My research area is in harmonic analysis. I am currently focusing my attention on the study of oscillatory and singular integrals.
My area of research is Dynamical Systems, particularly systems that exhibit chaotic (ergodic) behaviour. I work on geometric systems such as geodesic flow on surfaces and billiard motion. I study the mechanisms that produce chaotic behavior as well as trying to understand the transition from chaotic to non-chaotic motion.
I am interested in bringing the beauty and excitement of mathematics to the general public; Bryn Mawr students working with me have made a computer generated movie of the Costa Surface that was exhibited at the Maryland Science Center, an interactive web site about chaotic billiard motion and computer generated pictures of surfaces whose geodesic flow is chaotic. More information about these and other projects can be found at this website.
I am an applied mathematician who focuses on biological problems; the goal of my work is to develop mathematical models motivated by biological phenomena. Practical limitations in experimental procedures leave gaps in our understanding of important scientific questions, and mathematical models help to fill these holes by providing additional insights. I employ a variety mathematical tools and techniques to accomplish this, including nonlinear dynamical systems, stochastic processes, and numerical simulation. My current projects include modeling physiological mechanisms underlying type 2 diabetes, anticoagulation therapy, and female hormone regulation.
Helen G. Grundman
My mathematical research interests span a wide range of topics in the areas of algebra and number theory. Below are examples of some area in which I am currently working:
- Diophantine equations over number rings: using algebraic number theory and computational methods to solve Diophantine equations over number rings (rings of integers of algebraic number fields).
- Thue equations with few terms: bounding the number of integer solutions of Thue equations with a specified number of nonzero terms.
- Galois realizability problems: characterizing fields that have normal extensions with specified small 2-groups as Galois groups.
- Elementary number theory: discovering properties of the integers and interesting subsets of the integers, such as Harshad (Niven) numbers and happy numbers.
My research interests are in geometry and topology. More specifically, I work in symplectic topology and contact geometry. I have worked on the symplectic camel problem, symplectic homology, symplectic packings, and legendrian knots. To learn more about the types of problems that interest me, click on this link.
- Continuous dependence on modeling for ill-posed evolution problem, Matthew Fury, 2010. Advisor: Rhonda Hughes.
- Relative Knovanov-Jacobsson classes for spanning surfaces, Jonah Swann, 2010. Advisor: Paul Melvin.
- Bounding the number of solutions to tetranomial Thue equations, Daniel Wisniewski, 2010. Advisor: Helen Grundman.
- Lp Estimates for Oscillatory Singular Integral Operators and Marcinkiewicz Integral Operators, Ayako Fukui, 2009. Advisor: Leslie Cheng.
- Determining Lower Bounds for Packing Densitites of Non-Layered Patterns Using Weighted Templates, Cahtleen Battiste-Presuitti, 2008 Advisor: Walter Stromquist.
- Legendrian Torus Links, Jennifer Dalton, 2008. Advisor: Lisa Traynor.
- The Existence of Elliptic periodic Orbits in the Smoothed Bunimovich Stadium, Sherry Teti, 2008. Advisor: Victor Donnay.
- Continuous Dependence Results for Inhomogeneous Ill-Posed Problems, Beth Campbell-Hetrick, 2006. Advisor: Rhonda Hughes.
- Generating Family Invariants for Legendrian Links, Jill Jordan, 2005. Advisor: Lisa Traynor.
- Smooth Approximation of Singular Perturbations of the Laplacian, Walter Huddell, 2002. Advisor: Rhonda Hughes.
- The Arithmetic Genus of Threefolds Defined by Extended Hilbert Modular Groups, Amber Salzman, 2002. Advisor: Helen Grundman.
- Symplectic Packings of Cotangent Bundles of Tori, Jean Mastrangeli, 1997. Advisor: Lisa Traynor.
- Sewn up and Surgered Swen up Link Exteriors: Surgery Presentations & Formulas for Lescop’s Invariant, Gowri Meda, 1997. Advisor: Paul Melvin.
Recent M.A. Theses
- Expansion of Boundedness Results for Certain Oscillatory Integral Operators, Lise Chlebak, 2011. Advisor: Leslie Cheng.
- Galois Groups of Certain Polynomials of Prime Degree, Rebecca Rebhuhn-Glanz, 2011. Advisor: Helen Grundman.
- Mathematical Modeling of Malaria, Amy Veprauskas, 2010. Advisor: Victor Donnay.
- On the Boundedness of Certain Oscillatory Integral Operators, Manal Zaher, 2010. Advisor: Leslie Cheng.
- Properties of Class Groups of a Family of Cyclic Cubic Fields, Jaclyn Lang, 2009. Advisor: Helen Grundman.
- On the Boundedness of Oscillatory Integral Operators in Harmonic Analysis, Sarah Khasawinah, 2009. Advisor: Leslie Cheng.
- Solutions to a Parametrized Family of Relative Thue Equations, Amanda Hittson, 2009. Advisor: Helen Grundman.
- Aditi Vashist, 2008
- A New Proof of the Spectral Theorem, Milena Redzic, 2008. Advisor: Rhonda Hughes.
- Thomas's Work on Split Families of Cubic Thue Equations, Wen Gao, 2007. Advisor: Helen Grundman.
- Lp Boundedness of Oscillatory Integral Operators, Anna Gordon, 2007. Advisor: Leslie Cheng.
- Applications of Heatlets to Ill-Posed Problems, Emily McNabb, 2007. Advisor: Rhonda Hughes.
- Kirsten Kemp, 2006.
- Arnold's 4-Cusp Conjecture, Devasish Majumdar, 2005. Advisor: Lisa Traynor.
- Tangling Legendrian Knots, Cristina Nistor, 2005. Advisor: Paul Melvin.
- The Arnold Invariants of Plane Curves, Emi Arima, 2005. Advisor: Lisa Traynor.
- Class Numbers and Other Numerical Invariants of Imaginary Quadratic Fields, Laura Hall, 2004. Advisor: Helen Grundman.
- Morse Theory, Kim Urso, 2004. Advisor: Lisa Traynor.
- Producing Positive Lyapunov Exponents on the Sphere, Gina Calderaio, 2003. Advisor: Victor Donnay.
- A Simple-Homotopy Classification of Certain CW-Complexes by Reidemeister Torsion, Ben Allen, 2002. Advisor: Paul Melvin.
- Constructing Brownian Motion with Wavelets, Beth Campbell, 2002. Advisor: Rhonda Hughes.
- Chekanov’s Decomposition Invariant for Legendrian Knots, Jane Holsapple, 2002. Advisor: Lisa Traynor.
- Power Basis Generators in Cyclotomic Fields, Jill Jordan, 2002. Advisor: Helen Grundman.
- Weight Systems, Rachael Thomas, 2001. Advisor: Paul Melvin.
- On Galois Realizability of Groups of Order 32, Grisha Stewart, 2001. Advisor: Helen Grundman.