It is conjectured that many 2D lattice models of physical phenomena (percolation, Ising model of a ferromagnet, self avoiding polymers, ...) become invariant under rotations and even conformal maps in the scaling limit (i.e. when "viewed from far away"). A well-known example is the Random Walk (invariant only under rotations preserving the lattice) which in the scaling limit converges to the conformally invariant Brownian Motion.
Assuming the conformal invariance conjecture, physicists were able to make a number of striking but unrigorous predictions: e.g. dimension of a critical percolation cluster is almost surely 91/48; the number of simple length N trajectories of a Random Walk is about N11/32·mN, with m depending on a lattice, and so on.
We will discuss the recent progress in mathematical understanding of this area, in particular for the Ising model. Much of the progress is based on combining ideas from probability, complex analysis, combinatorics.
This is a survey of Lagrangian Floer homology which I developed together with Y.G.-Oh, Hiroshi Ohta, and Kaoru Ono. I will focus on its relation to (homological) mirror symmetry. The topic discussed include
Definition of filtered A infinity algebra associated to a Lagrangian submanifold and its categorification.
Its family version and how it is related to mirror symmetry.
Some example including toric manifold. Calculation in that case and how mirror symmetry is observed from calculation.
Simple singularities in dimension 2 have crepant resolutions and satisfy the McKay correspondence. But higher dimensional generalizations do not. We propose the categorical crepant resolutions of such singularities in the sense that the Serre functors act as fractional shifts on the added objects.
Tropical geometry studies an algebraic variety X by `tropicalizing' it into a polyhedral complex Trop(X) which retains much of the information about X. This technique has been applied successfully in numerous contexts in pure and applied mathematics.
Tropical varieties may be simpler than algebraic varieties, but they are by no means well understood. In fact, tropical linear spaces already feature a surprisingly rich and beautiful combinatorial structure, and interesting connections to geometry, topology, and phylogenetics. I will discuss what we currently know about them.
Micro-ElectroMechanical Systems (MEMS) and Nano-ElectroMechanical Systems (NEMS) are now a well established sector of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate that is held fixed along its boundary , where is a bounded domain in The plate, which lies below another parallel rigid grounded plate (say at level ) has its upper surface coated with a negligibly thin metallic conducting film, in such a way that if a voltage l is applied to the conducting film, it deflects towards the top plate, and if the applied voltage is increased beyond a certain critical value , it then proceeds to touch the grounded plate. The steady-state is then lost, and we have a snap-through at a finite time creating the so-called pull-in instability. A proposed model for the deflection is given by the evolution equation
Now unlike the model involving only the second order Laplacian (i.e., ), very little is known about this equation. We shall explain how, besides the above practical considerations, the model is an extremely rich source of interesting mathematical phenomena.
Let where is a sequence of independent and identically distributed (i.i.d.) of random variables with . According to the classical law of large number (LLN), the sum converges strongly to . Moreover, the well-known central limit theorem (CLT) tells us that, with and , for each bounded and continuous function we have with .
These two fundamentally important results are widely used in probability, statistics, data analysis as well as in many practical situation such as financial pricing and risk controls. They provide a strong argument to explain why in practice normal distributions are so widely used. But a serious problem is that the i.i.d. condition is very difficult to be satisfied in practice for the most real-time processes for which the classical trials and samplings becomes impossible and the uncertainty of probabilities and/or distributions cannot be neglected.
In this talk we present a systematical generalization of the above LLN and CLT. Instead of fixing a probability measure P, we only assume that there exists a uncertain subset of probability measures . In this case a robust way to calculate the expectation of a financial loss is its upper expectation: where is the expectation under the probability . The corresponding distribution uncertainty of is given by , . Our main assumptions are:
The distributions of are within an abstract subset of distributions , called the distribution uncertainty of , with and .
Any realization of does not change the distributional uncertainty of (a new type of `independence' ).
Our new LLN is: for each linear growth continuous function we have
Namely, the distribution uncertainty of is, approximately, .
In particular, if , then converges strongly to 0. In this case, if we assume furthermore that and , . Then we have the following generalization of the CLT:
Here stands for a distribution uncertainty subset and its the corresponding upper expectation. The number can be calculated by defining which solves the following PDE , with
An interesting situation is when is a convex function, with . But if is a concave function, then the above has to be replaced by . This coincidence can be used to explain a well-known puzzle: many practitioners, particularly in finance, use normal distributions with `dirty' data, and often with successes. In fact, this is also a high risky operation if the reasoning is not fully understood. If , then which is a classical normal distribution. The method of the proof is very different from the classical one and a very deep regularity estimate of fully nonlinear PDE plays a crucial role.
A type of combination of LLN and CLT which converges in law to a more general -distributions have been obtained. We also present our systematical research on the continuous-time counterpart of the above `G-normal distribution', called G-Brownian motion and the corresponding stochastic calculus of Itô's type as well as its applications.
Regular permutation groups are the `smallest' transitive groups of permutations, and have been studied for more than a century. They occur, in particular, as subgroups of automorphisms of Cayley graphs, and their applications range from obvious graph theoretic ones through to studying word growth in groups and modeling random selection for group computation. Recent work, using the finite simple group classification, has focused on the problem of classifying the finite primitive permutation groups that contain regular permutation groups as subgroups, and classifying various classes of vertex-primitive Cayley graphs. Both old and very recent work on regular permutation groups will be discussed.
Math Mania is a popular alternative math education event which PIMS runs in elementary schools. These are some photos taken at Ecole Glen in Coquitlam BC on Feb 16, 2011. Thank you to all the teachers, students and volunteers from Douglas college who participated!