## Number Theory

The Department of Mathematics at the University of Illinois at Urbana-Champaign has long been known for the strength of its program in number theory. The department has a large and distinguished faculty noted for their work in this area, and the graduate program in number theory attracts students from throughout the world. At present over twenty students are writing dissertations in number theory. Each semester upper level graduate courses are offered in a variety of topics in analytic, algebraic, combinatorial, and elementary number theory.

### Weekly Seminars

One or two regularly scheduled seminars are held each week, with lectures being given by faculty, graduate students, and visiting scholars. The lectures may be elementary introductions, surveys, or expositions of current research.

### Number Theory Conferences

Illinois has hosted a long running series of number theory conferences, the most recent being

- 2007 Illinois Number Theory Fest
- 2009 Illinois Number Theory Celebration
- 2010 Illinois Number Theory Conference
- 2011 Illinois Number Theory Conference
- 2014 Illinois Number Theory: A Conference in Memory of the Batemans and Heini Halberstam
- 2015 Illinois Number Theory Conference

In addition, every two of three years Illinois hosts the Midwest Graduate Number Theory conference for Graduate Students and recent PhDs, a unique type of conference organized almost entirely by graduate students.

### Number Theory faculty and their research interests

**Scott Ahlgren** [Homepage] — Modular forms and number theory.

**Patrick Allen** [Homepage] — Galois representations and automorphic forms.

**Bruce
Berndt** [Homepage] —
Ramanujan's notebooks, elliptic functions, theta-functions, *q*-series, continued fractions, character sums, classical analysis.

**Florin
Boca** [Homepage] —
Diophantine approximation, spacing statistics.

**Iwan
Duursma** [Homepage] —
Cryptography, algebraic coding theory, algebraic curves.

**Kevin
Ford** [Homepage] —
Arithmetic functions, probabilistic number theory, Weyl sums, comparative prime number theory, sieve theory, Riemann zeta function.

**Elena Fuchs**
[Homepage] —
Number theory and geometric group theory.

**Martin Luu** [Homepage] —
Various aspects of the Langlands program. This includes compatibility results between local and global Langlands correspondences, applications of the modularity lifting results introduced by Wiles, the role of p-adic deformations of the various notions appearing in the Langlands program, and also relations to mathematical physics in particular dualities of quantum field theories.

**Bruce Reznick ** [Homepage] —
Sums of squares of polynomials, combinatorial number theory.

**Alexandru Zaharescu** [Homepage] —
Number theory.

### Faculty in related areas

József Balogh — graph theory, extremal combinatorics, additive combinatorics

Nathan Dunfield — 3-dimensional geometry and topology, hyperbolic geometry, geometric group theory, experimental mathematics, connections to number theory.

Vesna Stojanoska — Homotopy theory and its relations to arithmetic.

### Postdoctoral Faculty in Number Theory

**Sun Kim** (Visiting Asst Prof) —
Analytic Number Theory and Enumerative Combinatorics.

**Detchat Samart** (Doob Postdoc) —
Mahler measures, Modular forms, L-functions, Arithmetic of algebraic varieties.

**Xin Zhang** (Doob Postdoc) —
Analytic number theory, spectral theory, homogeneous dynamics and the interplay between them.

### Emeritus Faculty in Number Theory

**Harold G. Diamond [Homepage]** —
Prime number theory, sieves, connections with analysis

** A.J. Hildebrand** [Homepage] —
Analytic and probabilistic number theory, asymptotic analysis

**Leon McCulloh ** —
Algebraic number theory, relative Galois module structure of rings
of integers, class groups of integral group rings, Stickelberger relations

**Ken
Stolarsky** [Homepage] —
Diophantine approximation, special functions, geometry
of zeros of polynomials

**Stephen Ullom **[Homepage] —
Galois modules, class groups

### Courses

Each year, the following one semester courses are offered:

**Math 453, Elementary Theory of Numbers**(upper undergraduate level)

**Math 530, Algebraic Number Theory**(beginning graduate course)

**Math 531, Analytic Theory of Numbers, I**(beginning graduate course)

In addition, at least four topics courses are also offered each year, with student input helping to determine the choice of the topics courses. In recent years, enrollment in these courses has been excellent, typically ranging from 10 to 25. We list below some of the topics courses taught between 2005 and 2013:

**Algebraic:**Class field theory, elliptic curves, modular forms, Iwasawa theory, algorithmic number theory, cryptography, Abelian varieties.

**Analytic:**Exponential sums, Asymptotic methods in analysis, Riemann zeta-function and L-functions, sieve methods, probabilistic number theory, elliptic functions, Continued fractions, Modular forms, Theory of partitions, additive number theory, Uniform distribution, Distribution of sequences, hypergeometric functions, Diophantine approximation, polynomials, elementary methods, anatomy of integers.

### Graduate Awards

The ** Bateman Prize ** and the ** Bateman Fellowship **
are given annually for
outstanding research in number theory. They are named for former
Professor Paul T. Bateman, who was on the faculty from 1950 to 1989
and continued being active in the group's activities until shortly
before his death in 2012. Bateman served
as Department Head for the years 1965-1980.

Recipients of the Bateman Prize in Number Theory

Recipients of the Bateman Fellowship in Number Theory

### Recent former postdocs in number theory

Francesco Cellarosi (Postdoc, 2012-2015)

Armin Straub (Postdoc, 2012-2015)

Xiannan Li (Postdoc, 2011-2013)

Jimmy Tseng (Postdoc, 2011-2012)

Youness Lamzouri (Postdoc, 2010-2012)

Paul Pollack (NSF Postdoc, 2008-2011)

Mathew Rogers (NSF Postdoc, 2008-2011)

Andrew Schultz (Postdoc, 2007-2010)

Jeremy Rouse (Postdoc, 2007-2010)

Sung-Geun Lim (Postdoc 2007-2009)

Maria Sabitova (Postdoc, 2006-2009)

Nayandeep Deka Baruah (Postdoc 2006-2007)

Emre Alkan (Postdoc, 2003-2006)

Matt Boylan (Postdoc, 2002-2005)

Ae Ja Yee (Postdoc, 2000-2003)