# Mathematics

## Emerging Aboriginal Scholars Summer Camp

From July 4 to August 5, 2011, the UBC First Nations House of Learning and PIMS ran a summer camp for grade 10 and 11 students with First Nations backgrounds. The camp combined academics and cultural components. In this video we meet some of the camp organizers and participants. Videography by Elle-Maija Tailfeathers.

## Perfect Crystals for Quantum Affine Algebras and Combinatorics of Young Walls

In this talk, we will give a detailed exposition of theory of perfect crystals, which has brought us a lot of significant applications. On the other hand, we will also discuss the strong connection between the theory of perfect crystals and combinatorics of Young walls. We will be able to derive LLT algorithm of computing global bases using affine paths. The interesting problem is how to construct affine Hecke algebras out of affine paths.

## Regular Permutation Groups and Cayley Graphs

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.

## Law of Large Number and Central Limit Theorem under Uncertainty, the related New Itô's Calculus and Applications to Risk Measures

- The distributions of $X_i$ are within an abstract subset of distributions $\{F_q(x):q \in Q\}$, called the distribution uncertainty of $X_i$, with $['(m)]=[\^(\mathbf{E})][X_i]=\sup_q\int_{-\infty}^\infty xF_q(dx)$ and $m=-[\^\,(\mathbf{E})][-X_i]=\inf_q \int_{-\infty}^\infty x F_q(dx)$.
- Any realization of $X_1, \ldots, X_n$ does not change the distributional uncertainty of $X_{n+1}$ (a new type of `independence' ).

## On Fourth Order PDEs Modelling Electrostatic Micro-Electronical Systems

Now unlike the model involving only the second order Laplacian (i.e., $d = 0$), 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.

## Geometry and analysis of low dimensional manifolds

In this talk, I will start with a brief tour on geometrization of 3-manifolds. Then I will discuss recent progresses on geometry and analysis of 4-manifolds.

## Categorical Crepant Resolutions of Higher Dimensional Simple Singularities

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.

## Linearity in the Tropics

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.

- Read more about Linearity in the Tropics
- 4921 reads

## Lagrangian Floer Homology and Mirror Symmetry

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.

## Conformal Invariance and Universality in the 2D Ising Model

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.