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Events for September 14, 2018 from General and Seminar calendars

Noncommutative Functions Seminar

Time: 10:00AM - 11:00AM

Location: BLOC 624

Speaker: Jurij Volcic, Texas A&M University

Title: An introduction to noncommutative function theory (2nd talk)

Abstract: The rise of noncommutative (nc) functions started in 1972 with the pioneering work of J. L. Taylor. Since then, there has been a vivacious development of this theory, fueled by free probability, operator spaces, control theory, complex analysis, and free real algebraic geometry. Each of these areas offer their own aspect of nc functions. In two lectures I will try to lay out a very basic framework following the exposition by Kaliuzhnyi-Verbovetskyi and Vinnikov. In the first (algebraic) lecture we will define nc sets,nc functions and nc difference-differential operators. The study of their properties will naturally lead to higher order nc functions and their difference-differential calculus, which will culminate with the Taylor-Taylor formula. The second (analytic) lecture will then introduce various topologies on nc sets, corresponding analyticities of nc functions, and of course a few fundamental theorems.

Mathematical Physics and Harmonic Analysis Seminar

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Joonhyun La, Princeton University

Title: Global well posedness of 2D diffusive Fokker-Planck-Navier-Stokes systems

Abstract: In this talk, we prove that there is a unique global strong solution to the 2D Navier-Stokes system coupled with diffusive Fokker-Planck equation of a Hookean type potential. This system regards a polymeric fluid as a dilute suspension of polymers in an incompressible solvent, which is governed by the Navier-Stokes equation, and distribution of polymer configuration is governed by the Fokker-Planck equation, where spatial diffusion effects of polymers are also considered. Well-known Oldroyd-B models can be rewritten in the form of this system. Main conceptual difficulties include multi-scale nature of the system. We discuss an appropriate notion for the solution for this multi-scale system, and approximation scheme.

Working Seminar in Groups, Dynamics, and Operator Algebras

Time: 2:00PM - 2:50PM

Location: BLOC 506A

Speaker: Jintao Deng, Texas A&M University

Title: Topological full groups of one-sided shifts of finite type II


Algebra and Combinatorics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Jurij Volcic, Texas A&M University

Title: Multipartite rational functions: the universal skew field of fractions of a tensor product of free algebras

Abstract: A commutative ring embeds into a field if and only if it has no zero divisors; moreover, in this case it admits a unique field of fractions. On the other hand, the problem of noncommutative localization and embeddings into skew fields (that is, division rings) is much more complex. For example, there exist rings without zero divisors that do not admit embeddings into skew fields, and rings with several non-isomorphic "skew fields of fractions". This led Paul Moritz Cohn (1924-2006) to introduce the notion of the universal skew field of fractions to the general theory of skew fields in the 70's. However, almost all known examples of rings admitting universal skew fields of fractions belong to a relatively narrow family of Sylvester domains. One of the exceptions is the tensor product of two free algebras. After a decent introduction, we will look at the skew field of multipartite rational functions, whose construction via matrix evaluations of formal rational expressions is inspired by methods in free analysis. This skew field turns out to be the universal skew field of fractions of a tensor product of free algebras (for arbitrary finite number of factors).