Long Road
Post-quantum cryptography
When quantum computers are operational, intruders are expected to be able to use them to easily break public-key systems that currently protect online transactions. There is a large effort underway to meet this challenge by developing secure protocols to protect information from attacks using quantum computers. Recently, one approach based on the theory of supersingular elliptic curves [1] was broken unexpectedly. In this project, we will consider a higher dimensional setting, where pencils of elliptic curves are mapped to neighbouring pencils through a translation operation in their symmetry group. The aim is to identify the analogues of supersingular curves and their isogeny maps to lay possible groundwork for future algorithms.
Discrete integrable systems
There has been a huge surge of interest in discrete integrable systems in the last two decades. These systems overlap with models in theoretical physics (random matrix theory, string theory, quantum gravity) and mathematics (algebra, algebraic geometry, topology, and differential and difference equations). A famous integrable system is the Korteweg-de Vries equation and the Nonlinear Schrodinger equation. Such PDEs have unexpected connections to the classical Painlevé equations. Their discrete integrable versions have deep mathematical properties, but many questions still remain open. What are the asymptotic behaviours of their solutions? How do we solve them through Riemann-Hilbert methods? What are the geometric properties of associated monodromy manifolds?These lead to projects that are suitable for methods of applied mathematics, dynamical systems theory, and geometry.
Blowing up the solution space of Integrable Systems
Imagine a dynamical system with points where the equations become ill-defined, so much so that no solutions can be defined there. What do you do? You blow it up (mathematically)! This technique (known to Newton but not widely used in modern analysis) leads to very detailed information about solutions of non-linear differential equations. In this project, you will learn how to calculate explicit, new information about solutions through such techniques and extend them to discrete equations. For example, there are classes of discrete Painlevé equations for which we know nothing about how their solutions behave as parameters go to special limits. There is a great opportunity in this project to explore such original questions.
Integrable Cellular Automata
Cellular automata (CA) are discrete systems where the values of solutions (or states)and their independent variables take on only discrete values. A famous example is Conway's Game of Life where each cell only has two states (black or white) and their states evolve according to four very simple rules. The evolution of the states of such CA only depend on the states of their immediate neighbours. It turns out that there are more general CA, called filter parity rules, whose evolution depends on a semi-infinite set of cells at each time step. In this project, you can carry out mathematical analysis of the states of integrable CA. If you are interested in programming, you can also write java applets to simulate the collection of distinct states of such CA.
Exponential Asymptotics
Near an irregular singular point of a differential equation, the solutions usually have divergent series expansions. Although these can be "summed" in some way to make sense as approximations to the solutions, they do not provide a unique way of identifying a solution. There is a hidden free parameter which has an effect like the butterfly in chaos theory. This problem has been well studied for many classes of nonlinear ODEs but almost nothing is known for PDEs and not much more is known for difference equations. This project would include studies of a model PDE, like the famous Korteweg-de Vries equation near infinity, or a difference equation like the string equation that arises in 2D quantum gravity.
Mathematical Immunology
After almost three decades of knowledge about HIV/AIDS, it was only recently discovered that a cellular automata model is capable of replicating the long three-phase cycle in clinical data on T-cell populations. However, agreement with reality is still lacking. Asymptotics of cellular automata appear to be crucial to understanding this cycle. However, currently no such theory is known. This project can go in many directions. One direction is to consider different cellular-automata models for disease transmission. A second direction is to develop mathematical methods to work out limiting behaviours of cellular automata.

Integrable Systems:
Integrable systems is a subject that is full of paradoxes and miraculous resolutions. They were discovered half a century ago, through a paradox uncovered by Fermi, Pasta and Ulam, who studied heat distribution in metals through a lattice model. They expected that an initial distribution of heat in such a model would die away with time, much like a metallic bar cools down over time. But to their surprise, they found recurrences of initial heat distributions. This paradox engaged many famous scientists of that time and was finally resolved by Kruskal and Zabusky who discovered astonishingly well-behaved solutions that they called "solitons" in the continuum limit of the lattice model. Their discoveries led to a completely new field: completely integrable systems, which is now regarded as one of the most profound advances of twentieth century mathematics. In this course, I will present this theory and its modern development for discrete non-linear systems.

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