Applications of Descriptive Set Theory in Dynamical systems

Speaker: 

Matt Foreman

Institution: 

UCI

Time: 

Friday, April 8, 2022 - 4:00pm to 5:00pm

Host: 

Location: 

MSTB 124

Many classical problems in dynamical systems have been studied for more than a century. Despite enormous progress, several have resisted solution. Descriptive Set Theory provides the tools to prove impossibility results, both in the quantitative and the qualitative behavior of dynamical systems.

What is Learning Theory?

Speaker: 

Paata Ivanisvili

Institution: 

UCI

Time: 

Friday, April 1, 2022 - 4:00pm

Host: 

Location: 

MSTB 124

I will talk about some challenging problems and the role of Fourier concentration in an emerging field called learning theory. Suppose we want to learn a certain data. The basic question we ask is what is the minimal amount of information needed  to learn (predict)  the data with high precision and probability, and what is the role of Fourier analysis in these questions. 

The almost-sure theories of finite graphs and finite metric spaces

Speaker: 

Isaac Goldbring

Institution: 

UCI

Time: 

Friday, January 28, 2022 - 4:00pm to 5:00pm

Host: 

Location: 

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

Given a property P of graphs, it is natural to wonder how likely a given finite graph satisfies property P, that is, given some fixed natural number n, what proportion of the n vertex graphs satisfies property P.  If P is a "first-order" property, then there is a 0-1 law that says the proportion of graphs of size n that satisfy P approaches either 0 or 1 as n tends to infinity.  A simple proof of this fact uses the model theory of the so-called random (or Rado) graph.  We will present the proof of this result and then discuss recent work, joint with Bradd Hart and Alex Kruckman, which show these ideas can be used to prove an approximate 0-1 law for finite metric spaces.  The metric space version of the random graph that is relevant turns out to be somewhat surprising!

A sketch of the neoclassical electromagnetic theory

Speaker: 

Alexander Figotin

Institution: 

UCI

Time: 

Friday, February 19, 2021 - 4:00pm to 5:00pm

Host: 

Location: 

Zoom

Abstract

This presentation is about our recently developed neoclassical theory of electromagnetic interactions. We demonstrate that the classical EM theory can be extended down to atomic scales so that many phenomena at atomic scales, usually explained in the quantum-mechanical framework, can be explained in our neoclassical framework. The proposed extension bridges the classical and quantum-mechanical approaches, so they are not separated by a gap but rather overlap in a large common domain. Our theory, though similar to QM in some respects, is markedly different from it. In particular: (i) there is no need in our theory for the correspondence principle and consequent quantization procedure to obtain the wave equation; (ii) the Heisenberg uncertainty principle, though quite often applicable, is not a universal principle; (iii) there is no configuration space; (iv) there is no probabilistic interpretation of the wave function.

Our theory features a new spatial scale - the size a_{\mathrm{e}} of a free electron. This scale is special to our theory and does not appear in either classical EM theory nor in the quantum mechanics where electron is always a point-like object. Our current assessed value for this scale is a_{\mathrm{e}}\approx100a_{\mathrm{B}} where a_{\mathrm{B}} is the Bohr radius, and consequently a_{\mathrm{e}}\approx5 nm. In our theory any elementary charge is a distributed in space quantity. Its size is understood as the localization radius which can vary depending on the situation. For instance, if an electron is bound to a proton in the Hydrogen atom then its the size of is approximately 1 Bohr radius, that is a_{\mathrm{B}}\approx0.05 nm, and when the electron is free its size is a_{\mathrm{e}}\approx100a_{\mathrm{B}}\approx5 nm.

Interestingly, the upper bound 25 nm is the skin depth and that implies that a nanosystem of size smaller than 25 nm is transparent to the external field. The same transparency should hold for a nanostructured surface indicating such a surface is better for nearly ideal field electron emission. There is an experimental evidence showing that the highest current densities were obtained for nanotips with sizes about 1nm yet another important fact supporting a possibility of a fundamental nanoscale.

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