Georgia Institute of Technology College of Sciences

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

Among n-dimensional regions with fixed volume, which one hasthe least boundary? This question is known as an isoperimetricproblem; its nature depends on what is meant by a "region". I willdiscuss variations of an isoperimetric problem known as thegeneralized Cartan-Hadamard conjecture: If Ω is a region in acomplete, simply connected n-manifold with curvature bounded above byκ ≤ 0, then does it have the least boundary when the curvature equalsκ and Ω is round? This conjecture was proven when n = 2 by Weil andBol; when n = 3 by Kleiner, and when n = 4 and κ = 0 by Croke. Injoint work with Benoit Kloeckner, we generalize Croke's result to mostof the case κ < 0, and we establish a theorem for κ > 0. It was originally inspired by the problem of finding the optimal shape of aplanet to maximize gravity at a single point, such as the place wherethe Little Prince stands on his own small planet.

Recently, a "symplectic duality" between D-modules on certainpairs of algebraic symplectic manifolds was discovered, generalizingclassic work of Beilinson-Ginzburg-Soergel in geometric representationtheory. I will discuss how such dual spaces (some known and some new) arisenaturally in supersymmetric gauge theory in three dimensions.

The Lickorish-Wallace theorem states that every closed, connected, orientable three-manifold can be expressed as surgery on a link in the three-sphere (i.e., remove a neighborhood of a disjoint union of embedded $S^1$'s from $S^3$ and re-glue). It is natural to ask which three-manifolds can be obtained by surgery on a single knot in the three-sphere. We discuss a new way to obstruct integer homology spheres from being surgery on a knot and give some examples. This is joint work with Jennifer Hom and Cagri Karakurt.