Oxford

Bourdon's building is a negatively curved 2-complex built out of hyperbolic right-angled polygons. Its automorphism group is large (uncountable) and remarkably rich. We study, and mostly answer, the question of when there is a discrete subgroup of the automorphism group such that the quotient is a closed surface of genus g. This involves some fun elementary combinatorics, but quickly leads to open questions in group theory and number theory. This is joint work with David Futer.

A behavioral mean-variance portfolio selection problem in continuous time is formulated and studied. Based on the standard mean-variance portfolio selection problem, the cumulative distribution function of the cash flow is distorted by a probability distortion function. Then the problem is no longer a convex optimization problem. This feature distinguishes it from the conventional linear-quadratic (LQ) problems.

The stochastic optimal LQ control theory no longer applies. We take the quantile function of the terminal cash flow as the decision variable.

The corresponding optimal terminal cash flow can be recovered by the optimal quantile function. Then the efficient strategy is the hedging strategy of the optimal terminal cash flow.

This talk I present a study of BSDEs with non-linear terms of quadratic growth by using Girsanov's theorem. In particular we are able to establish a non-linear version of the Cameron-Martin formula, which can be for example used to obtain gradient estimates for some non-linear parabolic equations.

Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos.

After hep-th/0909.0483

An emerging set of methods enables an experimental dialogue with biological systems composed of many interacting cell types---in particular, with neural circuits in the brain. These methods are sometimes called “optogenetic” because they employ light-responsive proteins (“opto-“) encoded in DNA (“-genetic”). Optogenetic devices can be introduced into tissues or whole organisms by genetic manipulation and be expressed in anatomically or functionally defined groups of cells. Two kinds of devices perform complementary functions: light-driven actuators control electrochemical signals; light-emitting sensors report them. Actuators pose questions by delivering targeted perturbations; sensors (and other measurements) signal answers. These catechisms are beginning to yield previously unattainable insight into the organization of neural circuits, the regulation of their collective dynamics, and the causal relationships between cellular activity patterns and behavior.