PhD - | Harvard University |
---|

**Interests**

- Though I like to learn about all kinds of mathematics and science (mostly physics, genetics), my main research interests are in number theory and arithmetic geometry, and my main work has been in developing various aspects of function field arithmetic. The personal home page should give some idea about my research work.

**Teaching**

- Diophantine Equations (568) (UR)
- Elliptic curves and modular forms (530) (UR)
- Function Field Arithmetic (569) (UR)
- Honors Calculus I-IV (171, 172, 173, 174) (UR)
- Function Field Arithmetic (519) (UA)
- Algebraic Number Theory (514AB, 531 at UR)
- Algebraic Geometry (536AB, 538 at UR)
- Abstract Algebra (415/515AB, 236 at UR)
- Algebraic Coding Theory (Math/ECE 539)
- Algebra (511AB)
- Complex analysis and Riemann surfaces (520AB)
- Game Theory (479/579)
- Linear Algebra (413/513)
- Real variables (265 at UR)
- Mathematical Cryptology (233 at UR)
- Financial Mathematics (210 at UR)
- Transition to higher mathematics (200W at UR)
- Ordinary Differential Equations (254)
- Calculus I and II (123, 124H, 125, 129, 129 H, and at Michigan, Harvard)
- Complex Variables (424/524, 467 at UR)
- Complex Variables with Applications (421/521)
- Theory of Numbers (446/546 and at Michigan, 230 at UR)
- Advanced Calculus (Minnesota)
- Linear and nonlinear multivariable analysis (Minnesota)
- Fundamental Structures of Algebra (Minnesota)
- Algebraic Number Theory (University of Bombay)
- Multivariable calculus and linear algebra (Harvard)

** PhD Students**

- Javier Diaz-Vargas: On zeros of characteristic p zeta functions(1996)
- Aaron Ekstrom: On the infinitude of elliptic Carmichael numbers(1999)
- Justin Miller: On p-adic continued fractions and quadratic irrationals(2007)
- Huei Jeng chen (Co-advised): Distribution of Diophantine approximation exponents for algebraic quantities in finite characteristic (2011)
- Aleksandar Petrov: On A-expansions of Drinfeld modular forms (2012)
- Alejandro Lara Rodriguez : Relations between multizeta values, and Bernoulli-Carlitz numbers (2013)
- Somjit Datta: On Jacobian Theta functions, Roger Series and Identities of Rogers-Ramanujan type (2013)
- George Todd: Linear relations between multizeta values (2015)
- Shuhui Shi: Multiple Zeta Values over F_q[t] (2018)
- Current student: Qibin Shen.
- (For abstracts of thesis, links to final versions, and papers based on them, see `abstracts' on my personal page. Official Thesis copies themselves are also publicly available through Math Sci Net, or the University.)

** Masters students**

- Jason Belnap: Ideal class groups (98)
- Aaron Ekstrom: Asymptotically good function field towers (99)
- Selin Kalaycioglu: Construction of expanders, Ramanujan graphs and their Applications (04),
- Justin Miller: q-definability and algebraicity of Laurent series over F_q(t) (05),
- Joshua Chesler: The four color theorem (06),
- Eric Rudd: Why the Borda count delivers the maximum efficiency (06),
- Alejandro Lara-Rodriguez: Some conjectures and results about multizeta values for F_q[t] (09).
- George Todd: Congruences for F_q[t] (12).
- Alex (Lap K) Tao: Class number problem and elliptic curves (13).
- Model theory and Diophantine geometry
- Computation, algebraicity, and transcendence
- ABC conjecture
- Child's drawings and curves over number fields
- Codes and curves with many rational points
- Continued fractions and exponential
- Finite automata and algebraic power series
- Old and new approaches to Fermat's last theorem
- Baker functions and Gauss sums
- Diophantine approximation exponents
- L-values, periods and Mahler measure
- Recognizing primes in polynomial time (Colloquium)
- Discriminants, class numbers and composition laws
- Proof of Catalan conjecture
- Absolute Galois group of rationals and Grothendieck-Teichmuller group
- Multizeta and Ihara Power series for function fields
- Introduction to rigid analysis and Tate curves
- Higher Diophantine approximation exponents for function fields
- Bernoulli numbers and class groups in function fields
- Modular forms and modular symbols in function fields
- Higher congruences, derivatives and zeta values (Colloquium)
- Fermat-Wilson congruences for function fields and number fields (UR)
- Power sums of polynomials, generalizations and applications (UR)
- Schmidt subspace theorem and automatic real numbers (UR)
- Surprising symmetries in distribution of prime polynomials (UR)
- Infinite families of finite dynamical systems (UR)
- Participated in Cryptography related outreach activities of the Southwest Regional Institute in Mathematical Sciences.
- Participating in Arizona Winter School since its inception
- Participating in training workshops for Olympiad and graduate students in India

**Selected Publications and Papers****Seminar Talks at UA (followed by) at UR****Outreach**