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
- 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.
- 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:
- 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