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The Fourier extension operator
The Fourier extension operator
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role on the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theoryand combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role on the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.
== Past Colloquia ==
== Past Colloquia ==
Revision as of 19:17, 27 October 2018
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
The calendar for spring 2019 can be found here.
|Sep 12 Room 911||Gunther Uhlmann (Univ. of Washington) Distinguished Lecture series||Harry Potter's Cloak via Transformation Optics||Li|
|Sep 14 Room 911||Gunther Uhlmann (Univ. of Washington) Distinguished Lecture series||Journey to the Center of the Earth||Li|
|Sep 21 Room 911||Andrew Stuart (Caltech) LAA lecture||The Legacy of Rudolph Kalman||Jin|
|Sep 28||Gautam Iyer (CMU)||Stirring and Mixing||Thiffeault|
|Oct 5||Eyal Subag (Penn State)||Symmetries of the hydrogen atom and algebraic families||Gurevich|
|Oct 12||Andrei Caldararu (Madison)||Mirror symmetry and derived categories||...|
|Oct 19||Jeremy Teitelbaum (U Connecticut)||Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist||Boston|
|Oct 26||Douglas Ulmer (Arizona)||Rational numbers, rational functions, and rational points||Yang|
|Nov 2||Ruixiang Zhang (Madison)||The Fourier extension operator|
|Nov 7||Reserved for job talk||TBA||hosting faculty|
|Nov 9||Reserved for job talk||TBA||hosting faculty|
|Nov 16||Reserved for job talk||TBA||hosting faculty|
|Nov 30||Reserved for job talk||TBA||hosting faculty|
|Dec 7||Reserved for job talk||TBA||hosting faculty|
Sep 12: Gunther Uhlmann (Univ. of Washington)
Harry Potter's Cloak via Transformation Optics
Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc. including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last fifteen years or so there have been several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion one of them, the so-called "traansformation optics" in a non-technical fashion n the so-called that has received the most attention in the scientific literature.
Sep 14: Gunther Uhlmann (Univ. of Washington)
Journey to the Center of the Earth
We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.
The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.
We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.
Sep 21: Andrew Stuart (Caltech)
The Legacy of Rudolph Kalman
In 1960 Rudolph Kalman published what is arguably the first paper to develop a systematic, principled approach to the use of data to improve the predictive capability of mathematical models. As our ability to gather data grows at an enormous rate, the importance of this work continues to grow too. The lecture will describe this paper, and developments that have stemmed from it, revolutionizing fields such space-craft control, weather prediction, oceanography and oil recovery, and with potential for use in new fields such as medical imaging and artificial intelligence. Some mathematical details will be also provided, but limited to simple concepts such as optimization, and iteration; the talk is designed to be broadly accessible to anyone with an interest in quantitative science.
Sep 28: Gautam Iyer (CMU)
Stirring and Mixing
Mixing is something one encounters often in everyday life (e.g. stirring cream into coffee). I will talk about two mathematical aspects of mixing that arise in the context of fluid dynamics:
1. How efficiently can stirring "mix"?
2. What is the interaction between diffusion and mixing.
Both these aspects are rich in open problems whose resolution involves tools from various different areas. I present a brief survey of existing results, and talk about a few open problems.
Oct 5: Eyal Subag (Penn State)
Symmetries of the hydrogen atom and algebraic families
The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system. No prior knowledge on quantum mechanics or representation theory will be assumed.
Oct 12: Andrei Caldararu (Madison)
Mirror symmetry and derived categories
Mirror symmetry is a remarkable phenomenon, first discovered in physics. It relates two seemingly disparate areas of mathematics, symplectic and algebraic geometry. Its initial formulation was rather narrow, as a technique for computing enumerative invariants (so-called Gromov-Witten invariants) of symplectic varieties by solving certain differential equations describing the variation of Hodge structure of “mirror" varieties. Over the past 25 years this narrow view has expanded considerably, largely due to insights of M. Kontsevich who introduced techniques from derived categories into the subject. Nowadays mirror symmetry encompasses wide areas of mathematics, touching on subjects like birational geometry, number theory, homological algebra, etc.
In my talk I shall survey some of the recent developments in mirror symmetry, and I will explain how my work fits in the general picture. In particular I will describe an example of derived equivalent but not birational Calabi-Yau three folds (joint work with Lev Borisov); and a recent computation of a categorical Gromov-Witten invariant of positive genus (work with my former student Junwu Tu).
Oct 19: Jeremy Teitelbaum (U Connecticut)
Lessons Learned and New Perspectives: From Dean and Provost to aspiring Data Scientist
After more than 10 years in administration, including 9 as Dean of Arts and Sciences and 1 as interim Provost at UConn, I have returned to my faculty position. I am spending a year as a visiting scientist at the Jackson Laboratory for Genomic Medicine (JAX-GM) in Farmington, Connecticut, trying to get a grip on some of the mathematical problems of interest to researchers in cancer genomics. In this talk, I will offer some personal observations about being a mathematician and a high-level administrator, talk a bit about the research environment at an independent research institute like JAX-GM, outline a few problems that I've begun to learn about, and conclude with a discussion of how these experiences have shaped my view of graduate training in mathematics.
Oct 26: Douglas Ulmer (Arizona)
Rational numbers, rational functions, and rational points
One of the central concerns of arithmetic geometry is the study of solutions of systems of polynomial equations where the solutions are required to lie in a "small" field such as the rational numbers. I will explain the landscape of expectations and conjectures in this area, focusing on curves and their Jacobians over global fields (number fields and function fields), and then survey the progress made over the last decade in the function field case. The talk is intended to be accessible to a wide audience.
Nov 2: Ruixiang Zhang (Madison)
The Fourier extension operator
I will present an integral operator that originated in the study of the Euclidean Fourier transform and is closely related to many problems in PDE, spectral theory, analytic number theory, and combinatorics. I will then introduce some recent developments in harmonic analysis concerning this operator. I will mainly focus on various new ways to "induct on scales" that played an important role on the recent solution in all dimensions to Carleson's a.e. convergence problem on free Schrödinger solutions.