Speaker: Prof. Chunjing Xie (Shanghai Jiaotong University)
Time: 16:00-17:00 April 30, 2024 (Tuesday)
Place: MCM110
Title: The rigidity of steady solutions of Navier-Stokes system and its applications
Abstract: The Liouville type theorem for stationary Navier-Stokes system in the whole space is longstanding open problem. In this talk, we first discuss the rigidity of steady Navier-Stokes system with dimension bigger than three in a class more general than self-similar solutions, where we do not need any type of self-similarity or smallness of solutions. Furthermore, this rigidity result is used to study the regularity and far field behavior of steady solutions of high dimensional Navier-Stokes system. Finally, we discuss the rigidity for steady Navier-Stokes system in domains with physical boundaries.
Speaker: Wooyeon Kim (ETH Zurich)
Time: 14:00-15:00 May 1st, 2024 (Wednesday)
Venue: MCM110
Title: Moments of Margulis functions and indefinite ternary quadratic forms
Abstract: We consider the moments of the Margulis \alpha-function integrating over expanding translates of a unipotent orbit in the space of 3-dimensional lattices. I will present a uniform boundedness result for the moments of the Margulis function over expanding translates of a unipotent orbit, under suitable Diophantine conditions of the initial unipotent orbit. I will also talk about its application in the distribution of values of an indefinite irrational ternary quadratic form at integral points.
Speaker: Dr. Andreas Wieser (Hebrew University of Jerusalem)
Time: 14:00-15:00 April 30, 2024 (Tuesday)
Place: MCM110
Title: Rational subspaces, shapes, and equidistribution
Abstract: Following works of Maass from the 50's and Schmidt from the 90's we study rational subspaces of Euclidean space and the shape of the integer lattice in them.
Here, we order the subspaces by height/discriminant. Conjecturally, the triples consisting of a subspace, the shape of the lattice in the subspace, and the shape in the orthogonal complement should equidistribute in the appropriate ambient space when the height goes to infinity. In this talk, we discuss the current knowledge on this conjecture based on works with Aka, Einsiedler, Luethi, Michel, and Musso. These works rely crucially on a variety of results in homogeneous dynamics.
If time permits, we will also discuss a recently discovered connection of shapes of subspaces to Gauss composition of quadratic forms and the role this connection plays in a problem from knot theory.
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