Zoom https://us02web.zoom.us/j/85954136465?pwd=RmZzNzU4TXVZOS9sSEhHRkFFa1RFUT09 Meeting ID: 859 5413 6465 Passcode: song
Corona problem was initiated from function algebra, then it became one of the most important problems in harmonic analysis and complex analysis. For the case of one complex variable, the problem was solved by L. Carleson in 1960s. In the several complex variables, the problem remains open. In this talk, I will introduce the development of the Corona Problems of one and several complex variables. I will also introduce the Hormander’s weighted L2-estimate for ∂, and demonstrate how to use the weighted L2 estimates to prove some old and new results on Corona Problem.
The goal of this talk is to explain how enumerative geometry can be used to simplify the solution of polynomials in one variable. Given a polynomial in one variable, what is the simplest formula for the roots in terms of the coefficients? Hilbert conjectured that for polynomials of degree 6,7 and 8, any formula must involve functions of at least 2, 3 and 4 variables respectively (such formulas were first constructed by Hamilton). In a little-known paper, Hilbert sketched how the 27 lines on a cubic surface should give a 4-variable solution of the general degree 9 polynomial. In this talk I’ll recall Klein and Hilbert's geometric reformulation of solving polynomials, explain the gaps in Hilbert's sketch and how we can fill these using modern methods. As a result, we obtain best-to-date upper bounds on the number of variables needed to solve a general degree n polynomial for all n, improving results of Segre and Brauer.
We provide a general introduction to the field of inverse boundary problems for elliptic PDE, with the celebrated Calderon problem serving as a prototypical example. The emphasis of the talk is on the techniques based on Carleman estimates and their role in the construction of complex geometric optics solutions. We also survey some of the more recent developments, including partial data problems, inverse boundary problems on manifolds, as well as inverse boundary problems for non-linear equations.
A typical result in graph/hypergraph theory has the following structure: Every G satisfying certain conditions must have some target property P. For example, a classical theorem by Dirac asserts that every graph on n vertices and with minimum degree at least n/2 must contain a hamiltonian cycle (that is, a cycle that passes through every vertex).
After establishing such a theorem, it is natural to ask how ``robust'' is G with respect to this property P. In this talk we discuss some possible measures of ``robustness'' and illustrate them with many examples.
iFEM is a MATLAB software package containing robust, efficient, and easy-following codes for the main building blocks of adaptive finite element methods on unstructured simplicial grids in both two and three dimensions. Besides the simplicity and readability, sparse matrixlization, an innovative programming style for MATLAB, is introduced to improve the efficiency. In this novel coding style, the sparse matrix and its operations are used extensively in the data structure and algorithms.