**UBC Mathematics Department**

*http://www.math.ubc.ca*

## Colloquium Abstract: Professor Aaron Bertram,
Department of Mathematics, University of Utah

*Quantum Products of Cohomology Classes*

Even in the case of the complex projective plane, the
result is stunning. Let N_d denote the number of rational
curves of degree d passing through 3d-1 general points
in {\bf CP}^2. Then the ``potential function'':
\Phi(y_0,y_1,y_2) :=
\frac 12(y_0^2y_2 + y_0y_1^2)
+ \sum_{d \ge 1} N_d
\frac{y_2^{3d-1}}{(3d-1)!} e^{dy_1}
satisfies the following partial differential equation:
\Phi_{222} = \Phi_{112}^2 - \Phi_{111}\Phi_{122}
which allows one to solve for the N_d.

Such a result, unimagined in the course of more than a hundred
years of intermittent study of such numbers by enumerative algebraic
geometers, is just one of many corollaries of the newly developed
quantum cohomology theory. This theory, which is a wonderful blend
of ``physical'' insights and proofs coming from algebraic geometry
and symplectic topology, has been extensively developed and
interpreted over the past several years. In this talk, I will explain
how potential functions and PDE's are associated to smooth complex
projective varieties by counting rational curves, and present some
of my favorite examples.

Return to this week's seminars