Research Overview

A large part of my research consists of studying families of exponential sums over finite fields. My approach mostly uses techniuqes of p-adic analysis, pioneered by Bernard Dwork in the 1960s.

Courses Offered (subject to change)

• MATH 174:  Honors Calculus IV
• MATH 486:  Guided Independent Study
• MATH 568:  Topics in Number Theory

Selected Publications

• Haessig. p-adic unit roots of L-functions over finite fields
• Haessig. Meromorphy of the rank one unit root L-function revisited, Finite Fields Appl. 30 (2014), 191–202.
• Haessig and Sperber. Families of generalized Kloosterman sums
• Haessig and Sperber. L-functions associated with families of toric exponential sums, J. Number Theory 144 (2014), 422–473.
• Haessig and Rojas-Leon. L-functions of symmetric powers of the generalized Airy family of exponential sums: ell-adic and p-adic methods., Int. J. Number Theory 7 (2011), no. 8, 2019–2064.
• Haessig, On the zeta function of divisors of projective varieties with large rank divisor class group, J. Number Theory. Vol 129, Issue 5, May 2009, Pages 1161-1177
• Haessig, L-functions of symmetric powers of cubic exponential sums. J. Reine Angew. Vol 2009, Issue 631, Pages 1 - 57
• Haessig. On the p-adic meromorphy of the function field height zeta function. J. Number Theory (2007) Vol 128/7, pp. 2063-2069.
• Haessig. Equalities, congruences, and quotients of zeta functions in Arithmetic Mirror Symmetry. Appendix to D. Wan's Mirror symmetry for zeta functions. Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics, Vol. 38, (2007) 159-184.
• D. Wan and C. D. Haessig. On the p-adic Riemann hypothesis for the zeta function of divisors. J. Number Theory. 104 (2004) 335-352.

Expositions

• Haessig. Exponentials sums and the Chevalley-Warning theorem (Michigan number theory days. 2010)

Past Student Projects

• Tim McCrossen (S10): An Introduction to Separation of Variables with Fourier Series, Fourier1.nb, Fourier2.nb, Fourier3.nb
• Corey Adams (S10): Special Relativity and Linear Algebras
• Kevin Tang (S10): Newton's method in Mathematica, Newton animation
• James Grotke (S10): An Introduction to the Mathematics of Value-at-Risk
• Jaime Sorenson:
  • Exploring p-adic numbers and Dirichlet characters
  • A review of prime patterns
• Elizabeth Munch: The Ihara zeta function for graphs and 3-adic convergence of the Sierpinski gasket
• Alex Halperin: Complex Multiplication: Kronecker's Jugendtraum for Q(i)
• Max Mikel-Stites: An overview of general encryption techniques with a focus on asymmetric key encryption
• Diane Panagiotopoulos: The mathematics of juggling
• Max Abernethy: RSA techniques
• Michael Weiss: Sylow Theorems