# Scientific

## Expanders, Group Theory, Arithmetic Geometry, Cryptography and Much More

This is a lecture given on the occasion of the launch of the PIMS CRG in "L-functions and Number Theory".

The theory of expander graphs is undergoing intensive development. It finds more and more applications to diverse areas of mathematics. In this talk, aimed at a general audience, I will introduce the concept of expander graphs and discuss some interesting connections to arithmetic geometry, group theory and cryptography, including some very recent breakthroughs.

## 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.

## 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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