Week of October 17, 2021

Mon Oct 18, 2021
12:00pm - Zoom - Probability and Analysis Webinar
Cristina Benea - (University of Nantes)
TBA

https://sites.google.com/view/paw-seminar

4:00pm - Zoom https://zoom.us/j/8473088589 - Applied and Computational Mathematics
Peter Hinow - (University of Wisconsin - Milwaukee)
Modeling and Simulation of Ultrasound-mediated Drug Delivery to the Brain

We use a mathematical model to describe the delivery of a drug to a specific region of the brain. The drug is carried by liposomes that can release their cargo by application of focused ultrasound. Thereupon, the drug is absorbed through the endothelial cells that line the brain capillaries and form the physiologically important blood-brain barrier. We present a compartmental model of a capillary that is able to capture the complex binding and transport processes the drug undergoes in the blood plasma and at the blood-brain barrier. We apply this model to the delivery of L-dopa (used to treat Parkinson's disease) and doxorubicin (an anticancer agent). The goal is to optimize the delivery of drug while at the same time minimizing possible side effects of the ultrasound. In a second project, we present a mathematical model for drug delivery through capillary networks of increasing complexity with the goal to understand the scaling behavior of model predictions on a coarse-to-fine sequence of grids.

Tue Oct 19, 2021
1:00pm - Zoom - Dynamical Systems
Michael Yampolsky - (Toronto University)
Chaotic dynamics meets computer science: a study of computability of Julia sets

Numerical simulation has played a key role in the study of dynamical systems, from modeling ecosystems to weather simulations. Archetypical examples of complex fractals generated by simple non-linear dynamical systems are Julia sets of quadratic polynomials. Computer-generated Julia sets are among the most familiar mathematical images, enjoyed both for their beauty and for the deep theory behind them. In a series of works with M. Braverman and others we have put to the test the modern paradigm of numerical simulation of chaotic dynamics, and asked whether images of Julia sets can always be computed if the parameters are known. My talk will describe some of the surprising results we have obtained, and several intriguing open problems.

4:00pm to 5:00pm - NS2 1201 - Differential Geometry
Teng Fei - (Rutgers University)
The Type IIA flow and its applications in symplectic geometry

The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. It can also be used to prove a stability result about Kahler structures. This is based on joint work with Phong, Picard and Zhang.

Wed Oct 20, 2021
2:00pm to 3:00pm - 510R - Combinatorics and Probability
Asaf Ferber - (UCI)
Odd subgraphs are odd

In this talk we discuss the problems of finding large induced subgraphs of a given graph G with some degree-constraints. We survey some classical results, present some intersting and challenging open problems, and sketch solutions to some of them. 

This is based on joint works with Liam Hardiman and Michael Krivelevich. 

Thu Oct 21, 2021
9:00am to 10:00am - Zoom - Inverse Problems
Jingni Xiao - (Rutgers University)
Nonscattering Wavenumbers

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

10:00am to 10:50am - https://uci.zoom.us/j/95642648816 - Number Theory
Natalie Evans - (King's College)
Correlations of almost primes

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

11:00am - zoom ID: 949 5980 5461. Password: the last four digits of the zoom ID in the reverse order - Harmonic Analysis
Stefan Steinerberger - (University of Washington)
The Nicest Average and New Uncertainty Principles for the Fourier Transform

Two years ago, a colleague from economics asked me for the ”best” way to compute the average income over the last year. At first I didn’t understand but then he explained it to me: suppose you are given a real-valued function f(x) and want to compute a local average at a certain scale. What we usually do is to pick a nice probability measure u, centered at 0 and having standard deviation at the desired scale, and convolve f ∗ u. Classical candidates for u are the characteristic function or the Gaussian. This got me interested in finding the ”best” function u – this problem comes in two parts: (1) describing what one considers to be desirable properties of the convolution f ∗ u and (2) understanding which functions u satisfy these properties. I tried a basic notion for the first part, ”the convolution should be as smooth as the scale allows”, and ran into lots of really funky classical Fourier Analysis that seems to be new: (a) new uncertainty principles for the Fourier transform, (b) that potentially have the characteristic function as an extremizer, (c) which leads to strange new patterns in hypergeometric functions and (d) produces curious local stability inequalities. Noah Kravitz and I managed to solve two specific instances on the discrete lattice completely, this results in some sharp weighted estimates for polynomials on the unit interval – both the Dirichlet and the Fejer kernel make an appearance. The entire talk will be completely classical Harmonic Analysis, there are lots and lots of open problems and I will discuss several.