INSTRUCTOR: Sasha Voronov
OFFICE: 467 Lake Hall
PHONE: (617) 373-5528
E-MAIL ADDRESS: voronov@math.umn.edu. You are welcome to use e-mail to send questions to me.
INTERNET: All class announcements and assignments will be posted on the class homepage http://www.math.umn.edu/~voronov/G375/index.html and NOT handed out in class.
OFFICE HOURS: 4:30-5:30 M, 11-noon Th (467 Lake), and by appointment. Or just stop by at any time.
TEXT (not required): Operads in Algebra, Topology and Physics by M. Markl, S. Shnider, and J. Stasheff, 2002. See also lecture notes on the course web page.
PREREQUISITES: Some familiarity with complex algebraic curves (or Riemann surfaces) and basic homology theory will be helpful. No knowledge of physics or operads is required - all necessary notions will be introduced along the way.
DESCRIPTION: In this one-semester course, I plan to give an introduction to operad theory in relation to topology and recent developments in mathematics coming from quantum field theory (QFT). Examples of QFT's include conformal field theory (CFT), topological CFT, cohomological field theory (CohFT), string theory, and quantum gravity. The theory of operads is an area of algebraic topology, which proved to be useful in the study of loop spaces in the seventies and has been undergoing a period of renaissance since the 90s, mainly because of recent applications to homotopy structures in algebra and topology, deformation quantization, and theoretical physics.
CONTENT: Introduction to operads; operads related to moduli spaces of algebraic curves; cyclic and modular operads; PROP's and algebraic theories; the little disks operad; the Fulton-MacPherson compactification; algebras over operads; algebras related to moduli spaces; homotopy algebras, including A_infty-, L_infty-, and G_infty-algebras; deformation quantization; introduction to CFT's, modular functors, and other 2d QFT's, such as topological QFT's, CohFT, CFT's, and quantum gravity; graph homology; string topology.
REQUIREMENTS: : There will be three homeworks throughout the semester, but no exams.GRADING: Based on your homework. Grades will be assigned on curve. I expect you to put enough hard work to earn grades not lower than a B. The curve does not exclude the possibility of everybody getting A's, though.
IMPORTANT DATES:
January 16 - Registration ends.
January 19 - Martin Luther King Day (holiday).
February 16 - Presidents Day (holiday).
March 1-6 - Spring Break.
April 15 - Last course meeting.