ATMW Recent advances in Operator Theory and Operator Algebras (2012)

The plan is to have a one week workshop followed by a week of conference, broadly in the area of operator theory and operator algebras with a special emphasis on quantum theory.  The workshop will consist of courses on three   topics.  This is aimed at students as well as researchers in  the area who are keen to learn these fields.

Main Speakers in Workshop and description of their courses
(A) Uwe Franz,  University of Franche­Comte,   Besançon:, France

Title: Independence and Levy processes in quantum probability (6 lectures)
Abstract : Quantum probability describes the probabilistic foundations of quantum physics. Many concepts from classical probability have their counterpart in quantum probability. In these lectures we will study the notions of independence and independent increment processes. One surprising feature of quantum probability is the existence of several different notions of (stochastic) independence, as, e.g., tensor independence (which generalizes the notion of stochastic independence in classical probability), freeness, Boolean independence, or monotone independence. These lectures will give an introduction to the most common notions of independence and their basic theory, including the description of their convolutions and infinitely divisible measures. Then we will study the axiomatic approach due to Speicher, Ben Ghorbal, Schurmann, and Muraki, and the classification of the five universal notions of independence. Finally, we will study the relations between these notions and the
theory of independent  increment processes.

(B)  Adam Skalski: Institute of Mathematics,  Polish Academy of Sciences , Warszawa.
Title: Quantum Processes (6 lectures ): 
Abstract:  This   will   be     on   applications   of   the   fundamental   concepts   of   the   classical  theory   of   dynamical   systems   to   the   study   of  endomorphisms of operator algebras.   Classical theory of continuous or measurable dynamics can be in a natural way extended to so­called noncommutative dynamical systems, i.e. endomorphisms of respectively C*­algebras or von Neumann algebras.  In these series of lectures we present how the fundamental concepts such as the topological entropy or ergodic­theoretical notions extend to the noncommutative framework.

(C )  Brett D. Wick, School of Mathematics, Georgia Institute of Technology, Atlanta,
Title: The Corona Problem  (6 lectures)
Abstract:  The space H∞ (D)  is the collection of bounded analytic functions on the unit disk D and has been well­studied from the viewpoint of complex and harmonic analysis and the interaction with operator theory. Under sup­norm it is a complex Banach algebra.  With this Banach algebra, it is possible to ask questions that are a blend of analysis and algebra. One such important question is the Corona Problem.   This  problem can be phrased purely as a function theoretic question.   Using functional analysis, it is possible to change this problem into an  algebraic question about the maximal ideals of H ∞ (D)   and a topological question about the density of the unit disk D in the maximal ideal  space of H∞ (D).

The famous Carleson Corona Theorem,  answered this question affirmatively.  The tools and techniques Carleson implemented have become an integral part of analysis and have served as an impetus for much research in function theory, complex analysis, harmonic analysis, and operator  theory for the past 50 years. The course presented by Wick will focus on certain aspects of the Corona problem for multiplier algebras of Besov­­Sobolev spaces.  The ultimate goal of the course will be to discuss the background and ideas in the proof of the Corona theorem for more general algebras.
Numerous exercises   will be provided during the course so that the interested student can learn the basics of the topics covered.   These  exercises will be designed so that an interested student can go from a limited background in the area to a relatively deep understanding of the material.  Additionally, open problems and future directions of research will be pointed out and highlighted during the course.
Remarks:  Currently all the three  main speakers  have confirmed  their participation and have given details of the courses  they will give  (attached). Six guest speakers in the Workshop could be among, Michael Skeide, Gadadhar Misra, K. B. Sinha, J. Martin Lindsay, K. R.  Parthasarathy, R. Bhatia, V. S. Sunder,  or from  other conference participants. Please note that we have not contacted guest speakers so far.  Every day there will be tutorials anchored by organizers covering all the talks of the day. Main speakers are required to be present here to clear  doubts of participants. The tutorial sessions may go beyond the scheduled time of  5.30 p.m.

Workshop schedule
Dec. 31, 2012­Jan. 5, 2013

Abbreviations: A: Uwe Franz B: Adam Skalski C: Brett Wick G: Guest lectures T: Daily Tutorials

The Conference ( Jan.7­11, 2013)  will be on a wider variety of topics  in operator theory, operator algebras and quantum probability. will be  survey lectures covering some recent developments and also presentations of research papers. The conference schedule will be web in due course.

Speakers

• By Air
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• By Rail
  
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