Superirreducibility of Polynomials, Binomial Coefficient Asymptotics and Stories from my Classroom

In the first main section of this thesis, I investigate superirreducible polynomials over fields of positive characteristic and also over Q and Z. An n-superirreducible polynomial f(x) is an irreducible polynomial that remains irreducible under substitutions f(g(x)) for g of degree at most n. I find asymptotics for the number of 2-superirreducible polynomials over finite fields. Over the integers, I give examples of both families of superirreducible polynomials and families of irreducible polynomials which have an obstruction to superirreducibility. The writing and results on finite fields in this section have come from a collaboration with Jonathan Bober, Dan Fretwell, Gene Kopp and Trevor Wooley. The results over Z and Q are my own independent work. In the second section I determine the asymptotic growth of certain arithmetic functions A(n), B(n) and C(n), related to digit sum expansions. I consider these functions as sums over primes p up to n. I obtain unconditional results as well as results with better error terms conditional on the Riemann Hypothesis. The results over primes have come from collaboration with Jeff Lagarias. I also independently solved the analogous problem of summing over all positive integers b ≤ n. Finally in the third section, I discuss mathematical education via the lens of interviews and interactions. I consider my role as a teacher through multiple real-life anecdotes and what those stories have taught me. My interviews were conducted with young mathemati- cians from Bronx, NY that I got the opportunity to talk to as a result of my employment with Bridge to Enter Advanced Mathematics during the summer of 2019. The anecdotes I give are from working with teenaged students from a variety of different cultural, socio- economical and mathematical backgrounds.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162876/1/hjkl_1.pd

Elliptic genera from multi-centers

I show how elliptic genera for various Calabi-Yau threefolds may be understood from supergravity localization using the quantization of the phase space of certain multi-center configurations. I present a simple procedure that allows for the enumeration of all multi-center configurations contributing to the polar sector of the elliptic genera\textemdash explicitly verifying this in the cases of the quintic in $\mathbb{P}^4$, the sextic in $\mathbb{WP}_{(2,1,1,1,1)}$, the octic in $\mathbb{WP}_{(4,1,1,1,1)}$ and the dectic in $\mathbb{WP}_{(5,2,1,1,1)}$. With an input of the corresponding `single-center' indices (Donaldson-Thomas invariants), the polar terms have been known to determine the elliptic genera completely. I argue that this multi-center approach to the low-lying spectrum of the elliptic genera is a stepping stone towards an understanding of the exact microscopic states that contribute to supersymmetric single center black hole entropy in $\mathcal{N}=2$ supergravity.Comment: 30+1 pages, Published Versio

Pairings in Cryptology: efficiency, security and applications

Abstract The study of pairings can be considered in so many di�erent ways that it may not be useless to state in a few words the plan which has been adopted, and the chief objects at which it has aimed. This is not an attempt to write the whole history of the pairings in cryptology, or to detail every discovery, but rather a general presentation motivated by the two main requirements in cryptology; e�ciency and security. Starting from the basic underlying mathematics, pairing maps are con- structed and a major security issue related to the question of the minimal embedding �eld [12]1 is resolved. This is followed by an exposition on how to compute e�ciently the �nal exponentiation occurring in the calculation of a pairing [124]2 and a thorough survey on the security of the discrete log- arithm problem from both theoretical and implementational perspectives. These two crucial cryptologic requirements being ful�lled an identity based encryption scheme taking advantage of pairings [24]3 is introduced. Then, perceiving the need to hash identities to points on a pairing-friendly elliptic curve in the more general context of identity based cryptography, a new technique to efficiently solve this practical issue is exhibited. Unveiling pairings in cryptology involves a good understanding of both mathematical and cryptologic principles. Therefore, although �rst pre- sented from an abstract mathematical viewpoint, pairings are then studied from a more practical perspective, slowly drifting away toward cryptologic applications

Bifurcation analysis of the Topp model

In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao