Geometry and topology at UH center around the study of manifolds, with the incorporation of methods from algebra and analysis.

The department conducts research into many topics in or related to geometry and topology. These include the following:

• low dimensional differential geometry and topology;
• knot theory;
• hyperbolic structures;
• infinite groups;
• the Novikov conjecture;
• the Baum-Connes conjecture;
• non-commutative differential geometry;
• geometric quantization;
• symplectic topology;
• index theory and spectra of operators associated with manifolds;
• pure and applied control theory;
• sub-Riemannian geometry;
• singularities;
• index theory of compact transformation groups;
• equivariant cobordism and surgery;
• equivariant cohomology.

Visitors and Friends

Colleagues who have visited us in recent years include: Greg Brumfiel, Robert Gardner, Richard Hamilton, Jerry Kaminker, Jerrold Marsden.

Faculty and students interested in the applications of mathematics are an integral part of the Department of Mathematics; there is no formal separation between pure and applied mathematics, and the Department takes pride in the many ways in which they enrich each other. We also benefit tremendously from close collaborations with faculty and students in other departments at UH.

The Department regularly offers courses in ordinary and partial differential equations and their numerical solution, discrete applied mathematics, the methods of mathematical physics, mathematical biology (including a
certificate in mathematical biology), the mathematical aspects of fluid and solid mechanics, approximation theory, scientific computing, numerical linear algebra, and mathematical aspects of computer science. Courses in probability theory, stochastic processes, data analysis are regularly offered as are courses in combinatorial and convex optimization. Our students are encouraged to take courses of mathematical interest in these and offering from other departments.

Visitors and Friends

Colleagues who have visited us in recent years include: John Chadam, Olga Cordero-Brana, Robert Gardner, Jerrold Marsden.

Mathematical analysis encompasses many areas involving complex function theory, differential equations, harmonic analysis, operator theory, probability theory and many other areas. Very roughly, mathematical analysis generally involves spaces of functions and their limits.

Visitors and Friends

Colleagues who have visited us in recent years include: Pat Ahern, Ann Castelfranco, Boo Rim Choe, Paul Cohen, Alex Gottlieb, Kathryn Hare, Walter Hayman, Hyungwoon Koo, Mark Lawrence, Walter Rudin, Joel Shapiro, Ken Stephenson, Alexander Volberg, Yitzhak Weit, Larry Zalcman.

Areas of active research in algebra and number theory in the department are generally situated within algebraic number theory, arithmetic geometry, the theory of modular forms, and universal algebra. Specific subfields include: arithmetic dynamics, arithmetic statistics, computational number theory, Iwasawa theory, and mock modular forms of integral and half-integral weight, primarily from an algebraic/p-adic prospective.

Visitors and Friends

Colleagues who have spent extended visits with us in recent years include: Khalid Benabdallah, Greg Brumfiel, Danielle Gondard, Keith Kearnes, Claude Levesque, Jon Merzel, Otto Mutzbauer, Nilakatan Sankaran, Tara Smith, Lutz Strüngmann and Charles Vinsonhaler.