Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Department Colloquium

Department Colloquium - Camille Horbez (CNRS-Universite Paris Sud)

Camille Horbez (CNRS-Universite Paris Sud)

Friday, October 19th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Camille Horbez (CNRS – Universite Paris Sud)

Title: Mostow-type rigidity and normal subgroups of automorphisms of free groups.

Abstract: The talk is based on a recent joint work with Richard D. Wade. Let Out(Fn) be the outer automorphism group of a finitely generated free group. In 2007, Farb and Handel proved that when n is at least 4, every isomorphism between two finite-index subgroups of Out(Fn) extends to an inner automorphism of Out(Fn). This rigidity statement, asserting that Out(Fn) has no more symmetries than the obvious ones, can be viewed as an analogue of the Mostow rigidity theorem for lattices in Lie groups, or of a result of Ivanov for mapping class groups of surfaces. We recently gave a new proof of Farb and Handel's theorem, which enabled us to also understand the symmetries of some natural normal subgroups of Out(Fn). In my talk, I will emphasize the analogies between Out(Fn), arithmetic groups and mapping class groups, and will present some general ideas behind these rigidity phenomena.

Camille Horbez obtained his Ph.D in at the Universite de Rennes in 2014, and after a year at the University of Utah, he became a Charge de Recherches for the CNRS at the Universite de Paris Sud (Orsay). In 2017, Camille was selected to give a Cours Peccot, a semester long course given at the College de France by a mathematician less than 30 year old.

Department Colloquium - Undergrad Summer Project Presentations

Undergrad Summer Project Presentations

Friday, October 12th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Undergraduate Summer Project Presentations

This week colloquium will consists of five short presentations by:

  • Riley Becker - On the Size of Diophantine $m$-tuples.
  • Fernando Camacho Cadena & Troy Giorshev - On a Construction of Surfaces with Distinct Hilbert Metrics and Identical Length Spectra.
  • Sean Monahan - The Cartan determinant conjecture.
  • Shikai Liu - The fluctuations of the Kesten-McKay law.
  • Linke Li - Kalman Filter and its variation.
  • Riley Becker - On the Size of Diophantine m-tuples.

    Riley Becker
    On the Size of Diophantine $m$-tuples.

  • Fernando Camacho Cadena & Troy Giorshev - On a Construction of Surfaces with Distinct Hilbert Metrics and Identical Length Spectra.

    Fernando Camacho Cadena & Troy Giorshev
    On a Construction of Surfaces with Distinct Hilbert Metrics and Identical Length Spectra.

  • Sean Monahan - The Cartan determinant conjecture.

    Sean Monahan
    The Cartan determinant conjecture.

  • Shikai Liu - The fluctuations of the Kesten-McKay law.

    Shikai Liu
    The fluctuations of the Kesten-McKay law.

  • Linke Li - Kalman Filter and its variation.

    Linke Li
    Kalman Filter and its variation.

Department Colloquium - Aaron Smith (Queen's University)

Aaron Smith, Queen's University

Friday, October 5th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Aaron Smith (Queen's University)

Title: Mixing and the Glass Transition.

Abstract: Supercooled liquid forms when a liquid is cooled below its usual freezing temperature without entering a crystalline solid phase. As supercooled liquids continue to get colder, they exhibit something called the glass transition: they remain disordered, but start to otherwise behave much like solids. This glass transition is important for many materials, including rubbers and colloids, but is not theoretically well-understood. In this talk I will introduce a simple model for the glass transition that is easy to understand but difficult to study. I will then introduce two related families of models, introduced by physicists, that seem to give similar "glassy" behaviour. Finally, I will present some heuristics and recent results on the relaxation and mixing behavior of these two models. My results in this talk are from joint and ongoing work with Paul Chleboun, Alessandra Faggionato, Fabio Martinelli, Natesh Pillai and Cristina Toninelli.

Aaron Smith works in the areas of applied probability, with a focus on Markov chain Monte Carlo and related methods from computational statistics or statistical physics. He also has interest in in data mining and machine learning. He obtained his Ph.D. at Stanford in Mathematics, and was an undergraduate student at Queen's University. He held short-term appointments at Federal government of Canada, Brown University (applied math), and Harvard (statistics). He is currently an assistant professor in the Department of Mathematics and Statistics at the University of Ottawa.

Department Colloquium - Jon Chaika (University of Utah)

Jon Chaika, University of Utah

Friday, September 28th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Jon Chaika (University of Utah)

Title: Horocycle orbits in strata of translation surfaces.

Abstract: Ergodic theory is concerned with describing the long term behavior of orbits as time evolves. Ratner, Margulis, Dani and many others, showed that the horocycle flow have strong measure theoretic and topological rigidity properties that allow a good understanding of every such orbit. Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, showed that the action of $SL(2,\mathbb{R})$, and its upper triangular subgroup, on strata of translation surfaces have similar rigidity properties. We will describe how some of these results fail for the horocycle flow on strata of translation surfaces. In particular, 1) There exist horocycle orbit closures with fractional Hausdorff dimension; 2) There exist points which do not equidistribute under the horocycle flow with respect to any measure; 3) There exist points which equidistribute under the horocycle flow with respect to a measure, but they are not in the topological support of that measure. No familiarity with these objects will be assumed and the talk will begin with motivating the subject of dynamics and ergodic. This is joint work with John Smillie and Barak Weiss.

Jon Chaika works in the field of Dynamical systems. He did his undergraduate at the University of Iowa, obtained his Ph.D. from Rice, then went to the University of Chicago before coming to the University of Utah.

Department Colloquium - Boris Levit (Queen's University)

Boris Levit, Queen's University

Friday, September 21st, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Boris Levit (Queen's University)

Title: Optimal Cardinal Interpolation in Approximation Theory, Nonparametric Regression, and Optimal Design

Abstract: For the Hardy classes of functions analytic in the strip around real axis of a size $2\beta$, an optimal method of cardinal interpolation has been proposed within the framework of Optimal Recovery. It will be shown that this method, based on the Jacobi elliptic functions, is also optimal according to the criteria of Nonparametric Regression and Optimal Design. In a stochastic non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from $0$. A pivotal role is played by the interference effect, in which the oscillations exhibited by the interpolant's bias and variance mutually cancel each other. In the limiting case $\beta \rightarrow \infty $, the optimal interpolant converges to the well known Nyquist-Shannon cardinal sampling series.

Department Colloquium - Anthony Bloch (University of Michigan)

Anthony Bloch, University of Michigan

Friday, September 14th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Anthony Bloch (University of Michigan)

Title: Control and Geometry of Quantum systems with Dissipation

Abstract: In this talk we discuss aspects of the mathematics, control and geometry of quantum control systems interacting with their environment. In particular we discuss the control of a finite-dimensional dissipative Lindblad system by considering its orbit and interorbit dynamics. This entails considering the geometry of the system and its unitary orbits, the structure of the Lindblad operator, and the convexity associated with the density equation. Applications are given to constructing pure states. We discuss controllability and also discuss optimality and optimal control in this setting.

Anthony Bloch (University of Michigan): Anthony M. Bloch is the Alexander Ziwet Collegiate Professor of Mathematics and current department chair at the University of Michigan. He received his B.Sc.~in Applied Mathematics and Physics from the University of the Witwatersrand, Johannesburg, an M.~S.~in Physics from CalTech, an M.~Phil in Control Theory and Operations Research from Cambridge and a Ph.~D in Applied Mathematics from Harvard. He has received many awards including a Presidential Young Investigator Award, a Guggenheim Fellowship and a Simons Fellowship and is Fellow of the IEEE, SIAM and the AMS.

Department Colloquium - Oleg Bogoyavlenskij (Queen's University)

Oleg Bogoyavlenskij, Queen's University

Friday, September 7th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Oleg Bogoyavlenskij (Queen's University)

Title: Counterexamples to Moffatt’s statements on vortex knots

Abstract: One of the well-known problems of hydrodynamics is studied - the problem of classification of vortex knots for ideal fluid flows. In the literature there are known Moffatt's statements that all torus knots $K_{m,n}$ for all rational numbers $m/n$ $(0 < m/n < \infty)$ are realized as vortex knots for each one of the considered axisymmetric fluid flows. We prove that actually such a uniformity does not exist because it does not correspond to the facts. Namely, we derive a complete classification of all vortex knots realized for the fluid flows studied by Moffatt and demonstrate that the real structure of vortex knots is much more rich because the sets of mutualy non-isotopic vortex knots realized for different axisymmetric fluid flows are all different.

Department Colloquium - Amie Wilkinson (University of Chicago)

Amie Wilkinson, University of Chicago

Friday, April 6th, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Amie Wilkinson (University of Chicago)

Title: Robust Mechanisms for Chaos

Abstract: What are the underlying mechanisms for robustly chaotic behavior in smooth dynamics? In addressing this question, I will focus on the study of diffeomorphisms of a compact manifold, where "chaotic" means "mixing" and and "robustly" means "stable under smooth perturbations." I will describe recent advances in constructing and using tools called "blenders" to produce stably chaotic behavior with arbitrarily little effort.

Amie Wilkinson (University of Chicago): Prof. Amie Wilkinson received her BA from Harvard and her Ph.D. from the University of California at Berkeley. After a post-doc at Harvard, she became a professor at North­ western University, where she stayed 13 years, before moving to the University of Chicago in 2012. Prof. Wilkinson is a leading researcher in ergodic theory and dynamical sys­ tems. Among other recognitions of her many achievements, she was an invited speaker at the International Congress of Mathematicians in 2010, was awarded the Satter Prize in Mathematics in 2011, and was elected a fellow of the AMS in 2014. Prof. Wilkinson has also been very active in public outreach, giving numerous public lectures, interviews, and recently publishing an article in the New York Times.

Fields Lecture Series - Amie Wilkinson (University of Chicago)

Amie Wilkinson, University of Chicago

Thursday, April 5th, 2018

Time: 5:30 p.m.  Place: Jeffery Hall 127

Speaker: Amie Wilkinson (University of Chicago)

Title: The Mathematics of Deja Vu

Abstract: Dynamics is an area of mathematics concerned with the motion of spaces (" dynamical systems") over time. Dynamics has its roots in the late nineteenth century, when it was developed as a tool to understand physical phenomena, such as the motion of gas molecules in a box and planets around the sun. A simple and yet powerful concept in dynamics is that of recurrence. In everyday language, recurrence is the mathematical version of deja vu: a motion of a space is recurrent if, given enough time, it eventually returns to its original configuration (allowing for a small amount of error). In this talk, I will describe how mathematical results about recurrence can be used to answer surprisingly disparate questions, from the mixing and unmixing of two ideaI gases in a box, to deep properties of the prime numbers, to the discovery of exoplanets in nearby solar systems.

Amie Wilkinson (University of Chicago): Prof Wilkinson (University of Chicago) is a leading researcher in ergodic theory and dynamical systems. Among other recognitions of her many achievements, she was an invited speaker at the International Congress of Mathematicians in 2010, was awarded the Satter Prize in Mathematics in 2011, and was elected a fellow of the AMS in 2014. Prof Wilkinson has also been very active in public outreach, giving numerous public lectures, interviews, and recently publishing an article in the New York Times.

Department Colloquium - Mihai Nica (University of Toronto)

Mihai Nica, University of Toronto

Friday, March 23rd, 2018

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Mihai Nica (University of Toronto)

Title: Phase transitions in random matrices and the spiked tensor model

Abstract: Given a matrix of noisy data, principal component analysis (PCA) can be viewed as "de-noising" technique that recovers the closest rank-one approximation. In certain matrix models, it is known that this procedure exhibits a phase transition: if the signal-to-noise ratio is below a critical value then PCA returns uninformative information. In this talk, we also consider a generalization of this problem to k-tensors (the matrix case corresponds to k=2). By studying the energy landscape of this model, we also find phase transitions akin to the matrix case. The proof of the results uses the Kac-Rice formula for the expected number of critical points of a random function and results about spiked random matrices. Based on joint work with Gerard Ben Arous, Song Mei and Andrea Montanari.

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