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Case Western Reserve University DEPARTMENT OF MATHEMATICS |
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DEPARTMENT OF MATHEMATICS SEMINARS and COLLOQUIAANALYSIS |
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Professor Stanislaw SZAREK ABSTRACT An important problem in probability, the so-called Bernoulli Conjecture (posed by M. Talagrand) has been very recently solved by W. Bednorz and R. Latala. Roughly speaking, the conjecture is about the relationship between families of Gaussian random variables (a.k.a. Gaussian processes) and related families involving Bernoulli random variables (= coin flips). In this talk I will offer an introduction to the topic, including the precise statement of the conjecture and its motivation. The talk should be accessible to students whose familiarity with probability goes beyond MATH 380, certainly to those who took MATH 491 or MATH 494 . We plan talks on the solution to the conjecture in the weeks to come.
Professor Stanislaw SZAREK ABSTRACT An important problem in probability, the so-called Bernoulli Conjecture (posed by M. Talagrand) has been very recently solved by W. Bednorz and R. Latala. Roughly speaking, the conjecture is about the relationship between families of Gaussian random variables (a.k.a. Gaussian processes) and related families involving Bernoulli random variables (= coin flips). In this talk I will offer an introduction to the topic, including the precise statement of the conjecture and its motivation. The talk should be accessible to students whose familiarity with probability goes beyond MATH 380, certainly to those who took MATH 491 or MATH 494 . We plan talks on the solution to the conjecture in the weeks to come.
Professor Stanislaw SZAREK ABSTRACT An important problem in probability, the so-called Bernoulli Conjecture (posed by M. Talagrand) has been very recently solved by This is the third talk on this topic, but it will be largely independent from the first two. The objective of the current talk
PhD DefenseJustin JENKINSON, Graduate Student Thesis Advisors: Professor Stanislaw Szarek and Professor Elisabeth Werner (Mathematics Dept.,CWRU) Committee: Professor Elizabeth Meckes(Mathematics Dept.,CWRU) Professor Kenneth Kowalski (Physics Dept., CWRU) ABSTRACT Convex geometry is a field of mathematics that emphasizes a geometric perspec- tive on functional analysis. Problems in many different fields can be interpreted geometrically which often leads to powerful and surprising results. This thesis establishes connections between convex geometry and both classical and quantum information theory. We introduce the mean width bodies and illustrate the geometric interpretation they provide for the relative entropy of cone measures of a convex body and its polar. We define relative entropy for convex bodies and its relation to affine isoperimetric inequalities is considered. Quantum states are fundamental objects in quantum information theory. Fairly sharp estimates are obtained regarding the geometry of quantum states using basic notions in convex geometry. In particular, the distance between the set of states with positive partial transpose and the set of separable states is explored. Finally, the optimal constants for the spherical isoperimetric inequality are pro- vided and generalizations of a concentration inequality are suggested.
Professor Dennis AMELUNXEN ABSTRACT Convex optimization provides a powerful approach to solving a wide range of problems under structural assumptions on the solutions. Examples include solving linear inverse problems or separating signals with mutually incoherent structures. A curious phenomenon arises when studying such problems; as the underlying parameters in the optimization program shift, the convex relaxation can change quickly from success to failure. We reduce the analysis of these phase transitions to a summary parameter, the statistical dimension, associated to the problem. We prove a new concentration of measure phenomenon for some integral geometric invariants, and deduce from this the existence of phase transitions for a wide range of problems; the phase transition being located at the statistical dimension. We furthermore calculate the statistical dimension in concrete problems of interest, and use it to relate previously existing—but seemingly unrelated—approaches to compressed sensing by Donoho and Rudelson & Vershynin. This is joint work with Martin Lotz, Michael B. McCoy, Joel A. Tropp.
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