Nonlinear Brownian motion and nonlinear Feynman-Kac formula of path-functions

Shige Peng
Shandon University
April 23, 2014
We consider a typical situation in which probability model itself has non-negligible cumulated uncertainty. A new concept of nonlinear expectation and the corresponding non-linear distributions has been systematically investigated: cumulated nonlinear i.i.d random variables of order \(1/n\) tend to a maximal distribution according a new law of large number, whereas, with a new central limit theorem, the accumulation of order \(1/\sqrt{n}\) tends to a nonlinear normal distribution.

Free entropy

Philippe Biane
Université Paris-Est Marne-la-Vallée
April 22, 2014
Free entropy is a quantity introduced 20 years ago by D. Voiculescu in order to investigate noncommutative probability spaces (e.g. von Neumann algebras). It is based on approximation by finite size matrices. I will describe the definition and main properties of this quantity as well as applications to von Neumann algebras. I will also explain a new approach based on work with Y. Dabrowski, using random matrices, which leads to the solution of some problems concerning this quantity.

Results and open problems in theory of quantum complexity

Andris Ambainis
University of Latvia; Member, School of Mathematics
April 22, 2014
I will survey recent results and open problems in several areas of quantum complexity theory, with emphasis on open problems which can be phrased in terms of classical complexity theory or mathematics but have implications for quantum computing:

1. Quantum vs. classical query complexity
2. Quantum vs. classical query complexity for almost all inputs
3. Quantum counterparts of Valiant-Vazirani theorem (reducing NP to unique-NP)

A Riemann-Roch theorem in Bott-Chern cohomology

Jean-Michel Bismut
Université Paris-Sud
April 21, 2014
If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By results of Bott and Chern, vector bundles have characteristic classes in Bott-Chern cohomology, which will be denoted with the subscript \(\mathrm{BC}\). Let \(p:M\to S\) be a proper holomorphic submersion of complex manifolds. Let \(F\) be a holomorphic vector bundle on \(M\) and let \(Rp_{*}F\) be its direct image.

True Randomness: Its Origin and Expansion

Yaoyun Shi
University of Michigan
April 21, 2014
How can we produce randomness of almost perfect quality, in large quantities, and under minimal assumptions? This question is fundamental not only to modern day information processing but also to physics. Yet a satisfactory answer is still elusive to both the practice and the theory of randomness extraction. Here we propose a solution through a new paradigm of extracting randomness from physical systems and basing security on the validity of physical theories, such as quantum mechanics and special relativity.

Limiting Eigenvalue Distribution of Random Matrices Involving Tensor Product

Leonid Pastur
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
April 16, 2014
We consider two classes of \(n \times n\) sample covariance matrices arising in quantum informatics. The first class consists of matrices whose data matrix has \(m\) independent columns each of which is the tensor product of \(k\) independent \(d\)-dimensional vectors, thus \(n=d^k\). The matrices of the second class belong to \(\mathcal{M}_n(\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}), \ n=d_1 d_2\) and are obtained from the standard sample covariance matrices by the partial transposition in \(\mathbb{C}^{d_2}\).

IP = PSPACE via error correcting codes

Or Meir
Institute for Advanced Study; Member, School of Mathematics
April 15, 2014
The IP theorem, which asserts that IP = PSPACE (Lund et. al., and Shamir, in J. ACM 39(4)), is one of the major achievements of complexity theory. The known proofs of the theorem are based on the arithmetization technique, which transforms a quantified Boolean formula into a related polynomial. The intuition that underlies the use of polynomials is commonly explained by the fact that polynomials constitute good error correcting codes. However, the known proofs seem tailored to the use of polynomials, and do not generalize to arbitrary error correcting codes.

Toroidal Soap Bubbles: Constant Mean Curvature Tori in \(S^3\) and \(R^3\)

Emma Carberry
University of Sydney
April 14, 2014
Constant mean curvature (CMC) tori in \(S^3\), \(R^3\) or \(H^3\) are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes. I will address these periodicity questions for the spherical and Euclidean cases, using Whitham deformations, which I will explain.

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