Friday, August 19, 2011, 2:10 p.m., in BA 6183, 40 St. George Street ON THE LOGARITHIMIC CALCULUS AND SIDORENKO's CONJECTURE Janet Li Joint work with Balazs Szegedy. We study a type of calculus for proving inequalities between subgraph densities which is based on Jensen's inequality for the logarithmic function. As a demonstration of the method we verify the conjecture of ErdÂ¨os-Simonovits and Sidorenko for new families of graphs. In particular we give a short analytic proof for a result by Conlon, Fox and Sudakov.

Friday, March 12, 2010,11:10 a.m. – 12:00 noon,in BA 2135, 40 St. George Street

Student: Alexander Dahl

Supervisor: Valentin Blomer

Thesis Title: On moments of class numbers of real quadratic fields

Thesis Abstract: Class numbers of algebraic number fields are central invariants. Once the underlying field has an infinite unit group they behave very irregularly due to a non-trivial regulator. This phenomenon occurs already in the simplest case of real quadratic number fields of which very little is known. Hooley derived a conjectural formula for the average of class numbers of real quadratic fields,$S(x) = \sum_{D\le x} \, h(D).$ In this thesis we extend his methods to obtain conjectural formulae and bounds for any moment,$S_\lambda(x) = \sum_{D\le x} \, h(D)^\lambda,$where $\lambda$ is an arbitrary real number. Our formulae and bounds are based on similar (quite reasonable) assumptions of Hooley’s work.

In the final chapter we consider the case $\lambda = -1$ from a numerical point of view and develop an efficient algorithm to compute $S_{-1}$ without computing class numbers.

A link to the thesis:http://www.math.toronto.edu/aodahl/files/alexanderdahl-mthesis.pdf