Forthcoming Seminars

It is a general belief that the heat kernels on fractals should exhibit highly oscillatory behaviors as opposed to the classical case of Riemannian manifolds.

For example, on a class of finitely ramified fractals, called (affine) nested fractals, a canonical ``Brownian motion" has been constructed and its transition density (heat kernel) $p_{t}(x,y)$ satisfies $c_{1} \leq t^{d_{s}/2} p_{t}(x,x) \leq c_{2}$ for $t \leq 1$ for any point $x$ of the fractal; here $d_{s}$ is the so-called spectral dimension. Then it is natural to ask whether the limit of this quantity as $t$ goes to 0 exists or not, and it has been conjectured NOT to exist by many people.

In this talk, I will present partial affirmative answers to this conjecture. First, for a general (affine) nested fractal, the non-existence of the limit is shown to be true for a ``generic" (in particular, almost every) point. Secondly, the same is shown to be valid for ANY point of the fractal in the particular cases of the $d$-dimensional standard Sierpinski gasket with $d\geq 2$ and of the $N$-polygasket with $N\geq 3$ odd, e.g. the pentagasket ($N=5$) and the heptagasket ($N=7$).

 Determining the price at which to conduct a trade is an age-old problem. The first (albeit primitive) pricing mechanism dates back to the Neolithic era, when people met in physical proximity in order to agree upon mutually beneficial exchanges of goods and services, and over time increasingly complex mechanisms have played a role in determining prices. In the highly competitive and relentlessly fast-paced markets of today’s financial world, it is the limit order book that matches buyers and sellers to trade at an agreed price in more than half of the world’s markets.  In this talk I will describe the limit order book trade-matching mechanism, and explain how the extra flexibility it provides has vastly impacted the problem of how a market participant should optimally behave in a given set of circumstances.

I will explain how moduli spaces of quadratic differentials on Riemann surfaces can be interpreted as spaces of stability conditions for certain 3-Calabi-Yau triangulated categories. These categories are defined via quivers with potentials, but can also be interpreted as Fukaya categories. This work (joint with Ivan Smith) was inspired by the papers of  Gaiotto, Moore and Neitzke, but connections with hyperkahler metrics, Fock-Goncharov coordinates etc. will not be covered in this talk.