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:	337 Swords Hall
		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, Mathemematical 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. In recent years, I have taught a year-long sequence in Montserrat on the history and mathematics of cryptology. This seminar consists of the following courses:

  Ciphers and Heroes (Fall)

  How are secret codes constructed? What weaknesses allow them to be cracked by clever analysts? Welcome to cryptology, the scientific study of encoding and decoding secret messages. We’ll explore the mathematics behind codes and analyze the strengths and weaknesses of historically significant cryptosystems, including the nomenclator used by Mary Queen of Scots in her quest to dethrone Queen Elizabeth, and most thrillingly, the ENIGMA machine used during World War II. Along the way, we’ll delve into the lives of code creators and the clever analysts who cracked the codes, including the Polish and British heroes who cracked the seemingly unbreakable ENIGMA. (Students in this course should have a strong competence in high school algebra, and an aptitude for analytical thinking.)

  Privacy in the Digital Age (Spring)

  How does Amazon.com keep your payment information secure when you order online? Does the nature of social media influence the way you present yourself and interact with others online? Is there a divide between your actual self, and your digital self? Secure electronic communication is vital to modern society in many ways, and cryptosystems are at the heart of this. These mathematical systems rely on the stunning development of public key cryptography, a concept born in the computer revolution of the 1970s. In this seminar, we will focus on these modern cryptosystems, the visionaries who created them, and the advances in computing that make them secure. We will also examine the significant impact that electronic communication has on the self. (Students in this course should have a strong competence in high school algebra, and an aptitude for analytical thinking.)

RESEARCH INTERESTS

• Number Theory: item Number theory, in particular modular forms, $L$-functions and multiple Dirichlet series, and both \\classical and finite-field hypergeometric functions.
• Algebraic combinatorics, as connected to representation theory and number theory.
• Elliptic curves and their connections to modular forms.

PUBLICATIONS

• " A Cubic Transformation Formula for Appell-Lauricella Hypergeometric Functions over Finite Fields," (with H. Swisher and F.-T. Tu), {\em Res. Number Theory,} 4 (2018), no. 2, Art. 27.

• 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