how they change our results.