Sharon M. Frechette

Asssociate Professor

		Email:	sfrechet@holycross.edu
		US Mail:	Dept. of Mathematics & Computer Science College of the Holy Cross 1 College Street Worcester, MA 01610
		Office:	Haberlin 309
		Phone:	(508) 793-2257
		Fax:	(508) 793-3040

EDUCATION

• Ph.D., Dartmouth College, 1997.
  Advisor: Thomas Shemanske.
  Thesis: "Decomposition of Spaces of Half-Integral Weight Cusp Forms".

• A.M.,Dartmouth College, 1994.

• B.A., Boston University, 1988.
  Senior Thesis Advisor: Paul Blanchard.
  

TEACHING EXPERIENCE AND INTEREST

• I have taught courses at all levels of Holy Cross' undergraduate curriculum, including Calculus, Principles of Real Analysis, Algebraic Structures, Linear Algebra, Modern Algebra, Number Theory, Cryptography, and Combinatorics.
• I have also developed specialty courses for the College's Montserrat Program, a first-year seminar program with an emphasis on writing and discussion, and the College Honors Program. Previous seminars have included some flavor of Mathematics of Art and Architecture. This year, I'm teaching a year-long sequence in Montserrat, on the history and mathematics of cryptology:

  Ciphers and Heroes, Fall 2014:

  How are secret codes constructed? What weaknesses allow many of them to be cracked by clever analysts? Welcome to cryptology, the scientific study of encoding and decoding secret messages. We will explore many cryptosystems, investigating their strengths and weaknesses, and surveying their historical developments, setbacks, and implications. This semester we focus on cryptosystems such as the shift ciphers used by Caesar, the Vigenere cipher used during the Victorian era, and most thrillingly, the ENIGMA cipher used during World War II. Along with the mathematics of these ciphers, we will discover fascinating facts about their creators and the clever analysts who crack the codes, including the Polish and British heroes who cracked the seemingly unbreakable ENIGMA.

  Privacy in the Digital Age, Spring 2015:

  How does Amazon.com keep your credit card information secure when you order online? What weaknesses can hackers exploit, in their quest to steal your identity online? Secure electronic communication is vital to today's society, and modern cryptosystems are at the heart of this enterprise. Most of these systems are based on the mathematics of elementary number theory, and the stunning development of public key cryptography, a revolutionary concept born in the computer revolution of the 1970s. This semester we focus on these modern cryptosystems, the visionaries who created them, and the advances in computing that have made them secure.

RESEARCH INTERESTS

• Number Theory: Specifically modular forms, multiple Dirichlet series, special values of L-functions, and finite-field hypergeometric functions as related to traces of Hecke operators.

PUBLICATIONS

• "Gaussian Hypergeometric Functions and Rational Points on Calabi-Yau $n$-folds,'' (with Matthew Papanikolas, Jonathan Root, and Valentina Vega), in preparation.
• Shalika Germs for $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$ are Motivic," (with Julia Gordon and Lance Robson), accepted for publication, Proceedings of the Women in Numbers--Europe Conference, October 2013m, CIRM. [arxiv version]

• "Newly Irreducible Iterates in Families of Quadratic Polynomials," (with Katherine Chamberlin, Emma Colbert, Patrick Hefferman, Rafe Jones, and Sarah Orchard), Involve, 5 (no. 4), (2012), 481--495.   [Preprint version]

• " A Crystal Definition for Symplectic Multiple Dirichlet Series," (with Jennifer Beineke and Ben Brubaker), Multiple Dirichlet series, L-functions and automorphic forms, 37--63, Progr. Math., 300, Birkhauser/Springer, New York, 2012.   [Preprint version]
• " Weyl Group Multiple Dirichlet Series for Type $C$," (with Jennifer Beineke and Ben Brubaker), Pacific J. Math., 254 (1), (2011), 11--46.   [Preprint version]
• " An Interdisciplinary Course: 'On Beauty: Perspective, Proportion, and Rationalism in Western Culture,' (with Alison Fleming and Sarah Luria), refereed Conference Proceedings for Renaissance Banff: Mathematical Connections in Art, Music, and Science (July 2005), accepted for publication, 4 pages.
• "Determinants Associated to Zeta Matrices of Posets," (with Cristina Ballantine and John Little), Linear Algebra Appl., 411C (2005), 364-370.   [Preprint version]
• "Nonvanishing Twists of $GL(2)$ Automorphic $L$-functions," (with Alina Bucur, Ben Brubaker, Gautam Chinta, and Jeffrey Hoffstein), Internat. Math. Res. Notices, 78 (2004), 4211-4239.   [Preprint version]
• "The Combinatorics of Traces of Hecke Operators," (with Ken Ono and Matthew Papanikolas), Proc. of the National Academy of Sciences, USA, 101 (2004), 17016-17020.   [Preprint version]
• "Gaussian Hypergeometric Functions and Traces of Hecke Operators," (with Ken Ono and Matthew Papanikolas), Internat. Math. Res. Notices, 60 (2004), 3233-3262.  [Preprint version]
• "A Classical Characterization of Newforms with Equivalent Eigenforms in S_{k + 1/2}(4N,\chi)," J. London Math. Soc., (3) 68 (2003), 563--578.   [Preprint version]
• "Nonvanishing of Special Values of $L$-functions for Quadratic Twists of Newforms," preprint.
• "Hecke Structure of Spaces of Half-Integral Weight Cusp Forms," Nagoya Math. J., 159 (2000), 53--85.   [Preprint version]

CONFERENCES

I was one of the co-organizers for the 18th Annual Workshop on Automorphic Forms and Related Topics, which was held March 21-24, 2004 at the University of California, Santa Barbara.

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Created by Sharon M. Frechette on Jan. 24, 2004
Last modified by Sharon M. Frechette on October 14, 2013