J. E. Pascoe's homepage

About me

Photo by Ivonne Vetter at MFO in 2013.

I am a National Science Foundation Mathematical Science Postdoctoral Research Fellow (DMS-1606260) and William Chauvenet Postdoctoral Lecturer at Washington University in St. Louis, officially mentored by John McCarthy. I obtained my PhD in Mathematics at the University of California, San Diego under Jim Agler in Spring 2015. My thesis was about geometric aspects of the boundary behavior of complex analytic functions in several variables. Prior to my PhD, I obtained a Masters in Applied Math from UCSD in 2011, and my BS at the University of North Texas in 2010. My email is pascoej@math.wustl.edu.

Research

I am generally interested in functional analysis and its various applications to matrix inequalities, moment problems, several complex variables, noncommutative function theory, multivariable operator theory, real algebraic geometry, and free probability. Much of my work is inspired by problems and methods in systems engineering as they apply to pure mathematics. Some of my past work has dealt with:

Boundary behavior of holomorphic functions: analogues of Julia-Caratheodory type theorems, Lowner type theorems.
Matrix inequalities. Analogues of the Positivstellensatz
Noncommutative real algebraic geometry.
Operator monotone functions in several variables.
Invariant theory for free polynomials.
The Jacobian conjecture for free polynomials.
Cauchy transforms in operator-valued free probability.

See my papers for details.

CV
Research statement

I recommend reading about the wedge-of-the-edge theorem if you are looking for some accessible entertainment.

Papers

Free functions with symmetry Mathematische Zeitschrift (With David Cushing and Ryan Tully-Doyle) (to appear)
A noncommutative Julia inequality. Mathematische Annalen (to appear) (With John McCarthy)
Derivatives of rational inner functions: geometry of singularities and integrability at the boundary. Proceedings of the London Mathematical Society (to appear)(With Kelly Bickel and Alan Sola)
A wedge-of-the-edge theorem: analytic continuation of multivariable Pick functions in and around the boundary. Bulletin of the London Mathematical Society (to appear)
Positivstellensatze for noncommutative rational functions. Proceedings of the American Mathematical Society (to appear)
An inductive Julia-Caratheodory theorem for Pick functions in two variables. Proceeding of the Edinburgh Mathematical Society (to appear)
Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables. Journal of Functional Analysis 273 (1) 283-328 (2017)(With Ryan Tully-Doyle)
The Julia-Caratheodory theorem on the bidisk revisited. Acta Sci. Math. (Szeged) 83:1-2, 165-175, (2017) (With John McCarthy)
Regular and positive noncommutative rational functions. Journal of the London Mathematical Society 95 (2) 613-632, (2017) (With Igor Klep and Jurij Volcic)
Convex entire noncommutative functions are polynomials of degree two or less. Integral Equations and Operator Theory 86 (2), 151-163, (2016). (With Bill Helton, Ryan Tully-Doyle, Victor Vinnikov)
The inverse function theorem and the Jacobian conjecture for free analysis. Mathematische Zeitschrift 278 (3-4) 987-994, (2014).

Preprints

The wedge-of-the-edge theorem: edge-of-the-wedge type phenomenon within the common real boundary Submitted
The noncommutative Loewner theorem for matrix monotone functions over operator systems. Submitted
(With Ryan Tully-Doyle) Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not. Submitted
(With Benjamin Passer and Ryan Tully-Doyle) Representation of free Herglotz functions Submitted
Note on Löwner's theorem on matrix monotone functions in several commuting variables of Agler, McCarthy and Young submitted

Slides

Regular and positive noncommutative rational functions Workshop on noncommutative analysis in Iowa June 2016
A wedge of the edge of the wedge ICMS 2014: Function theory in several complex variables in relation to modelling uncertainty
Hankel vector moment sequences Great Plains Operator Theory Symposium 2014
Matrix monotonicity in several noncommuting variables St. Louis AMS Sectional Meeting 2013
Injective free polynomials Hilbert Function Spaces 2013