By: Matt Sourisseau (matt.sourisseau@mail.utoronto.ca)
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.