Mathematics Department Colloquia take place on Friday afternoons from
4:00-5:00PM in the Hill Center, Room 705, on Busch Campus.
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the Hill Center (Victory Cabs, (732) 545-6666). The
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Unfortunately, colloquium cancellations do occur from time to time.
Please feel free to call our department (732)-445-3921 before embarking on your journey.
Abstract: Modeling of a wide range of physical phenomena leads to tracking fronts moving with curvature-dependent speed. A particularly natural example is where the speed is the mean curvature. If the movement is monotone inwards, then the arrival time function is the time when the front arrives at a given point. It has long been known that this function satisfies a natural differential equation in a weak sense but one wonders what is the regularity. It turns out that one can completely answer this question. It is always twice differentiable and the second derivative is only continuous in very rigid situations that have a simple geometric description. The proof weaves together analysis and geometry.
Past Talks
Friday, April 21st
Jeremy Kahn , Brown University
"Applications and frontiers in surface subgroups "
Time: 4:00 PM
Location: Hill 705
Abstract:
In 2009 V. Markovic and the speaker proved that there are ubiquitous nearly geodesic subgroups in the fundamental groups of closed hyperbolic 3-manifolds. Since then there have been many attempts (some successful) to extend these results to other settings, including lattices in other Lie groups, nonuniform lattices, delta-hyperbolic groups, and the mapping class group. After a review of the fundamental principles and methods, I will try to describe some of the successes, some of the difficulties, and some of the applications of these kinds of results.
Friday, April 14th
François Treves , Rutgers University
"A brief overview of the history of the solvability of PDEs "
Time: 4:00 PM
Location: Hill 705
Abstract: The talk is purely historical, starting with PDE's in the 19-th century (Fourier transform and the heat equation, Cauchy-Kowalewski Theorem, harmonic functions, Maxwell equations) to the 20-th through Hadamard's work, with particular attention to developments in the second half of the last century. I will try to explain the transition from linear differential operators to pseudodifferential operators and the successful application of the latter to the complete analysis of linear PDE with simple real characteristic (basic definitions will be provided under assumption that the audience knows little about the whole subject). I hope to have time to indicate some glaring open problems.
Friday, April 7th
Denis Auroux, UC Berkeley
"New constructions of monotone Lagrangian tori"
Time: 4:00 PM
Location: Hill 705
Abstract: A basic open problem in symplectic topology is to classify
Lagrangian submanifolds (up to, say, Hamiltonian isotopy) in a given
symplectic manifold. In recent years, ideas from mirror symmetry have
led to the realization that even the simplest symplectic manifolds (eg.
vector spaces or complex projective spaces) contain many more Lagrangian
tori than previously thought. We will present some of the recent
developments on this problem, and discuss some of the connections
(established and conjectural) between Lagrangian tori, cluster
mutations, and toric degenerations, that arise out of this story.
Friday, March 31st
Mohammed Abouzaid, Columbia University
"Symplectic topology, mirror symmetry, and rigid analytic geometry "
Time: 4:00 PM
Location: Hill 705
Abstract: Strominger, Yau, and Zaslow proposed a geometric explanation for mirror symmetry via a dualization procedure relating symplectic manifolds equipped with Lagrangian torus fibration with complex manifolds equipped with totally real torus fibrations. By considering the family of symplectic manifolds obtained by rescaling the symplectic form, one obtains a degenerating family of complex manifolds, which is expected to be the mirror.
Because of convergence problems with Floer theoretic constructions, it is difficult to make this procedure completely rigorous. Kontsevich and Soibelman thus proposed to consider the mirror as a rigid analytic space, defined over the field C((t)), equipped with the non-archimedean t-adic valuation, or more generally over the Novikov field. This is natural because the Floer theory of a symplectic manifold is defined over the Novikov field.
After explaining this background, I will give some indication of the tools that enter in the proof of homological mirror symmetry in the simplest class of examples which arise from these considerations, namely Lagrangian torus fibrations without singularities.
Friday, March 24th
Lauren Williams, Berkeley
"From hopping particles to Macdonald-Koornwinder polynomials "
Time: 4:00 PM
Location: Hill 705
Abstract: The asymmetric simple exclusion process (ASEP) is a Markov chain describing particles hopping on a 1-dimensional finite lattice. Particles can enter and exit the lattice at the left and right boundaries, and particles can hop left and right in the lattice, subject to the condition that there can be at most one particle per site. The ASEP has been cited as a model for traffic flow, protein synthesis, the nuclear pore complex, etc. In my talk I will discuss joint work with Corteel and with Corteel-Mandelshtam, in which we describe the stationary distribution of the ASEP and the 2-species ASEP using staircase tableaux and rhombic tilings. I will also discuss the link between these models and Askey-Wilson polynomials and Macdonald-Koornwinder polynomials.
Friday, March 10th
Special Colloquium
Jean Bricmont, University of Louvain, Belgium (NOTE: SPECIAL TIME!!)
" What is the meaning of the wave function?"
Time: 2:00 PM
Location: Hill 705
Abstract:
In quantum mechanics, the wave function or the quantum state has a perfectly well defined meaning as an instrument to predict results of measurements. But what does it mean outside of measurements? To this question, no clear answer is given in quantum mechanics textbooks. We will first give a naive interpretation of what the wave function could mean outside of measurements and show that it is mathematically inconsistent. Then, we will briefly explain how the de Broglie-Bohm theory solves that problem.
Friday, March 3rd
Tim Austin, Courant Institute
"Ergodic theory and the geometry of high-dimensional metric spaces"
Time: 4:00 PM
Location: Hill 705
Abstract: The most basic examples of shift-systems with positive entropy are the Bernoulli shifts, under which the coordinates are independent. In the special case of Bernoulli shifts, it was shown by Ornstein that entropy is actually a complete invariant. In order to prove this, Ornstein developed a concrete necessary and sufficient condition for a general shift-system to be isomorphic to a Bernoulli shift. We also know that Bernoulli shifts often appear as images of other, more complex systems under equivariant maps: by a theorem of Sinai, this is true whenever the necessary inequality between their entropies is satisfied.
The proofs of these more advanced results requires a delicate investigation of the finite-dimensional marginals of the shift-system, regarded as a sequence of discrete probability spaces endowed with their Hamming metrics. It turns out that other ergodic theoretic consequences are related to open problems on the possible structure of such discrete `metric probability spaces'. After sketching the history of this area, I will describe some of these connections.
This talk will require a knowledge of basic real analysis and some measure theory, and some simple probability theory will be helpful. I will not assume anything from dynamics or ergodic theory.
Friday, February 24th
Richard Schwartz, Brown University
"5 points on the sphere "
Time: 4:00 PM
Location: Hill 705
Abstract: Thomson's problem, which goes back to 1904, asks how N points will arrange themselves on the sphere so as to minimize their electrostatic potential. A more general problem asks what happens for other power law potentials. In spite of quite a bit of experimental evidence accumulated over the past century, and some spectacular results for values of N associated with highly symmetric polyhedra, there have been few rigorous results for the modest case N=5. In my talk I will explain my recent proof that, for N=5, the triangular bi-pyramid is the minimizer with respect to all power laws up to a constant S=15.04808..., and then the minimizer changes to a pyramid with square base. My talk will have some nice computer animations.
Friday, February 17th
Lenhard Ng , Duke University
"Studying knots through symplectic geometry and cotangent bundles "
Time: 4:00 PM
Location: Hill 705
Abstract: Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll focus on one particular application of this approach that yields strong invariants of knots. I'll discuss a mysterious connection between these knot invariants and string theory, as well as a recent result (joint with Tobias Ekholm and Vivek Shende) that the invariants completely determine the underlying knot.
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