Events for 11/10/2021 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 10:00AM - 11:00AM

Location: Zoom

Speaker: Christian Brennecke, University of Bonn

Title: Bogoliubov Theory for Trapped Bosons in the Gross-Pitaevskii Regime

Abstract: In this talk I present a rigorous derivation of Bogoliubov theory for systems of $N$ trapped bosons in $\mathbb{R}^3$ in the so called Gross-Pitaevskii regime, characterized by a scattering length of order $N^{-1}$. We prove complete Bose-Einstein condensation for approximate ground states with optimal rate and determine the low-energy excitation spectrum of the system up to errors vanishing in the limit $N\to \infty$. The talk is based on joint work with S. Schraven and B. Schlein.

Number Theory Seminar

Time: 11:00AM - 12:00PM

Location: BLOC 302

Speaker: Yen-Tsung Chen, National Tsing Hua University

Title: Linear equations on Drinfeld modules

Abstract: Let K be a number field and E be an elliptic curve defined over K. Given finitely many K-rational points P_1,..., P_n, how can we decide if they are linearly dependent in the sense that there exist integers a_1,...,a_n, not all zero, such that a_1P_1+...+a_nP_n=0 on E(K)? In 1988, Masser gave an answer to this question using techniques from the geometry of numbers. Let L be a global function field. In this talk, we would like to present an analogue of Masser's result for L-rational points on a given Drinfeld module defined over L.

Numerical Analysis Seminar

Time: 1:00PM - 2:00PM

Location: Zoom

Speaker: Maryam Parvizi, Leibniz University of Hannover

Title: On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

Abstract: In this talk, we consider locally L2(Ω)-stable operators mapping into spaces of continuous piecewise polynomial set on shape regular meshes with certain approximation properties in L2(Ω). For such operators, we discuss the following stability results: •These operators are stable mappings H3/2(Ω) →B3/2 2,∞(Ω). •If the mesh is additionally quasi-uniform, for the space of continuous piecewise polynomials on this mesh, we have a sharper stability estimate B3/2 2,∞(Ω) →B3/2 2,∞(Ω). Given a mesh T obtained by Newest Vertex Bisection (NVB) refinement from a regular triangulation ̂T0 and ̂T` as the sequence of uniformly refined NVB-generated meshes, we introduce the finest common coarsening (fcc) of two meshes ̃T` := fcc(T, ̂T`). For the space of continuous piecewise polynomials defined on the mesh hierarchy ̃T`, we construct the modified Scott-Zhang operator ̃ISZ` in such a way that for continuous piecewise polynomials on T, this operator coincides with the Scott-Zhang operator̂ISZ` on ̂T`. Since the Scott-Zhang operators are local, L2(Ω)-stable operators with certain approximation prop- erties in L2(Ω), therefore these operators admit the above stability results. Taking advantage of the stability results and the mentioned property of the modified Scott-Zhang operators, we present multi- level norm equivalences in the Besov spaces B3θ/2 2,q (Ω), θ ∈(0,1), q ∈[1,∞]. As an application, we present a local multilevel diagonal preconditioner for the integral fractional Laplacian (−∆)s for s ∈ (0,1) on adaptively refined meshes and prove this multilevel diagonal scaling gives rise to uniformly bounded condition number for the integral fractional Laplacian. To prove the main result, we apply the norm equivalence of the multilevel decomposition.

URL: Event link

Noncommutative Geometry Seminar

Time: 2:00PM - 3:00PM

Location: ZOOM

Speaker: Bo Zhu, University of Minnesota

Title: Geometry of positive scalar curvature on complete three-dimensional manifolds

Abstract: One of the basic questions related to positive scalar curvature is how the positive scalar curvature controls the size of geometry, many concepts and conjectures have been introduced by Gromov and many questions remain conjectural level. In this talk, we will discuss the interplay of geometry and positive scalar curvature on a complete manifold. Particularly, we will talk about volume growth of geodesic balls, an estimate of integral of scalar curvature and Uryson width on complete manifolds with nonnegative Ricci curvature and strictly positive scalar curvature based on my recent progress.

URL: Event link

Topology Seminar

Time: 4:00PM - 4:50PM

Location: zoom

Speaker: Dylan Wilson, Harvard University

Title: Higher Bott periodicities in Algebraic K-theory

Abstract: Algebraic K-theory is a powerful invariant that encodes a lot of information in number theory and geometric topology. One of the deepest theorems about the algebraic K-theory of number rings is that it approximately behaves like complex topological K-theory, which famously satisfies Bott periodicity (this is the Lichtenbaum-Quillen conjecture, now resolved by work of Voevodsky and many others). I will describe joint work with Jeremy Hahn where we exhibit an analog of this result involving periodicities of larger and larger 'wavelength', thus affirming the Redshift Conjecture of Ausoni-Rognes for a large class of examples.