# Scientific

## 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.

## Cloaking and Transformation Optics

We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. Maxwell's equations have transformation laws that allow for design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved, especially the use of singular transformations.

## What I am Doing in Australia

Jonathan Borwein talks about his current research and the Priority Research Center for Computer Assisted Research Mathematics and its Applications (CARMA). Professor Borwein is both a Laureate Professor and the Director at CARMA which is located at the University of Newcastle in New South Wales, Australia.

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## New geometric and functional analytic ideas arising from problems in symplectic geometry

The study of moduli spaces of holomorphic curves in symplectic geometry is the key ingredient for the construction of symplectic invariants. These moduli spaces are suitable compactifications of solution spaces of a first order nonlinear Cauchy-Riemann type operator. The solution spaces are usually not compact due to bubbling-off phenomena and other analytical difficulties.

## Frozen Boundaries and Log Fronts

In this talk, based on joint work with Richard Kenyon and Grisha Mikhalkin, Andrei Okounkov discusses a binary operation on plane curves which

- generalizes classical duality for plane curves and
- arises naturally in probabilistic context,

namely as a facet boundary in certain random surface models.

## IGTC Summit, 2011

The 2011 IGTC (International Graduate Training Centre) Summit was held as part of the Applied Mathematics Perspectives thematic program at the University of Victoria. There was a poster session, research discussion session where students and experts interacted, and an education session for IGTC students.

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## Hyperplane Arrangements and Applications

Some photos from the Hyperplane Arrangements and Applications conference which took place at UBC Vancouver, August 8-12. This conference was held in honour of Hiroaki Terao.

## Applied Mathematics Perspectives

Photos taken during the Applied Mathematics Perspectives workshops, July 12-17, 2011 at the University of British Columbia, Vancouver.

## PIMS Board Meeting - Fall 2011

The PIMS 2011 Fall board meeting was held at the University of Saskatchewan. In addition to the board meeting, board members toured the Canadian Light Source facility.