Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Fall 2017

Date	Speaker	Title	Host(s)
September 8	Tess Anderson (Madison)	A Spherical Maximal Function along the Primes	Yang
September 15
September 22, 9th floor	Jaeyoung Byeon (KAIST)	Patterns formation for elliptic systems with large interaction forces	Rabinowitz & Kim
September 29		TBA
October 6, 9th floor	Jonathan Hauenstein (Notre Dame)	Real solutions of polynomial equations	Boston
October 13, 9th floor	Tomoko L. Kitagawa (Berkeley)	A Global History of Mathematics from 1650 to 2017	Max
October 20	Pierre Germain (Courant, NYU)	Stability of the Couette flow in the Euler and Navier-Stokes equations	Minh-Binh Tran
October 27	Stefanie Petermichl (Toulouse)	Higher order Journé commutators	Stovall, Seeger
We, November 1, B239	Shaoming Guo (Indiana)	Parsell-Vinogradov systems in higher dimensions	Seeger
November 3	TBA
November 10	Reserved for possible job talks	TBA
November 17	Reserved for possible job talks	TBA
November 24	Thanksgiving break	TBA
December 1	Reserved for possible job talks	TBA
December 5	Ryan Hynd (U Penn)	TBA
December 8	Reserved for possible job talks	TBA
December 11	Connor Mooney (ETH Zurich)	Finite time blowup for parabolic systems in the plane

Fall Abstracts

September 8: Tess Anderson (Madison)

Title: A Spherical Maximal Function along the Primes

Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior. The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example. In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to. We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory. This is joint work with Cook, Hughes, and Kumchev.

September 22: Jaeyoung Byeon (KAIST)

Title: Patterns formation for elliptic systems with large interaction forces

Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions. The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.

October 6: Jonathan Hauenstein (Notre Dame)

Title: Real solutions of polynomial equations

Abstract: Systems of nonlinear polynomial equations arise frequently in applications with the set of real solutions typically corresponding to physically meaningful solutions. Efficient algorithms for computing real solutions are designed by exploiting structure arising from the application. This talk will highlight some of these algorithms for various applications such as solving steady-state problems of hyperbolic conservation laws, solving semidefinite programs, and computing all steady-state solutions of the Kuramoto model.

October 13: Tomoko Kitagawa (Berkeley)

Title: A Global History of Mathematics from 1650 to 2017

Abstract: This is a talk on the global history of mathematics. We will first focus on France by revisiting some of the conversations between Blaise Pascal (1623–1662) and Pierre de Fermat (1607–1665). These two “mathematicians” discussed ways of calculating the possibility of winning a gamble and exchanged their opinions on geometry. However, what about the rest of the world? We will embark on a long oceanic voyage to get to East Asia and uncover the unexpected consequences of blending foreign mathematical knowledge into domestic intelligence, which was occurring concurrently in Beijing and Kyoto. How did mathematicians and scientists contribute to the expansion of knowledge? What lessons do we learn from their experiences?

October 20: Pierre Germain (Courant, NYU)

Title: Stability of the Couette flow in the Euler and Navier-Stokes equations

Abstract: I will discuss the question of the (asymptotic) stability of the Couette flow in Euler and Navier-Stokes. The Couette flow is the simplest nontrivial stationary flow, and the first one for which this question can be fully answered. The answer involves the mathematical understanding of important physical phenomena such as inviscid damping and enhanced dissipation. I will present recent results in dimension 2 (Bedrossian-Masmoudi) and dimension 3 (Bedrossian-Germain-Masmoudi).

October 27: Stefanie Petermichl (Toulouse)

Title: Higher order Journé commutators

Abstract: We consider questions that stem from operator theory via Hankel and Toeplitz forms and target (weak) factorisation of Hardy spaces. In more basic terms, let us consider a function on the unit circle in its Fourier representation. Let P_+ denote the projection onto non-negative and P_- onto negative frequencies. Let b denote multiplication by the symbol function b. It is a classical theorem by Nehari that the composed operator P_+ b P_- is bounded on L^2 if and only if b is in an appropriate space of functions of bounded mean oscillation. The necessity makes use of a classical factorisation theorem of complex function theory on the disk. This type of question can be reformulated in terms of commutators [b,H]=bH-Hb with the Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such as in the real variable setting, in the multi-parameter setting or other, these classifications can be very difficult.

Such lines were begun by Coifman, Rochberg, Weiss (real variables) and by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of spaces of bounded mean oscillation via L^p boundedness of commutators. We present here an endpoint to this theory, bringing all such characterisation results under one roof.

The tools used go deep into modern advances in dyadic harmonic analysis, while preserving the Ansatz from classical operator theory.

November 1: Shaoming Guo (Indiana)

Title: Parsell-Vinogradov systems in higher dimensions

Abstract: I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions. Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed. Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.

December 5: Ryan Hynd (U Penn)

Title: TBA.

December 11: Connor Mooney (ETH Zurich)

Title: Finite time blowup for parabolic systems in the plane

Abstract: Hilbert's 19th problem asks about the smoothness of solutions to nonlinear elliptic PDE that arise in the calculus of variations. This problem leads naturally to the question of continuity for solutions to linear elliptic and parabolic systems with measurable coefficients. We will first discuss some classical results on this topic, including Morrey's result that solutions to linear elliptic systems in two dimensions are continuous. We will then discuss surprising recent examples of finite time blowup from smooth data for linear parabolic systems in two dimensions, and important open problems.

Spring 2018

date	speaker	title	host(s)
March 16	Anne Gelb (Dartmouth)	TBA	WIMAW
April 4 (Wednesday)	John Baez (UC Riverside)	TBA	Craciun
April 6	Reserved	TBA	Melanie
April 13	Jill Pipher (Brown)	TBA	WIMAW
April 25 (Wednesday)	Hitoshi Ishii (Waseda University) Wasow lecture	TBA	Tran
date	person (institution)	TBA	hosting faculty
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