We will discuss the category small functors from a combinatorial model category to simplicial sets and introduce a model structure in which the fibrant objects are the levelwise fibrant homotopy functors. This model structure is called the homotopy model structure.

Next, we will discuss the conditions under which a Quillen equivalence of model categories induces a Quillen equivalence of the homotopy model structures. If the time permits, then we will describe an alternative model structure Quillen equivalent to the homotopy model structure.

This work is joint with D. White.

Symmetric decreasing rearrangement replaces a given function $f$ on $\mathbb{R}^d$ by a radially decreasing function $f^*$ that is equimeasurable to $f$. Symmetrization techniques have been used to determine the sharp constants in classical functional inequalities such as the Sobolev inequality, and for solving minimization problems in Geometry and Mathematical Physics. Symmetrization also motivates the definition of rearrangement-invariant function spaces.

I will describe recent work with A. Ferone on the extremals of the Polya-Szego inequality. The inequality says that the $p$-norms of the gradient decrease under symmetrization. It is known that there are non-trivial cases of equality, even when $p>1$. We use Ryff's polar factorization to describe these equality cases.

Speaker's homepage: http://www.math.toronto.edu/almut/

I will construct the Joyal model structure.

I will discuss how the geometry of an algebraic variety restricts the possible p-group actions on it, especially concerning fixed points."