**Address:**

Department of Mathematics - Hill Center

Rutgers, The State University Of New Jersey

110 Frelinghuysen Rd.

Piscataway, NJ 08854-8019

**Office:**

Hill Center 624, Busch Campus

**Email Addresses:**

jcmiller at math or my Johns Hopkins email.

- I graduated in May 2015 and will be with the Department of Applied Mathematics and Statistics at Johns Hopkins University from August 2015.
- Class numbers of totally real fields and applications to the Weber class number problem.
164, no. 4 (2014), 381-397. arXiv version.*Acta Arithmetica* - Real cyclotomic fields of prime conductor and their class numbers. Published electronically in
. arXiv version. Related blog post.*Mathematics of Computation* - Class numbers of real cyclotomic fields of composite conductor.
17, Special Issue A (2014), 404-417, presented at the*LMS Journal of Computation and Mathematics*held in Gyeongju, Korea on August 7-11, 2014.*11th Algorithmic Number Theory Symposium* - Class numbers in cyclotomic
-extensions.**Z**_{p}150 (2015), 47-73. arXiv version. Editor's choice!*Journal of Number Theory* - Real cyclotomic fields of prime conductor and their class numbers II. Submitted.
- Toral automorphisms and applications to norm-Euclidean domains.
- Spring 2014: Math 244
- Fall 2013: Math 152 Section H1: Congratulations! Section H1's median score on the final exam was 90%, versus a median of 60% across all the sections in the course.

**Advisor:**

Henryk Iwaniec

**Research Interests:**

Number theory - especially the Weber problem and other class number problems, cyclotomic fields and * Z_{p}*-extensions, norm-Euclidean domains, the Minkowski conjecture, the principal ideal problem and related lattice problems, class group computation methods, and low-lying zeros of Dirichlet L-functions and Dedekind zeta functions. Recently I have begun to investigate the class groups of ray class fields of imaginary quadratic fields, and the

**Publications:**

**Preprints:**

Previous semesters:

**Research Talks:**

April 23, 2015, ICERM Topical Workshop: Mathematics of Lattices and Cybersecurity, Brown University: "Class numbers of cyclotomic fields."

February 16, 2015, Ph.D. Thesis Defense at Rutgers University: "Class numbers of totally real number fields."

December 5, 2014, RIMS Workshop on Algebraic Number Theory and Related Topics, Kyoto, Japan: "On class numbers of cyclotomic fields and * Z_{p}*-extensions."

September 16, 2014, Number Theory Seminar at Rutgers: "Class numbers of cyclotomic * Z_{p}*-extensions."

August 11, 2014, Eleventh Algorithmic Number Theory Symposium (ANTS-XI) in Gyeongju, Korea: "Class numbers of real cyclotomic fields of composite conductor."

July 17, 2014, Kobe University, Kobe, Japan: "Upper bounds on class numbers and applications to cyclotomic fields."

January 16, 2014, Joint Mathematics Meetings AMS Special Session on Analytic Number Theory: "Computing class numbers beyond Odlyzko's bounds: Real cyclotomic fields."

November 22, 2013, Algebra-Number Theory Seminar at University of Maryland: "Upper bounds on class numbers and applications to cyclotomic fields."

October 27, 2013, MAA New Jersey Section: "Computing class numbers beyond Odlyzko's bounds: Real cyclotomic fields."

October 25, 2013, Collaborative Number Theory Seminar at the CUNY Graduate Center: "Calculating class numbers beyond Odlyzko's bounds: Real cyclotomic fields."

October 10, 2013, Dartmouth College: "Breaking Odlyzko's bounds: Class numbers of real cyclotomic fields."

September 24, 2013, Number Theory Seminar at Rutgers: "Breaking Odlyzko's bounds: Class numbers of real cyclotomic fields.."

August 23, 2013, Graduate Student Research Glimpse, Special Pre-enrollment Program in Mathematics at Rutgers: "Class Numbers of Cyclotomic Fields."

August 10, 2013, Waseda University, Tokyo, Japan: "New results on class numbers of real cyclotomic fields of prime conductor and power of 2 conductor."

December 11, 2012, Number Theory Seminar at Rutgers: "Applications of toral automorphisms."

**Expository Talks:**

March 10, 2015, Financial Mathematics Seminar at Johns Hopkins University: "Cliquet pricing and forward volatility skew."

March 6, 2015, Graduate Pizza Seminar at Rutgers: "The Riemann Zeta Function."

December 11, 2014, Student Number Theory Seminar at Rutgers: "Ranks of elliptic curves."

September 25, 2014, Student Number Theory Seminar at Rutgers: "An introduction to Artin L-functions via examples."

July 16, 2014, Kobe University, Kobe, Japan: "A brief introduction to the Riemann zeta function and other L-functions."

May 2, 2014, Graduate Student Elliptic Curves Seminar at Rutgers: "Elliptic curves of everywhere good reduction."

February 10, 2014, Rutgers Basic Notions and Research Perspectives Seminar at Rutgers: "Class number problems in number theory."

November 20, 2013, Graduate Student Algebraic Geometry Seminar at Rutgers: "Automorphic forms and algebraic groups."

April 5, 2013, Graduate Pizza Seminar at Rutgers: "Who hid my subgroup?"

February 15, 2012, Number Theory Basic Knowledge Seminar at Rutgers: "Cyclotomic Units and Class Groups."

December 6, 2011, Number Theory Basic Knowledge Seminar at Rutgers: "Applications of Gauss Sums."

October 25, 2011, Number Theory Basic Knowledge Seminar at Rutgers: "More on Hecke's Lemma."

March 29, 2011, Number Theory Basic Knowledge Seminar on Homogeneous Spaces at Rutgers: "Laplace-Beltrami Operators."

March 22, 2011, Number Theory Basic Knowledge Seminar on Homogeneous Spaces at Rutgers: "Riemannian manifolds and Laplacians."

February 1, 2011, Number Theory Basic Knowledge Seminar on Homogeneous Spaces at Rutgers: "Galois correspondence for covering spaces."

November 12, 2010, Graduate Pizza Seminar at Rutgers: "Farey sequences."

September 28, 2010, Number Theory Basic Knowledge Seminar at Rutgers: "Stickelberger's Factorization of Gauss Sums."

March 25, 2010, Working Seminar on Rational Points on Surfaces at Rutgers: "Rational points on Cubic Surfaces."

February 15, 2010, Mathematical Finance Career Workshop at Rutgers: "Searching for, Applying to and Getting the Job."

November 3, 2009, Working Seminar on Rational Points on Curves at Rutgers: "Relative Primality and solutions to Ax^{p} + By^{q} = Cz^{r}."

Maintained by jcmiller and last modified May 27, 2015.