Mathematics Colloquium Spring 2019
For further information please call the Mathematics Department at 459-2969
Tuesday, April 2nd, 2019
Alex Vladimirsky, Cornell University
Agreeing to Disagree in Anisotropic Crowds
How do the choices made by individual pedestrians influence the large-scale crowd dynamics? What are the factors that slow them down and motivate them to seek detours? What happens when multiple crowds pursuing different targets interact with each other? We will consider how answers to these questions shape a class of popular PDE-based models, in which a conservation law models the evolution of pedestrian density while a Hamilton-Jacobi PDE is used to determine the directions of pedestrian flux. This presentation will emphasize the role of anisotropy in pedestrian interactions, the geometric intuition behind our choice of optimal directions, and connections to the non-zero-sum game theory. Joint work with Elliot Cartee.
Tuesday, April 9th, 2019
Cancelled Kiyokazu Nagatomo, Osaka Cancelled
Tuesday, April 16th, 2019
TBA
Tuesday, April 23rd, 2019
TBA
Tuesday, April 30th, 2019
TBA
Tuesday, May 7th, 2019
Daniel Cristofaro-Gardiner, University of California Santa Cruz
Subleading asymptotics of ECH capacities
In previous work, Hutchings, Ramos and I studied the embedded contact homology (ECH) spectrum for any closed three-manifold with a contact form, and proved a "volume identity" showing that the leading order asymptotics recover the contact volume. I will explain recent joint work that sharpens this asymptotic formula by estimating the subleading term. The main technical point needed in our work is an improvement of a key spectral flow bound in Taubes' proof of the three-dimensional Weinstein conjecture; the main goal of my talk will be to explain the ideas that go into this improvement. I will also discuss some possibilities for obtaining sharp asymptotics.
Tuesday, May 14th, 2019
Victor Dods, LedgerDomain
Recent Results in the Kepler-Heisenberg Problem; From Numerics to Proofs
Mathematics is privileged relative to the empirical sciences in that it can be done in the abstract, proving theorems with pencil and paper, without need for empirical observation. While constructing proofs this way is certainly feasible, a hybrid approach can be used to add an empirical stepping stone in between the blank slate and the finished proof.
In particular, recent work in the Kepler-Heisenberg problem (orbital dynamics research done in collaboration with Corey Shanbrom (CSU Sacramento, graduated UCSC 2013)), involved using numerical integration to generate "empirical observations" from which to form and refine hypotheses in service of finally forming proofs of the results. This process will be discussed, along with specific numerical methods used, including symplectic integration, shooting methods, and the fast Fourier transform.
Numerical methods and observations were implemented in Python and released as an open source project which fully reproduces the results of the corresponding paper. The code can be found at https://github.com/vdods/heisenberg
Tuesday, May 21st, 2019
Sarah Scherotzke, Bonn, MRSI
The Chern character and categorification'
The Chern character is a central construction which appears in topology, representation theory and algebraic geometry. In algebraic topology it is for instance used to probe algebraic K-theory which is notoriously hard to compute, in representation theory it takes the form of classical character theory. Recently, Toen and Vezzosi suggested a construction, using derived algebraic geometry, which allows to unify the various Chern characters. We will categorify this Chern character. In the categorified picture algebraic K-theory is replaced by the category of non-commutative motives. It turns out that the categorified Chern character has many interesting applications. For instance we show that the DeRham realisation functor is of non-commutative origin.
Tuesday, May 28th, 2019
Francois Monard, University of California Santa Cruz
Inversion of abelian and non-abelian ray transforms in the presence of statistical noise
In this talk we will discuss two problems associated with ray transform type of problems on 'simple' Riemannian surfaces:
(1) how to reconstruct a function from its noisy geodesic X-ray transform (with applications to X-ray tomography)
(2) how to reconstruct a skew-hermitian Higgs field from its noisy scattering data (with applications to Neutron Spin Tomography)
By 'noisy' here I mean that the measurements are corrupted, and that no strategy can ever recover the truth 'exactly'.
How can one then claim that the reconstructions provided make any sense ?
To answer this question, any inversion strategy must (i) address whether the reconstructions converge to the truth somehow as noise level approaches zero, or in the large sample size limit; (ii) provide uncertainty quantification bounds (e.g., can I tell how large my sample must be, to guarantee that my reconstruction is within a prescribed error margin to the truth ?).
Our ability to answer (i) and (ii) is in fact intimately connected to our ability to understand the noiseless measurement operator (proving statements of 'injectivity', 'stability', and sharp mapping properties of the measurement operator), and I will present some recent results in this direction, and discuss how these results provide the background facts needed to address (i) and (ii) above.
Numerical illustrations will be presented.
Joint works with Gabriel Paternain and Richard Nickl (Cambridge).
Tuesday, June 4th, 2019
Anthony Tromba, University of California Santa Cruz
On a Resolution of Hilbert's 19th Problem
At the International Congress of Mathematicians in Paris in 1900 David Hilbert who was, at the time, among the world's most prominent mathematicians, gave a lecture listing 23 problems which he believed to be central to further mathematical progress as well as being among the greatest intellectual challenges of the 20th Century. The 19th, 20th and 23rd problems had to do with the Calculus of Variations. The 20th was settled by the 1940's. Major steps towards a solution of the 19th were taken in the 1950's and 60's by Ennio De Giorgi, John Nash, Jürgen Moser, Olga Ladyzhenskaya and Nina Ural'steva all of whom contributed to a solution to what we now call the scalar case. The Non-Scalar Case remained open.
This talk will focus on the history of this beautiful problem and on a resolution 120 years after it was first posed
