Browsing by Author "Dykema, Kenneth"

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Fundamental Decomposition Results in Finite Von Neumann Algebras
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/397; National Science Foundation
The study of operators on Hilbert space became important with the advent of Quantum Mechanics, but in addition, understanding of these operators has proven to be vital to progress in many areas of mathematics. Historically, a method of studying and understanding such operators is to break them down into simpler components, based on spectral decomposition. This consists of describing parts of the operator that behave like multiplication by certain numbers, and to explain how these parts assemble into the whole. One major goal of this project is to advance such understanding of large classes of operators. Another major goal is to study families of operators that arise in various quantum mechanical models, in light of certain deep mathematical conjectures regarding finite dimensional approximations of infinite dimensional objects. More specifically, the principal investigator, together with collaborators, has made advances in recent years on spectral decomposition results for non-selfadjoint elements of finite von Neumann algebras. These are centered around upper triangular forms, analogous to the classical results of Issai Schur for matrices, and both utilize and extend results about hyperinvariant subspaces found recently by Haagerup and Schultz. Particular proposed projects include (a) studying norm convergence properties of bounded operators and (b) extending spectral distribution and upper-triangular form results to unbounded affiliated operators. In related directions, the principal investigator will work on the hyperinvariant subspace problem for elements of tracial von Neumann algebras and to investigate the Murray--von Neumann puzzle, which is akin to the Heisenberg relations. A second area of proposed research concerns the notion of bi-freeness and bi-free products. A third area of proposed research concerns quantum correlations. Recent results of the principal investigator showing non-closedness of the set of quantum correlations for five inputs and two outputs open the door to new understanding of these small cases, which the principal investigator proposes to pursue. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Great Plains Operator Theory Symposium 2019-
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/397; National Science Foundation
This award provides funding for US participation in the 39th annual Great Plains Operator Theory Symposium that will be held May 28 to June 1, 2019 at Texas A&M University. The conference focuses on recent developments in Operator Algebras and Operator Theory. This symposium aims to be a conference covering broad areas with large participation, including many young people, who are encouraged to give talks. A number of distinguished mathematicians have already agreed to attend and speak at this conference. This award gives early career researchers, members of underrepresented groups, researchers not funded by NSF and the like an opportunity to attend and participate in this conference. The organizing committee will strive to make this funding opportunity known to target groups through a number of different activities.
Travel Support for US Participants in Focus Program "New Developments in Free Probability and Applications-at the Centre de Recherche Mathematiques."
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/397; National Science Foundation
This award provides funding for US participation in the conference that will be held during the entire month of March 2019, at the Centre de Recherches Mathematiques(CRM) in Montreal, Canada. The conference focuses on recent developments in Free Probability Theory, which is deeply related to topics in several areas of Analysis, including Operator Algebras and Random Matrices, and also to other areas of mathematics such as Combinatorics. Free Probability was created in the 1980s by Dan Voiculescu. His fundamental insight was to treat phenomenon in operator algebras related to free groups in a noncommutative probabilistic framework, with his notion of freeness taking the place of independence. Free Probability Theory has developed extensively; the parallels with usual probability theory are quite far reaching. This has led to fundamental insights and far reaching developments in diverse areas, including Operator Algebras and Random Matrices. The program at CRM includes two workshops, one on theoretical aspects and another on applied aspects of Free Probability. A number of distinguished mathematicians have agreed to attend and speak at this conference. This award gives early career researchers, members of underrepresented groups, researchers not funded by NSF and the like an opportunity to attend and participate in this conference. The organizing committee will strive to make this funding opportunity known to target groups through a number of different activities. More information is available at: http://www.crm.math.ca/crm50/en/march-1-31-2019-new-developments-in-free-probability-and-applications/. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.