Talks Given

• Joint Mathematics Meetings
  Location: San Francisco, California
  January 16, 2010
  Title: Ramanujan-type congruences in partition-theoretic counting functions.
  (10 minutes)
  
  

• Midwest Number Theory Conference for Graduate Students VI
  Location: University of Wisconsin-Madison
  November 7-8, 2009
  Title: Ramanujan Congruences in Partition-Theoretic Functions
  (20 minutes)
  
  
• Number Theory Seminar, University of Wisconsin-Madison
  September 29, 2009
  Title: Ramanujan-type congruences in partition-theoretic counting functions
  (60 minutes)
  
  
• Number Theory Seminar, University of Illinois at Urbana-Champaign
  Tuesday, September 22, 2009
  Title: The number of modular forms with Ramanujan-type congruences
  Abstract: Ramanujan famously proved three congruences for the partition function like p(5n+4) = 0 modulo 5. He speculated there were no other such congruences and in 2003 Ahlgren and Boylan proved there were indeed no other such congruences. We place this result in context by providing the exact probability that a modular form has this type of congruence.
  
  
• Palmetto Number Theory Series X
  Invited Speaker
  Title: Non-existence of Ramanujan congruences in partition theoretic functions
  Abstract: We use modular forms to answer some fundamental questions on congruences in a class of generating functions from combinatorial number theory. In particular, we show how to find all Ramanujan congruences in some counting functions related to the partition function. In 1984, George Andrews introduced 2-colored F-partitions and showed that their counting function $c\phi_2 (n)$ exhibited congruences $c\phi_2 (2n+1) = 0 mod 2$ and $c\phi_2 (5n+3) = 0 mod 5$ analogous to Ramanujan's congruences for the usual partition function. We prove there are no other such congruences and provide a framework to find all Ramanujan type congruences for a class of combinatorial generating functions which also includes overpartitions, overpartition pairs, and crank differences.
  (45 minutes)
  
  
• 23rd Annual Workshop on Automorphic Forms and Related Topics
  Location: Bucknell University, Lewisburg, Pennsylvania
  March 10-13, 2009
  Title: Ramanujan Congruences in Inverses of Modular Forms
  Abstract: Ramanujan famously found congruences in the partition function of the form $p(n\ell + a) = 0 \mod \ell$. Since the partition generating function is essentially the inverse of a modular form, Ahlgren and Boylan (following Kiming and Olsson) use Tate cycles of level one mod \ell modular forms to find all such congruences. We generalize this and provide a method to find all of the finitely-many Ramanujan congruences in the inverse of any modular form of level four which is non-vanishing on the upper half plane.
  (30 minutes)
  
  
• Number Theory Seminar, University of Illinois at Urbana-Champaign
  Tuesday, March 3, 2009
  Title: Simple congruences in the coefficients of modular forms
  Abstract: It is well-known that the partition function has the congruence p(5n+4) = 0 mod 5. Although there are also simple congruences like this for the primes 7 and 11, Ahlgren and Boylan have proven there are no others. In this talk we show how Ramanujan's theta operator can be used to prove the non-existence of simple congruences (except at small primes) in many modular forms which vanish only at the cusps. Applications include overpartitions, crank differences, and 2-colored F-partitions.
  (50 minutes)
  
  

• Number Theory Seminar, University of Illinois at Urbana-Champaign
  Thursday, April 10, 2008
  Title: Overpartitions and Maass forms
  Abstract: An overpartition of $n$ is a partition in which the first occurrence of a part may be overlined. The overpartition rank generating function for $n$ lying in an arithmetic progression is not quite modular, but it is the holomorphic part of a Maass form. By computing the nonholomorphic part explicitly, we find linear combinations of the rank generating functions which are modular. Overpartitions are just one of many combinatorial objects which may be studied by looking at their shadows.
  (50 minutes)
  
  

• Fifth Annual Midwest Number Theory Conference for Graduate Students
  Location: University of Wisconsin-Madison
  November 3-4, 2007
  Title: Drinfeld Modular Forms
  
  

• Polynomials over Finite Fields and Applications
  Location: BIRS, Banff, Canada
  November 18-23, 2006
  Title: When do pentanomials divide trinomials over F_2?
  Abstract:Over F_2, up to reciprocals, no pentanomial of degree m divides a trinomial of degree at most 2m except for 25 specific exceptions, all with degree m < 14, and one infinite family of pentanomials. A careful case analysis reveals that for large degree the coefficients cancel in a “staircase”-like manner. This divisibility property allows the construction of orthogonal arrays of strength 3.
  [This is a joint work with L. Moura, D. Panario, B. Stevens and Q. Wang.]
  
  

• Graduate Student Seminar, Carleton University, Canada
  February 6, 2004
  Title: The Wonders of the Transformation Shift Register
  
  

• Fq7: Seventh International Conference on Finite Fields and Applications
  Location: Toulouse, France
  May 5-9, 2003
  Title: Linear transformation shift registers
  [This is a joint work with D. Panario]
  
  

Conferences Attended

• Sixth Annual Midwest Number Theory Conference for Graduate Students
  Location: University of Wisconsin-Madison
  November 7-8, 2009
  
  
• Palmetto Number Theory Series X
  Location: Armstrong Atlantic State University, Savannah, GA
  September 19-20, 2009
  
  
• International Conference: Mock theta functions and applications in combinatorics, algebraic geometry, and mathematical physics
  Location: Max Planck Institute for Mathematics, Bonn
  May 25-29, 2009
  
  
• 23rd Annual Workshop on Automorphic Forms and Related Topics
  Location: Bucknell University Lewisburg, Pennsylvania
  March 10-13, 2009
  
  

• CMI/MSRI Workshop: Modular Forms and Arithmetic
  Mathematical Sciences Research Institute, Berkeley, CA
  June 28 to July 2, 2008
  
  
• The Hawaii workshop on the arithmetic of modular forms
  University of Hawaii at Manoa
  May, 21-24(25), 2008.
  
  
• Conference on Partitions, q-series, and Modular Forms
  Location: University of Florida, Gainesville, FL.
  March 8-16, 2008
  The student workshop (March 8-11) on Mock Theta Functions was given by Sharon Garthwaite (Bucknell).
  
  

• Fifth Annual Midwest Number Theory Conference for Graduate Students
  Location: University of Wisconsin-Madison
  November 3-4, 2007
  
  

• Polynomials over Finite Fields and Applications
  Location: BIRS, Banff, Canada
  November 18-23, 2006
  
  

• Tenth Seminar on Analysis of Algorithms
  Location: MSRI, Berkeley, CA
  June 14-18, 2004
  
  

• Fq7: Seventh International Conference on Finite Fields and Applications
  Location: Toulouse, France
  May 5-9, 2003
  
  
• 15th Applied Algebra, Algebraic Algorithms, and Error Correcting Codes Symposium
  Location: Toulouse, France
  May 12-16, 2003