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Vasily Dolgushev, Temple University
I will talk about adjoint functors, limits and colimits. I hope to give many examples. If time will permit, I will start talking about monoidal categories and monoidal functors.
Blair Davey, City College of New York
In the late 1960s, E.M. Landis made the following conjecture: If $u$ and $V$ are bounded functions, and $u$ is a solution to $\Delta u = V u$ in $\mathbb{R}^n$ that decays like $|u(x)| \le c \exp(- C |x|^{1+})$, then $u$ must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded functions $u, V: \mathbb{R}^2 \to \mathbb{C}$ that solve $\Delta u = V u$ in $\mathbb{R}^2$ and satisfy $|u(x)| \le c \exp(- C |x|^{4/3})$. The result of Meshkov was accompanied by qualitative unique continuation estimates for solutions in $\mathbb{R}^n$. In 2005, J. Bourgain and C. Kenig quantified Meshkov's unique continuation estimates. These results, and the generalizations that followed, have led to a fairly complete understanding of the complex-valued setting. However, there are reasons to believe that Landis' conjecture may be true in the real-valued setting. We will discuss recent progress towards resolving the real-valued version of Landis' conjecture in the plane.
Marcus Michelin, UPenn
Given an infinite rooted tree, how might one sample, nearly uniformly, from the set of paths from the root to infinity? A number of methods have been studied including homesick random walks, or determining the growth rate of the number of self-avoiding paths. Another approach is to use percolation. The model of invasion percolation almost surely induces a measure on such paths in Galton-Watson trees, and we prove that this measure is absolutely continuous with respect to the limit uniform measure as well as other properties of invasion percolation. This work in progress is joint with Robin Pemantle and Josh Rosenberg.
Max Reinhold Jahnke, University of São Paulo, Brazil
First, in order to understand the statement of a theorem by Bott we will see a brief exposition of the theory of cohomology of Lie algebras. As an application, we will see how to use it to prove that the study of the Dolbeault cohomology of left-invariant complex structures on semisimple compact Lie groups can be reduced to the study a purely algebraic problem: the study of Dolbeault cohomology of complex structures on semisimple compact Lie algebras. This approach was first used by Pittie.
Samuel Taylor, Temple University
We’ll explore the following two questions:
1) What geometric structures can be associated to a random 3-dimensional manifold?
2) How likely is it that two random elements of a group G commute?
Although these questions appear to be quite different, we’ll see how they fit into the common framework of counting problems in geometry group theory. Our talk will be a friendly overview of this framework and stress many problems — some of which have been recently answered and some of which are still open.
Heike Faßbender, Technical University Braunschweig, Germany
There are no conferences this week.