Speaker: Professor Palle Jorgensen

Title: Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure

Awarded paper

Jorgensen, Palle E. T.; Tian, James Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure. Adv. Oper. Theory 9 (2024), no. 4, Paper No. 77, 17 pp.

Abstract

Motivated by questions in operator theory/operator algebras, and such applications as stochastics and quantum information, the authors introduce a new and general framework for operator-valued positive definite kernels, and associated Hilbert space-valued Gaussian processes. The applications include a variety of settings for dilation theory, and for stochastic analysis, commutative and non-commutative.

Bio

Professor Palle Jorgensen, University of Iowa.

Scopus ID: 55580299600
Web of Science profile: link
GoogleScholar profile: link
ORCID ID: http://orcid.org/0000-0003-2681-5753

Palle Jorgensen is a Professor of mathematics at the University of Iowa. He is in the faculty of both Math and Applied Math (AMCS). He is also an elected Fellow of the Amer.Math.Society. Before coming to the University of Iowa, he taught at Stanford University, and at the University of Pennsylvania.

Through his career, he has taught at all levels, covering multiple areas of math, both pure and applied. His research publications, and his books, are highly cited. His papers have appeared in major journals, both in pure and applied math journals. He advised more than 30 PhD students. And he has many research collaborators from Universities around the World.

Research interests

His research directions are diverse and interdisciplinary, covering, in math, operators in Hilbert space, spectral theory, harmonic analysis, PDE, wavelets, infinite-dimensional analysis. And more applied: probability, stochastic processes, Ito-calculus and mathematics of financial derivatives/financial mathematics, dynamical systems, deep learning, and mathematical physics (statistical mechanics, quantum fields, quantum information, entanglement). His more recent research includes analysis and probability on graphs, large networks, subdivision algorithms, image-processing, spectral-tile duality, scaling algorithms, and Fourier series on fractals.

He serves on the editorial board of 8 leading mathematics journals. He has held multiple research grants.

Recent books by the co-authors

1 Jorgensen, Palle; Rao, Murali; Tian, James

Brownian motion and potential theory, modern and classical.

Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes 9. Singapore: World Scientific (ISBN 978-981-12-9431-0/hbk; 978-981-12-9477-8/pbk; 978-981-12-9433-4/ebook). (2024).

2 Jorgensen, Palle E. T.; Song, Myung-Sin

Mathematics of multilevel systems. Data, scaling, images, signals, and fractals.

Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes 8. Singapore: World Scientific (ISBN 978-981-12-6897-7/hbk; 978-981-12-6901-1/ebook). xv, 253 p. (2023).

3 Jorgensen, Palle E. T.; Pearse, Erin P. J.

Operator theory and analysis of infinite networks. Theory and applications.

Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes 7. Singapore: World Scientific (ISBN 978-981-12-6551-8/hbk; 978-981-12-6553-2/ebook). lv, 392 p. (2023).

4 Cho, Ilwoo; Jorgensen, Palle

Comparing Banach spaces for systems of free random variables followed by the semicircular law.

Aron, Richard M. (ed.) et al., Operator and norm inequalities and related topics. Cham: Birkhäuser. Trends Math., 773-822 (2022).

5 Jorgensen, Palle; Tian, James

Infinite-dimensional analysis. Operators in Hilbert space; stochastic calculus via representations, and duality theory.

Hackensack, NJ: World Scientific (ISBN 978-981-12-2577-2/hbk; 978-981-12-2579-6/ebook). xvii, 234 p. (2021).

6 Jorgensen, Palle E. T.

Harmonic analysis: smooth and non-smooth.

CBMS Regional Conference Series in Mathematics 128. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4880-6/pbk; 978-1-4704-4978-0/ebook). xi, 266 p. (2018).

Drs Jorgensen and Tian have many research papers. Here are a few.

Jorgensen, Palle (1-IA); Stewart, David E. (1-IA)
Approximation properties of ridge functions and extreme learning machines. SIAM J. Math. Data Sci. 3 (2021), no. 3, 815–832.

Jorgensen, Palle E. T.; Tian, James Hilbert space valued Gaussian processes, their kernels, factorizations, and covariance structure. Adv. Oper. Theory 9 (2024), no. 4, Paper No. 77, 17 pp (won a best paper award.)

Jorgensen, Palle E. T.; Tian, James F. Polymorphisms, their associated operator theory, self-similar fractals, and harmonic analysis. Recent developments in fractal geometry and dynamical systems, 37–59, Contemp. Math., 797, Amer. Math. Soc., [Providence], RI, [2024], ©2024.

Jorgensen, Palle E. T.; Song, Myung-Sin; Tian, James Conditional mean embedding and optimal feature selection via positive definite kernels. Opuscula Math. 44 (2024), no. 1, 79–103.