Week of February 28, 2021

Thu Mar 4, 2021
9:00am to 10:00am - Zoom - Inverse Problems
Samuli Siltanen - (University of Helsinki)
Learning from electric X-ray images: the new EIT

https://sites.uci.edu/inverse/

10:00am to 11:00am - https://uci.zoom.us/j/93076750122?pwd=Y3pLdndoQTBuNUhxQUxFMkQ2QnRFQT09 - Mathematical Physics
C. Fischbacher - (UCI)
Entanglement Entropy Bounds in the Higher Spin XXZ Chain
This is the second part of a series of two talks. We consider the Heisenberg XXZ spin-$J$ chain ($J\in\mathbb{N}/2$) with anisotropy parameter $\Delta$. Assuming that $\Delta>2J$, and introducing threshold energies $E_{K}:=K\left(1-\frac{2J}{\Delta}\right)$, we show that the bipartite entanglement entropy (EE) of states belonging to any spectral subspace with energy less than $E_{K+1}$ satisfy a logarithmically corrected area law with prefactor $(2\lfloor K/J\rfloor-2)$. This generalizes previous results by Beaud and Warzel as well as Abdul-Rahman, Stolz, and CF who covered the spin-$1/2$ case.
3:00pm to 4:00pm - Zoom: https://uci.zoom.us/j/99322295399 - Number Theory
Fatma Karaoğlu - (Tekirdağ Namık Kemal University)
Smooth Cubic Surfaces with at Least 9 Lines

 

A cubic surface is an algebraic variety of degree three in projective three space.  We will study cubic surfaces over different fields.  We are interested in the number of points and lines on a smooth cubic surface.  In this talk, we will focus on smooth cubic surfaces with at least 9 lines.  There are three cases with 27, 15 and 9 lines, respectively.  We will describe these surfaces in terms of normal forms, each of which involves either 4 or 6 parameters over the given field.  Using birational maps, the rational pooints on these normal forms will be described explicitly.

Fri Mar 5, 2021
3:00pm to 4:00pm - Zoom - Nonlinear PDEs
Jürgen Saal - (Heinrich-Heine-Universität Düsseldorf (Germany))
Fluid Flow in General Domains
(Incompressible) Fluid flow in a domain is described by the fundamental Stokes (linear) and Navier-Stokes (nonlinear) equations. The Helmholtz decomposition into solenoidal and gradient fields serves as a helpful tool to analyze these systems. It has been an open question for some decades, whether the existence of the Helmholtz decomposition (which is equivalent to weak well-posedness of the Neumann problem) is necessary for well-posedness of Stokes and Navier-Stokes equations in the $L^q$-setting for $q\in(1,\infty)$. Note that by a classical result of Bogovski\u{i} and Maslennikova there are uniformly smooth domains, so-called non-Helmholtz domains, such that the Helmholtz decomposition does not exist. In my talk, I intent to present positive and negative results on well-posedness of the Stokes and Navier-Stokes equations in $L^q$ for a large class of uniform $C^{2,1}$-domains. In particular, classes of non-Helmholtz domains are addressed. This will include a comprehensive answer to the open question for the case of partial slip type boundary conditions.
The project is a joint work with Pascal Hobus.
Zoom link
4:00pm to 5:00pm - Zoom - Graduate Seminar
Richard Schoen - (UC Irvine)
From lines and circles to soap films and bubbles

We can think of a line as the shortest curve joining its endpoints and of a circle as the shortest closed curve enclosing a fixed area (isoperimetric problem). In this talk we will discuss what happens when curves are replaced by surfaces in such problems, and how the solutions of these problems can shed light on the differential geometry of curved spaces. 

Location: Zoom Address https://zoom.us/j/8473088589