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Texas A&M University
Mathematics

Graduate Student Organization Seminar

Date: January 22, 2020

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Jacob Mashburn

  

Title: Non-crossing Partitions and Free Probability Theory

Abstract: In the mid-1980s, Dan Virgil Voiculescu introduced free probability primarily as a tool for solving the free group factor isomorphism problem, but throughout the 1990s, connections were made to random matrix theory, combinatorics, representation theory, classical probability, etc. For the first few years of its existence, results were proven mostly using techniques in operator theory, but in the early 1990s, Roland Speicher provided an alternate proof to Voiculescu's Free Central Limit Theorem using non-crossing partitions, which motivated his later definition of free cumulants (another analogy to classical probability), which plays a crucial role in free probability. I will begin by covering the basics of free probability and noncrossing partitions, then introducing an equivalent definition of free independence in terms of free cumulants. Finally, I will sketch Speicher's proof of the Free Central Limit Theorem. Throughout, comparisons between free and classical probability will be discussed. Thanks to the combinatorial nature of these topics, a background in operator theory (or even analysis more advanced than undergraduate level) is not needed.