# Scientific

## 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.

## Warming Caused by Cumulative Carbon Emissions: the Trillionth Tonne

The eventual equilibrium global mean temperature associated with a given stabilization level of atmospheric greenhouse gas concentrations remains uncertain, complicating the setting of stabilization targets to avoid potentially dangerous levels of global warming. Similar problems apply to the carbon cycle: observations currently provide only a weak constraint on the response to future emissions. These present fundamental challenges for the statistical community, since the non-linear relationship between quantities we can observe and the response to a stabilization scenario makes estimates of the risks associated with any stabilization target acutely sensitive to the details of the analysis, prior selection etc. Here we use ensemble simulations of simple climate-carbon-cycle models constrained by observations and projections from more comprehensive models to simulate the temperature response to a broad range of carbon dioxide emission pathways. We find that the peak warming caused by a given cumulative carbon dioxide emission is better constrained than the warming response to a stabilization scenario and hence less sensitive to underdetermined aspects of the analysis. Furthermore, the relationship between cumulative emissions and peak warming is remarkably insensitive to the emission pathway (timing of emissions or peak emission rate). Hence policy targets based on limiting cumulative emissions of carbon dioxide are likely to be more robust to scientific uncertainty than emission-rate or concentration targets. Total anthropogenic emissions of one trillion tonnes of carbon (3.67 trillion tonnes of CO2), about half of which has already been emitted since industrialization began, results in a most likely peak carbon-dioxide induced warming of 2○C above pre-industrial temperatures, with a 5-95% confidence interval of 1.3-3.9○C.

## Discrete Stochastic Simulation of Spatially Inhomogeneous Biochemical Systems

In microscopic systems formed by living cells, the small numbers of some reactant molecules can result in dynamical behavior that is discrete and stochastic rather than continuous and deterministic. An analysis tool that respects these dynamical characteristics is the stochastic simulation algorithm (SSA), which applies to well-stirred chemically reacting systems. However, cells are hardly homogeneous! Spatio-temporal gradients and patterns play an important role in many biochemical processes. In this lecture we report on recent progress in the development of methods for spatial stochastic and multiscale simulation, and outline some of the many interesting complications that arise in the modeling and simulation of spatially inhomogeneous biochemical systems.

## 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.

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## 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.