Research

Advisor:
Henryk Iwaniec (Rutgers University)

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 Z_p-extensions built out of these. I am also working on problems associated to real quadratic fields, as well as elliptic curves over real quadratic fields.

Publications:

• Class numbers of totally real fields and applications to the Weber class number problem.   Acta Arithmetica 164, no. 4 (2014), 381-397. arXiv version.


• Real cyclotomic fields of prime conductor and their class numbers.   Published electronically in Mathematics of Computation. arXiv version.  Related blog post.


• Class numbers of real cyclotomic fields of composite conductor.   LMS Journal of Computation and Mathematics 17, Special Issue A (2014), 404-417, presented at the 11th Algorithmic Number Theory Symposium held in Gyeongju, Korea on August 7-11, 2014.


• Class numbers in cyclotomic Z_p-extensions.  Journal of Number Theory 150 (2015), 47-73. arXiv version.  Editor's choice!

Preprints:

• Real cyclotomic fields of prime conductor and their class numbers II.   Submitted.
• Toral automorphisms and applications to norm-Euclidean domains.

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 John Miller and last modified August 12, 2015.