The inverse problem we address is whether we can determine the structure of a region in space-time by measuring point light sources coming from the region. We can also observe gravitational waves since the LIGO detection in 2015. We will also consider inverse problems for nonlinear hyperbolic equations, including Einstein's equations, involving active measurements.
Polynomials and piecewise polynomials are most commonly used function classes in analysis and applications. In this talk, I will use these function classes to motivate and to present some remarkable properties of the function classes given by (both shallow and deep) neural networks. In particular, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.
We will discuss the critical almost Mathieu operator: Azbel/Hofstadter/Harper model of an electron on the square lattice in a magnetic field. When the commensurability parameter between the lattice and the magnetic field is irrational, the spectrum of the model is a zero-measure Cantor set and its Hausdorff dimension is not larger than 1/2. We will emphasize the significance of the two-dimensionality of the problem, which was used in recent work of the speaker with S. Jitomirskaya. We will also discuss some similarities with integrable two-dimensional statistical models: the Ising model and the dimer problem.
I will show how basic properties of polynomials can be used
in the study of problems in Combinatorics, Additive Number Theory,
Combinatorial Geometry and Graph Theory, describing recent and
less recent results, problems and algorithmic challenges.
Paraproducts are building blocks of many singular integral operators and the main instrument in proving ``Leibniz rule" for fractional derivatives (Kato--Ponce). Multi-parameter paraproducts are tools to prove more complicated Leibniz rules that are also widely used in well posedness questions for various PDEs. Alternatively, multi-parameter paraproducts appear naturally in questions of embedding of spaces of analytic functions in polydisc into Lebesgue spaces with respect to a measure in the polydisc.
Those Carleson embedding theorems often serve as a first building block for interpolation in complex space and also for corona type results. The embedding of spaces of holomorphic functions on n-polydisc can be reduced to problem (without loss of information) of boundedness of weighted dyadic n-parameter paraproducts.
We find the necessary and sufficient condition for this boundedness in n-parameter case, when n is 1, 2, or 3. The answer is quite unexpected and seemingly goes against the well known difference between box and Chang--Fefferman condition that was given by Carleson quilts example of 1974.
A random nxm matrix gives a random linear transformation from \Z^m to \Z^n (between vectors with integral coordinates). Asking for the probability that such a map is injective is a question of the non-vanishing of determinants. In this talk, we discuss the probability that such a map is surjective, which is a more subtle integral question. We show that when m=n+u, for u at least 1, as n goes to infinity, the surjectivity probability is a non-zero product of inverse values of the Riemann zeta function. This probability is universal, i.e. we prove that it does not depend on the distribution from which you choose independent entries of the matrix, and this probability also arises in the Cohen-Lenstra heuristics predicting the distribution of class groups of real quadratic fields. This talk is on joint work with Hoi Nguyen.