v
 Case Western Reserve University DEPARTMENT OF MATHEMATICS

### ANALYSIS

Professor Stanislaw SZAREK
Department of Mathematics
Case Western Reserve University
The Bernoulli Conjecture (background and introduction)
Date: Tuesday, February 19, 2013
Time: 2:45-3:45 p.m.
Location: CWRU Department of Mathematics Yost Hall - Room 335 (formerly Yost 321B)

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
Department of Mathematics
Case Western Reserve University
The Bernoulli Conjecture (background and introduction) II
Date: Tuesday, February 26, 2013
Time: 2:45-3:45 p.m.
Location: CWRU Department of Mathematics Yost Hall - Room 335 (formerly Yost 321B)

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
Department of Mathematics
Case Western Reserve University
The Bernoulli Conjecture (background and introduction) III
Date: Tuesday, March 26, 2013
Time: 2:45-3:45 p.m.
Location: CWRU Department of Mathematics Yost Hall - Room 335 (formerly Yost 321B)

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 was about the relationship between families of Gaussian random variables
(a.k.a. Gaussian processes) and related families involving Bernoulli random variables (= coin flips).

This is the third talk on this topic, but it will be largely independent from the first two. The objective of the current talk
will be to present two preliminary results, which are of independent interest. Future plans include a sketch of the solution to the conjecture.

## PhD Defense

Department of Mathematics
Case Western Reserve University
"Convex Geometric Connections to Information Theory"
Date: Wednesday , March 27, 2013
Time: 4:00-6:00 p.m..
Location: CWRU Department of Mathematics Seminar Room, Yost 306 (formerly Yost 300)

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.
Methods include affine invariance, duality, concentration of measure, entropy, and random matrices.

Professor Dennis AMELUNXEN
Department of Mathematics
Cornell University
Phase Transitions in Convex Geometry and Optimization
Date: Tuesday, April 16, 2013
Time: 2:45-3:45 p.m.
Location: CWRU Department of Mathematics Yost Hall - Room 306 (formerly Yost 300)

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.

Case Western Reserve University
Department of Mathematics
10900 Euclid Avenue
Cleveland, Ohio 44106
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