In this talk we will give an overview of the recent research on global quantizations on spaces of different types: compact and nilpotent Lie groups, general locally compact groups, compact manifolds with boundary.
The talk will focus on continuous time random walk with unbounded i.i.d. random conductances on the grid $\mathbb{Z}^d$ In the first place, in a joint work with Kumagai and Mathieu, we obtain Gaussian heat kernel bounds and also local CLT for bounded from above and not bounded from below conductances. The proof is given at first in a general framework, then it is specified in the case of plynomial lower tail conductances. It is essentially based on percolation and spectral analysis arguments, and Harnack inequalities. Then we will discuss the same questions for the same model with i.i.d. random conductances, bounded from below and with finite expectation.
A diophantine equation is an algebraic equation, or system of equations, in several unknowns and with integer (or rational) coefficients, which one seeks to solve in integers (or rational numbers). The study of such equations goes back to antiquity. Their name derives from the mathematician Diophantus of Alexandria, who wrote a treatise on the subject, entitled Arithmetica.
Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.
We want to discuss a collection of results around the following Question: Given a smooth compact manifold $M$ without boundary, does $M$ admit a Riemannian metric of positive scalar curvature?
We focus on the case of spin manifolds. The spin structure, together with a chosen Riemannian metric, allows to construct a specific geometric differential operator, called Dirac operator. If the metric has positive scalar curvature, then 0 is not in the spectrum of this operator; this in turn implies that a topological invariant, the index, vanishes.
We use a refined version, acting on sections of a bundle of modules over a $C^*$-algebra; and then the index takes values in the K-theory of this algebra. This index is the image under the Baum-Connes assembly map of a topological object, the K-theoretic fundamental class.
The talk will present results of the following type:
If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has non-trivial index, what conditions imply that $M$ does not admit a metric of positive scalar curvature? How is this related to the Baum-Connes assembly map?
We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$), Engel and new generalizations. Moreover, we will show how these results fit in the context of the Baum-Connes assembly maps for the manifold and the submanifold.
We will prove an upper bound for the volume of a minimal
hypersurface in a closed Riemannian manifold conformally equivalent to
a manifold with $Ric > -(n-1)$. In the second part of the talk we will
construct a sweepout of a closed 3-manifold with positive Ricci
curvature by 1-cycles of controlled length and prove an upper bound
for the length of a stationary geodesic net. These are joint works
with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).
Motivated by work of Borel and Serre on arithmetic groups, Bestvina and Feighn defined a bordification of Outer space; this is an enlargement of Outer space which is highly-connected at infinity and on which the action of $Out(F_n)$ extends, with compact quotient. They conclude that $Out(F_n)$ satisfies a type of duality between homology and cohomology. We show that Bestvina and Feighn’s bordification can be realized as a deformation retract of Outer space instead of an extension, answering some questions left open by Bestvina and Feighn and considerably simplifying their proof that the bordification is highly connected at infinity.
