• Speaker: Alex Waldron, MSU
• Title: Yang-Mills flow in dimension four
• Date: 09/12/2018
• Time: 4:00 PM - 4:50 PM
• Place: C517 Wells Hall

Among the classical geometric evolution equations, YM flow is the least nonlinear and best behaved. Nevertheless, curvature concentration is a subtle problem when the base manifold has dimension four. I'll discuss my proof that finite-time singularities do not occur, and briefly describe the infinite-time picture. This talk will be more analytic and contains <50% overlap with my talk last Thursday.

• Speaker: N. K. Nikolski, University of Bordeaux
• Title: V.Ya.Kozlov's completeness problem
• Date: 10/03/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

In 1948-1950, V.Ya.Kozlov (1914-2007) stated a series of interesting geometric properties of dilated systems D(f)= {f(kx): k= 1,2,...} in the spaces L^p(0,1). Since that, no proofs were published. In particular, for a Rademacher-Haar-Walsh type generator f= 2-periodic odd extension of the indicator function of (0,a), 0<a<1, the system D(f) was claimed to be complete/incomplete for many particular values of a. We prove all Kozlov's statements and several new, as well as discuss other geometric properties of D(f).

• Speaker: Maxim Gilula, MSU
• Title: l^2 decoupling with vanishing curvature
• Date: 10/24/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

I will discuss recent progress on decoupling for curves with vanishing curvature.

• Speaker: Guozhen Lu, University of Connecticut; Nobody Else
• Title: Fourier analysis on hyperbolic spaces and sharp higher order Hardy-Sobolev-Maz'ya inequalities
• Date: 11/07/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

In this talk, we will describe some recent works on the sharp higher order Hardy-Sobolev-Maz'ya and Hardy-Adams inequalities on hyperbolic balls and half spaces. The relationship between the classical Sobolev inequalities and the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives will be established. Our main approach is to use the Fourier analysis on hyperbolic spaces and Green's function estimates.

• Speaker: Alexey Karapetyants
• Title: On Bergman type spaces of functions of nonstandard growth and some related questions.
• Date: 11/14/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

Abstracts: We study various Banach spaces of holomorphic functions on the unit disc and half plane. As a main question we investigate the boundedness of the corresponding holomorphic projection. We exploit the idea of V.P.Zaharyuta, V.I.Yudovich (1962) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We treat the cases of variable exponent Lebesgue space, Orlicz space, Grand Lebesgue space and variable exponent generalized Morrey space. The major idea is to show that the approach can be applied to a wide range of function spaces. This opens a door in a sense for introducing and studying new function spaces of Bergman type in complex analysis. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollifying dilations.

• Speaker: Alexander Volberg, MSU
• Title: Improving L^1 Poincar\'e inequality on Hamming cube
• Date: 11/28/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

L^1 Poincar\'e inequality on hypercube is related to many interesting questions in random graph theory (like Margoulis graph connectivity theorem, e.g.). The sharp constant is unknown, but I will show how to improve the previously known constant \pi/2 obtained by Ben Efraim and Lust--Piquard by using non-commutative harmonic analysis. The approach will be probabilistic and luckily commutative. For Gaussian space the constant is known, it is \sqrt{\pi/2}, and the short proof belong to Maurey--Pisier.