The Vanier CGS program aims to attract and retain world-class doctoral students by supporting students who demonstrate both leadership skills and a high standard of scholarly achievement in graduate studies in social sciences and humanities, natural sciences and engineering, and health.

Louis-Phillipe is a second year PhD student of Ragnar-Olaf Buchweitz and is working on problems in noncommutative algebraic geometry and homological algebra.

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**Marcin and Michal Kotowski**

The Connaught International Scholarship for Doctoral Students provides financial support to outstanding international doctoral students and assists Graduate Units in recruiting and supporting top international graduate students.

Marcin and Michal are brothers and work in the fields of discrete probability and geometric group theory. They are graduates from the University of Warsaw.

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This past Saturday marked the end of the 2013 Canadian Undergraduate Math Conference, held this year at the beautiful hilltop campus of the Université de Montréal. Thanks to generous funding from the Department of Mathematics and the Arts & Science Student’s Union, we had a particularly strong contingent of students. Amongst the budding mathematicians present were the University of Toronto’s Dylan Butson, Changho Han, Max Klambauer, Tomas Kojar, Fangda Li, Christopher Mahadeo, Seong Hyun (Daniel) Park, Samer Seraj, Olivia Simmons, Matt Sourisseau, and Freid Tong. The majority of our delegation consisted of fresh graduates or students entering their fourth year, so beginning with Professor Dror Bar-Natan’s opening keynote on knot theory, U of T was well represented when it came time for presentation.

- Dylan Butson rigorously discussed classical mechanics in the context of symplectic and Poisson manifolds. Providing plently of examples, he outlined the description of physical systems by series of time-parametrized transformations of a space, which evolve by least action (or equivalently, by a geodesic on the group of transformations). After this, he proceeded to reduce the geodesic flow on the cotangent bundle of this group to one on the dual of its Lie algebra, obtaining equations describing the infinitesimal evolution of the system. Dylan concluded with a nod to the physical problems inspiring this mathematical formulation, such as the Euler top, ideal fluid flow, and magnetohydrodynamics.
- Changho Han began with the problem of analyzing the fundamental group of a topological space, providing motivation by discussing the winding number (an invariant under path homotopy). He then used the universal covering space of a circle to explain how different loops have different winding numbers, and provided a proof of the fundamental theorem of algebra using topological methods. It turns out that covering spaces can be used to understand locally defined functions, and this was elaborated on and used during a construction of the Riemann surface of $\sqrt{z}$. With the help of lots of well-drawn pictures and numerous examples, Changho concluded his talk with the topic of branched covering spaces.
- Max Klambauer introduced C*-algebras and discussed how they could arise as operator algebras, stressing the importance of the C*-identity. After providing various examples and exposing the notion of a representation, he discussed three characterizations of irreducibility of representations, and mentioned Kadison’s transitivity theorem. States were then defined and the G.N.S. construction was sketched. Functional analysis was swept under the rug as the connection between pure states and irreducible representations was briefly discussed, but towards the end the talk, the G.N.S. construction was used to show that every C*-algebra can be isometrically represented on a Hilbert space.
- Christopher Mahadeo provided an exposition of the non-linear Schrödinger (NLS) equation, $iu_t = \Delta u + gu|u|^{p−1}$, an important partial differential equation that arises in many physicals contexts: for example, this PDE describes the propagation of laser beams in a non-linear medium. He discussed the possible formation of singular solutions, which physically correspond to the focusing of the laser beam, and mathematically is associated to certain norms of the solution becoming infinite in a finite time. After discussing conservation of the $L^2$ norm and of the Hamiltonian, Chris presented a concise exposition of R. Glassey’s 1977 proof of the existence of singular solutions.
- Although given a shorter-than-expected timeslot, Seong Hyun Park valiantly constructed the Fredholm determinant as an extension of the ordinary determinant to infinite dimensional spaces; namely, to integral operators on the space of compactly supported continuous functions. He concluded with a discussion of eigenfunctions of such operators, and presented an application of the Fredholm determinant towards Brownian motion.
- Samer Seraj captivated his audience with a discussion of his recent work with Professor Barbeau on Diophantine equations, which gives a definitive answer to the question “Which sets $S \subset \mathbb{Z}$ exhibit the property that $\sum_{s \in S} s^3 = \left( \sum_{s \in S} s \right)^2$?”.
- • Olivia Simmons provided insight into a book by Mandelbrot, entitled “The (Mis)Behavior of Markets: A Fractal View of Financial Turbulence”. She illustrated how Mandelbrot’s ideas are more applicable than ever in the wake of the 2008 credit crunch, and discussed his applications of fractal geometry to statistical physics, information technology, meteorology, cosmology, and of course, economics. Olivia’s talk managed to cover everything from the basics of Fractals to the past and present conditions of financial markets, as well as an application of fractals to market scaling.
- Matt Sourisseau discussed a surprising confluence of complex analysis and probability theory by presenting a peculiar proof of Picard’s Little Theorem via Brownian motion. A result of Lévy provided the basic connection between nonconstant entire functions and Brownian motion, from which the proof of Little Picard proceeded by finding a topological contradiction. This hinged on understanding the long term behaviour of Brownian motion in the twice-punctured plane, as well as developing a way to circumvent the failure of point-recurrence for planar Brownian motion. Talk slides are available here.

The complete abstracts of the above students (and more!) can be accessed here.

Between conversations ranging from optimal cake-cutting strategies to Lie algebras, Dehn surgery, and arithmetic progressions in the primes, all of us learned a great deal of interesting mathematics. But perhaps more important was the palpable sense of community and camaraderie formed amongst those present over such a short time.

On a personal note, I very much regret waiting until my last year as an undergraduate to attend such a vibrant and exciting conference. I strongly recommend current undergraduates to avoid making my mistake, and instead attend next year’s CUMC at Ottawa’s Carlton University!

Students interested in attending are encouraged to practise giving presentations at the Mini Undergraduate Math Seminars, which are set to resume in September.

]]>- Professor Jim Arthur has been invited to give a plenary lecture
- Professor Balint Virag has been invited to speak in the Special Section on Probability and Statistics
- Professors Robert Jerrard and Robert McCann have been invited to speak in the Special Section on Partial Differential Equations

Being invited to speak at the ICM is often regarded as one of the highest honours that a mathematician can receive. It is a truly international recognition of the depth and ground breaking impact of their research.

In addition, two graduates of our doctoral program have been invited to speak as well.

- Professor Izabella Laba received her Ph.D. under the direction of Professor Michael Sigal and will be speaking in the Special Section on Analysis and its Applications.
- Professor Ilijas Farah received his Ph.D. under the direction of Professor Stevo Todorcevic and will be speaking in the Special Section on Logic and Foundations.

They are now faculty members at UBC and York University respectively.

Our congratulations to all the speakers!

]]>The March Break Math Camp, open to students in Grades 4 – 10, has seen 57 students participate in a week of hands-on, interactive activities designed to introduce and excite them about the various fields of mathematics. From number theory to geometry to building geodesic domes and optimizations. The camp has also drawn from a partnership with the Canadian Math Kangaroo Competition to provide students with competition based problems and content. Today they celebrate Pi with discussions of circles and the fascinating number ratio itself.

In addition, the camp will see guest speakers from the Department of Mathematics give talks to students on optimization, Mobius strips, origami and math, the board game Blockus and how Google works.

So far the topics covered have been far ranging and of various interests. A sampling of topics:

- Optimal Transport
- Topics in abstract algebra starting with some basic groups and rings theory building to some notions of geometry (e.g., projective spaces and affine varieties over complex numbers) with the ultimate goal to learn the intuition behind blowing-up and resolution of singularities, and producing graphs or animations of simple resolutions of curve singularities
- Issues in the foundations of mathematics such as the difference between countable and uncountable sets, and “constructing” most of the basic mathematical objects (the integers, rational numbers, real numbers, complex numbers) starting from just the empty set
- Number theory starting from the basics of congruences, Little Fermat Theorem, Euler function, Integer points inside polygons and ellipses and some other relatively elementary theorems
- Calculus of Young tableaux with applications to combinatorics and symmetric polynomials

… and much more.

We are looking forward to seeing their poster presentations in April on the fascinating mathematics they have learned.

]]>Further information on the seminar, including Professor Nachman’s talk, can be found at the SIAM website (http://www.siam.org/meetings/jmm13/index.htm).

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This is the 3rd year in a row a member of the Department of Mathematics at the University of Toronto has been awarded this prize by the CRM and the 6th time since it’s inception in 1992.

The official announcement from the CRM can be found here

Full information on the prize and it’s history can be found here

Our congratulations to Professor Alexakis on this remarkable achievement!

]]>The Fulkerson Prize is one of the most prestigious prizes in discrete math and it is awarded every three years.

From their website: “This award is sponsored jointly by the Mathematical Optimization Society (formerly the Mathematical Programming Society) and the American Mathematical Society (AMS). Up to three awards of US$1500 are presented at each (triennial) International Symposium of the MPS.”

This year it went to László Lovász and Balázs Szegedy for their paper on the “Limits of dense graph sequences”

A full list of winners, including this year’s, can be found here

The prize was announced on August 18, 2012 at the International Symposium on Mathematical Programming in Berlin.

]]>Last month, five U of T students, myself included, attended the 2012 CUMC at UBC Okanagan campus, in quiet, clement and panoramic Kelowna.We were Reza Asad, Dylan Butson, Anne Dranovski, Mike Hongyoul Park, and Jonathon Zung. (Years 4, 3, 4, 2 and 3, respectively.)

In the three days leading up to the CUMC, three of us participated in an Optimization Workshop organized by UBC Okanagan’s Department ofMathematics and Computer Science — the University is known for its unrivaled graduate programs (MSc and PhD) in Optimization and Convex Analysis (OCANA).

The workshop was a very interesting, concise and fast-paced introduction to major topics in optimization. Namely, monotone operators, derivative free optimization, and variational analysis.

During the CUMC, all five of us gave talks. For most of us this was a first talk. Audience turnout and feedback was extremely positive. Reza Asad’s talk was even attended by Professor Heinz Bauschke of the workshop. The subjects of our talks were as follows.

- Reza Asad presented the Stiener symmetrization, which is a rearrangement or transformation of a set in the plane that comes in handy whenproving the isoperimetric inequality, as well as other functional inequalities, when applied to functions’ level sets, in mathematical physics and elsewhere.
- Dylan Butson introduced the stochastic integral, the heart of the stochastic calculus, which extends the Riemann-Stieltjes integral to random processes such as Brownian motion, and has important applications in mathematical finance. To learn more about topics in stochastic calculus and, more generally, in mathematical probability, follow the previous link.
- Mike Park reviewed Diophantine approximations, constructing examples of numbers which have very good rational approximations and, therefore,could not be algebraic. He also explained how to find good rational approximations using the theory of continued fractions.
- In a crafty application of the Borsuk-Ulam theorem, (following Alon and West 1986,) Jonathon Zung showed how two topologically inclinedthieves, having stolen a necklace with k different types of jewels, could cut up the necklace so that each receives the same number of jewels of each type.
- I gave a description of random polarizations, or two-point symmetrizations, on the sphere, which are also useful for proving inequalities in mathematical physics, and admit convergence results which generalize to more complicated rearrangements such as the Stiener symmetrization.

Collected speakers’ abstracts can be viewed here

On our second last day, Dylan Butson and I presented a bid to host next year’s CUMC. We were well-received, but lost honorably to UMontreal.

The CUMC was an incredible learning experience. I only wish more students from U of T were able to share in the week’s worth of non-stop math-musement. The good news is there will be ample opportunity for you to share math in conference-like settings with your peers before CUMC 2013, starting with a Mini Undergraduate Math Seminar (MUMS), to be held early September.

Students will speak about topics of interest in 25 minute long presentations. Please check the wiki for updates, and e-mail me your abstract and/or slides by August 31st if you would like to present.

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