RFG – The geometry and combinatorics of branched covers
2009-2010

RFG leader: Brian Osserman

Additional faculty: Michael Kapovich, Fu Liu, Motohico Mulase, Jennifer Schultens.

Hurwitz theory – the study of branched covers of the Riemann sphere – has a long history, and a very interdisciplinary nature. Basic questions can be stated equivalently in topology, elementary combinatorial group theory, complex geometry, and algebraic geometry, so they can be understood and studied even at the undergraduate level. Moreover, applications range from these areas to string theory and number theory. The purpose of this Research Focus Group is to introduce undergraduate and graduate students to the field, and to provide a range of research problems suitable for participants at all levels.

In addition to the organized activities listed below, participants will meet regularly with faculty to receive guidance on research. Anyone interested in participating in or in finding out more about the RFG is encouraged to email Brian Osserman at .


Activities

Courses and organized seminars are listed below. In addition, anyone interested in pursuing research during the year is encouraged to contact us as early as possible to discuss possible projects, some of which could be started with essentially no prerequisites. Math 248AB is in fact independent of the RFG, but RFG participants may find it helpful to attend the course.

Fall — Math 248A: Algebraic Geometry

Taught by Brian Osserman. An introduction to classical algebraic geometry. Topics include:
  • Affine and projective varieties
  • Gluing and abstract varieties
  • Morphisms of varieties
  • Smooth varieties
  • Smooth curves and divisors on them
  • Covers of curves

Fall — Math 280: Hurwitz numbers and Hurwitz theory

Taught by Motohico Mulase. Without assuming backgrounds in algebraic geometry or moduli theory, will offer a survey of current areas of excitement in Hurwitz theory. Topics include:
  • What are Hurwitz numbers?
  • Combinatorics of Hurwitz numbers
  • Representation theory and Hurwitz numbers
  • Integrable PDEs and Hurwitz numbers a la Okounkov
  • Moduli theory, localization, and Hodge integrals
  • From Hurwitz theory to Witten-Kontsevich theory
  • Further topics

Winter — Math 180: Surfaces

Taught by Jennifer Schultens. Without prerequisites in topology or algebra, this elementary introduction to surfaces and coverings of surfaces will treat the following topics:
  • Classification of surfaces
  • Euler characteristic
  • Ramified coverings
  • The Riemann-Hurwitz formula
  • Monodromy of covers
  • Hurwitz numbers

Winter — Math 248B: Algebraic Geometry

Taught by Brian Osserman. An introduction to complex algebraic geometry and the modern language of sheaves and schemes. Topics include:
  • Complex varieties and the analytic topology
  • Sheaves and schemes
  • Fiber products
  • Properness
  • Functors and moduli spaces
  • Non-reduced schemes and deformation theory

Winter — Math 290: Research Seminar

Organized by Michael Kapovich, Fu Liu and Brian Osserman. Students and faculty will give presentations concentrating on background material in various subareas of Hurwitz theory. Occasional outside speakers will lecture on their work in the subject.

Spring — Math 290: Research Seminar

Organized by Michael Kapovich, Fu Liu and Brian Osserman. Students and faculty will give presentations on recent and current research, both their own and that of other researchers in the field. Occasional outside speakers will lecture on their work in the subject.