I'm a Professor at Washington University in St. Louis.

I was previously at the University of Alabama (2010-2013) and UC Irvine (2007-2010).

You can email me at geknese shift2 wustl dot edu.

My office is 214 Cupples I

My CV

polynomial zero sets and sums of squares decompositions,

and harmonic analysis.

NSF analysis grant 2019-2023 Operator theory and stable polynomials

NSF analysis grant 2014-2018 Harmonic analysis and spaces of analytic functions in several variables

NSF analysis grant 2010-2014: "Operator related function theory and algebraic varieties"

On arXiv and MathSciNet

- Three radii associated to Schur functions on the polydisk 2024.
- Rational inner functions on the polydisk -- a survey, 2024.
- (with K. Bickel, J.E.Pascoe, A.Sola) Stable polynomials and admissible numerators in product domains, 2024, to appear in BLMS.
- Boundary local integrability of rational functions in two variables 2024.
- (with K. Bickel, J.E.Pascoe, A.Sola) Local theory of stable polynomials and bounded rational functions of several variables. 2021. To appear in Annales Polonici Mathematici.
- A simple proof of necessity in the McCullough-Quiggin theorem. Proc. Amer. Math. Soc. 148 (2020), no. 8, 3453–3456.
- Kummert’s approach to realization on the bidisk. Indiana Univ. Math. J. 70 (2021), no. 6, 2369–2403.
- (with E.D. Knese, Keel, Bennert, Moiseev, Grokhovskaya, Dodonov) An [O III] search for extended emission around AGN with H I mapping: a distant cloud ionized by Mkn 1 . Monthly Notices of the Royal Astronomical Society, Volume 496, Issue 2, August 2020, pages 1035-1050. Note: This is an astronomy paper with a spherical version of the Buffon needle problem in the appendix. Available here
- Global bounds on stable polynomials , Complex Anal. Oper. Theory 13 (2019), no. 4, 1895-1915. DOI
- Extreme points and saturated polynomials. Illinois J. Math. 63 (2019), no. 1, 47-74.
- (with Kosinski, Ransford, and Sola) Cyclic polynomials in anisotropic Dirichlet spaces. J. Anal. Math. 138 (2019), no. 1, 23-47.
- The von Neumann inequality for 3x3 matrices , Bull. Lond. Math. Soc. 48 (2016), no. 1, 53-57.
- Integrability and regularity of rational functions , Proc. Lond. Math. Soc. (3) 111 (2015), no. 6, 1261-1306.
- (with Beneteau, Kosinski, Liaw, Seco, and Sola) Cyclic polynomials in two variables , Trans. Amer. Math. Soc. 368 (2016), no. 12, 8737-8754.
- (with K. Bickel) Canonical Agler decompositions and transfer function realizations , Trans. Amer. Math. Soc. 368 (2016), no. 9, 6293-6324.
- Determinantal representations of semi-hyperbolic polynomials , Michigan Math. J. 65 (2016), no. 3, 473-487.
- (with J. McCarthy and K. Moen) Unions of Lebesgue spaces and A_1 majorants , Pacific J. Math. 280 (2016), no. 2, 411-432.
- (with J. Geronimo and P. Iliev) Polynomials with no zeros on a face of the bidisk , J. Funct. Anal. 270 (2016), no. 9, 3505-3558.
- (with P.M. Gauthier) Zero-free polynomial approximation on a chain of Jordan domains , Ann. Sci. Math. Qu\'ebec 36 (2012), no.1, 107-112.
- (with K. Bickel) Inner functions on the bidisk and associated Hilbert spaces , J. Funct. Anal. 265 (2013), no. 11, 2753-2790.
- Uchiyama's lemma and the John-Nirenberg inequality , Bull. Lond. Math. Soc. 45 (2013), no. 4, 683-692.
- (with J. Geronimo and P. Iliev) Orthogonality relations for bivariate Bernstein-Szego measures , Recent advances in orthogonal polynomials, special functions, and their applications, 119-131, Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012. Also here
- Schur-Agler class rational inner functions on the tridisk , Proc. Amer. Math. Soc., 139(2011), 4063-4072. Also here .
- A refined Agler decomposition and geometric applications , Indiana Univ. Math. J. 60 (2011), 1831-1842. Also here
- (with M. Jury and S. McCullough) Nevanlinna-Pick interpolation on distinguished varieties in the bidisk , 2012, J. Functional Analysis, 262(9), 3812-3838.
- Stable symmetric polynomials and the Schur-Agler class , Illinois J. Math. 55 (2011), no. 4, 1603-1620 (2013).
- Rational inner functions in the Schur-Agler class of the polydisk , 2011, Publ. Mat., 55(2), 343-357.
- Kernel decompositions for Schur functions on the polydisk, 2011, Complex Analysis and Operator Theory, 5(4), 1093-1111. See also here
- Polynomials with no zeros on the bidisk, , Analysis and PDE, Vol. 3 (2010) No. 2, 109-149. See also here.
- Polynomials defining distinguished varieties, 2010, Trans. Math. Amer. Soc., 362(11), 5635-5655.
- (with M.Jury and S.McCullough) Agler
interpolation families of kernels, 14 pages,

Operators and Matrices, Vol. 3 (2009) No. 4, 571-587. - (with J. Agler and J. McCarthy) Algebraic pairs of isometries, 2012, J. Operator Theory, 67(1), 215-236. Also here .
- Bernstein-Szegõ measures on the two dimensional torus, Indiana Univ. Math. J. 57 No. 3 (2008), 1353-1376. Also here .
- Function theory on the Neil parabola, Michigan Math. J., Vol. 55, Issue 1 (2007), 139-154.
- A Schwarz lemma on the polydisk, Proc. Amer. Math. Soc., 135 (2007), 2759-2768.

Analytic combinatorics. Fall 2023.

Graduate Topics course: analysis of Dirichlet series. Spring 2023.

Introduction to Combinatorics, Fall 2022.

(Graduate topics course) Analytic combinatorics. Spring 2022.

Mathematical geodesy, Fall 2021.

Complex Analysis II, Spring 2021.

Differential Equations, Fall 2019.

Topics course on Reproducing kernel Hilbert space, Spring 2019

Partial Differential Equations, Math 415, Fall 2018

Analytic Combinatorics, Math 470, Fall 2018

Function theory on the polydisk, Math 522, Spring 2018

Measure Theory and Functional Analysis II Math 5052 Spring 2018

Measure Theory and Functional Analysis I Math 5051 Fall 2017

Measure Theory and Functional Analysis II Math 5052 Spring 2017

Geometry I Math 5041 Fall 2016

Calculus of Several Variables Math 318 Spring 2016

Complex Analysis II Math 5022 Spring 2016

Complex Analysis I Math 5021 Fall 2015

Measure theory and functional analysis II Math 5052 Spring 2015

Measure theory and functional analysis I Math 5051 Fall 2014

Matrix Algebra Math 309 Fall 2013

Older teaching

Operator theory and the singular value decomposition Oberwolfach snapshot.

Slides from AMS/MAA joint meetings in San Diego January 2013.

Texas A and M 2012 slides on Uchiyama's lemma

IWOTA 2012 slides on Distinguished Varieties

Slides from SEAM 27.

Richmond AMS meeting slides.

My Youtube calculus videos

Slides and audio for the presentation "Rational inner functions on the bidisk" presented at the Fields institute.

LaTeX style file , metafont file , and user's guide to a font/notation I created for two of the above papers.

My math genealogy