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Mathematics & Statistics

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Power Spectrum Analysis in Quantum Chaology and its Relation to Toeplitz Determinants

Roman Riser

Holon Institute of Technology, Israel

3:00pm-4:00pm

CMC 130

Seung-Yeop Lee

**Abstract**

By the Bohigas-Giannoni-Schmit conjecture (1984), the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behavior are described by random matrix theory. An alternative characterization of eigenvalue fluctuations was suggested where a long sequence of eigenlevels has been interpreted as a discrete-time random process. It has been conjectured that the power spectrum of energy level fluctuations shows \(1/\omega\) noise in the chaotic case, whereas, when the classical analog is fully integrable, it shows \(1/\omega^2\) behavior. This is expected for frequencies \(1\ll\omega\ll n\) where $n$ is the number of eigenlevels. We consider the power spectrum of the circular unitary ensemble with an additional fixed charge at 1. We will show that when the frequency gets of order \(n\) there is a correction to the \(1/\omega\) law which is described by a Painlevé V equation. Further we show a relation to Toeplitz determinants.

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Invariants in the Bergman and Szegö kernels in strictly pseudoconvex domains in \(\mathbb C^2\)

Peter Ebenfelt

University of California at San Diego

3:00pm-4:00pm

CMC 130

Dima Khavinson

**Abstract**

The Bergman and Szegö kernels in a bounded domain \(\Omega\subset\mathbb{C}^n\) are the reproducing kernels for the holomorphic functions in \(L^2(\Omega,dV)\) and \(L^2(\partial\Omega,d\sigma)\), respectively, where \(dV\) denotes the standard Lebesgue measure in \(\mathbb{C}^n\) and \(d\sigma\) a surface measure on the boundary \(\partial\Omega\). Their restrictions to the diagonal are known to have asymptotic expansions of the form: $$ K_B\sim \frac{\phi_B}{\rho^{n+1}}+\psi_B\log\rho,\quad K_S\sim \frac{\phi_S}{\rho^{n}}+\psi_S\log\rho, $$ where \(\phi_B,\phi_S,\psi_B,\psi_S\in C^\infty(\overline{\Omega})\) and \(\rho>0\) is a defining equation for \(\Omega\). The functions \(\phi_B,\phi_S,\psi_B,\psi_S\) encode a wealth of information about the biholomorphic geometry of \(\Omega\) and its boundary \(\partial\Omega\). In this talk, we will discuss this in the context of bounded strictly pseudoconvex domains in \(\mathbb C^2\) and pay special attention to the lowest order invariants in the log term and a strong form of a conjecture of Ramadanov.

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Turbulence in various environments — from planets to laboratory

Boris Galperin

USF College of Marine Science

3:00pm-4:00pm

CMC 130

Yumcheng You

**Abstract**

Geophysical and planetary flows feature various types of dispersive waves caused by stable stratification, planetary rotation, variation of the Coriolis parameter with latitude, topography, etc. The presence of the waves sets planetary turbulence apart from the usual Navier-Stokes turbulence. By virtue of featuring directions with zero phase speed, the waves render turbulence anisotropic and multiscale. The commingling of turbulence and waves changes flowsâ€™ transport properties. Mathematical treatment of anisotropic turbulence with waves is complicated and is in its early development. It will be shown how a combination of analytical theories, numerical simulations, dimensional analysis and laboratory experiments can be used to understand and quantify processes in such flows across different scales and different environments, from planetary scale macro- to laboratory scale micro-turbulence.

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Phase Transitions in Community Detection

Cris Moore

Santa Fe Institute

3:00pm-4:00pm

CMC 130

Razvan Teodorescu

**Abstract**

There is a deep analogy between statistical inference and statistical physics. I will give a friendly introduction to both of these fields, describing how the posterior distribution of a model given data is treated as the Boltzmann distribution of an appropriate physical system. I will then discuss phase transitions in community detection in networks, and clustering of sparse high-dimensional data, where if our data becomes too sparse or too noisy it suddenly becomes impossible to find the underlying pattern, or even tell if there is one. Along the way, I will visit ideas from computational complexity, random graphs, random matrices, and spin glass theory.

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Structured Decision-Making with Multiple Objectives

Lu Lu

3:00pm-4:00pm

CMC 130

**Abstract**

With increasingly constrained budgets, it is desirable to extract more information with limited data collection resources and make key decisions by balancing multiple objectives. A structured decision-making process using the Pareto front approach is presented to facilitate informed decision-making based on considering all possible solutions and trade-offs as well as understanding potential impacts from subjective choices. The method uses a two-stage process, where the first objective stage identifies the collection of all superior choices on the Pareto Front with an optimization search, and the second stage uses a rich set of graphical tools to quantitatively assess the trade-offs and robustness of different solutions to user priorities captured in weighting, scaling, and metric form choices. The final solution is a justifiable choice tailored to match what is important to the user. Applications in cost-quality choices and design of experiments serve as illustrating examples. Other applications in reliability analysis, resource allocation, as well as multiple response optimization are explored as well. The methodology is very general and can be applied to many applications with flexible choices of metrics for quantifying objectives of interest.