Dr. Claus Köstler
Visiting Assistant Professor

 Department of Mathematics, Computer Science and Statistics St. Lawrence University 23 Romoda Drive, Canton, New York, 13617, USA E-mail:   Phone: +1 (315) 229-5976 Fax: +1 (315) 229-7413 Office: 210-1 Valentine Hall

### Research

Research interests: Operator Algebras, Noncommutative Probability, Quantum Dynamics

My research lies in the areas of noncommutative probability and quantum dynamics. It combines techniques from functional analysis, probability theory, as well as dynamical systems and ergodic theory. It is guided from a probabilistic point of view onto operator algebras and, vice versa, an operator algebraic approach to probability theory.

Central for my research are suitable operator algebraic concepts of:
• noncommutative independence and distributional symmetries;
• Brownian motion and Levy processes;
• Bernoulli shifts in discrete and continuous time;
• white noise as a dynamical system in continuous time.
• My work contributes to the foundation and development of these theories, as well as to the construction of new examples for them. This establishes contacts with fascinating areas of research, among which are:
• Arveson's classification program of E_0-semigroups and continuous tensor products of Hilbert spaces;
• Jones' subfactor theory;
• Pisier- Xu's work on non-commutative martingales and their inequalities;
• Tsirelson-Vershik's approach to continuous products of probability spaces and stochastic flows;
• Voiculescu's free probability theory.

• #### Recent Preprints

• A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation. (With R. Speicher). To appear in Commun. Math. Phys. (Preprint Version 2008)
• Noncommutative independence from the braid group $B_\infty$. (With R. Gohm.) To appear in Commun. Math. Phys. (Preprint Version February 2009)
• On Lehner's `free' noncommutative analogue of de Finetti's theorem. To appear in Proc. Amer. Math. Soc. (Preprint Version 2008)
• A noncommutative extended de Finetti theorem. Preprint 2008.
• On the structure of noncommutative white noises. (With R. Speicher) Trans. Amer. Math. Soc. 359 (2007), no. 9, 4325--4338. (Preprint Version 2007)
• Noncommutative continuous Bernoulli shifts. (With J. Hellmich, and B. Kümmerer.) Preprint 2006.

• ### 2008 - 2009 Teaching at St. Lawrence University (Canton, New York, U.S.A.)

 Spring 2009 MATH 205 A: Multivariable Calculus MATH 217 A: Linear Algebra MATH 348 A: Fourier Series
 Fall 2008 MATH 135 A & B: Calculus I MATH 217 A: Linear Algebra

### 2007 - 2008 Teaching at UIUC (Urbana, Illinois, U.S.A.)

 Spring 2008 MATH 124: Finite Mathematics MATH 231: Calculus II
 Fall 2007 MATH 255: Introductory Matrix Theory MATH 386: Introduction to Differential Equations Plus

### 2005 - 2007 Teaching at Carleton University (Ottawa, Ontario, Canada)

 Winter Term 2007 MATH 2404 A: Ordinary Differential Equations I MATH 3101 B: Algebraic Structures with Computer Applications
 Fall Term 2006 MATH 2004 A: Multivariable Calculus for Engineers MATH 2004 C: Multivariable Calculus for Engineers
 Winter Term 2006 MATH 2004 D: Multivariable Calculus for Engineers MATH 2404 A: Ordinary Differential Equations I
 Fall Term 2005 MATH 2004 B: Multivariable Calculus for Engineers MATH 2454 A: Ordinary Differential Equations