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Jiří Lebl

Assistant Professor

Department of Mathematics

Oklahoma State University

Stillwater, OK 74078

**Office:**
MSCS 505

**Office hours:**
3:30-4:30pm Tu Th, 11:30-12:30pm We,
and by appointment at other times.

**Office phone:** (405) 744-7750

**Email:**

Mathematician at OSU, wearer of hats (or hats) and colored socks (odd pairs only). Degrees: PhD from UCSD (2007), my BA and MA are from SDSU (2001, 2003). I spent 2007-2010 as a postdoc at UIUC, the 2010-2011 year visiting UCSD, and 2011-2013 postdocing again at UW-Madison. Here is my partial mathematics genealogy. I usually write my name without the diacritical signs as Jiri Lebl. I sometimes use the name George to ease pronunciation. And finally, Markéta's page.

Current interests: Complex analysis and CR geometry. I am also interested in the connections to other fields such as computer science, quantum computing, algebraic geometry, and commutative algebra. In particular in complex analysis I study real analytic CR submanifolds, Levi-flat hypersurfaces, singular real and complex varieties, proper holomorphic mappings between balls and hyperquadrics. MSC: 32, 14. I am also interested in and/or have published papers that touched on 5, 11, 12, 13, 26, 30, 35, 68, 81.

See my Curriculum Vitae, my Research Statement, and my Teaching Statement.

I am the PI on
NSF grant DMS-1362337 for 2014-2017 titled *Complexity in CR Geometry* (\$141,000).
In 2009-2013 I was the PI on NSF grant DMS-0900885.

Math 4143/5043 Fall 2017, Math 4233 Fall 2017.

Past:

Math 4263 Spring 2017,
Math 4013 Spring 2017,
Math 4283 Fall 2016,
Math 4403 Fall 2016,
Math 6283 Spring 2016,
Math 4153/5053 Spring 2016,
Math 5193 Fall 2015,
Math 4143/5043 Fall 2015,
Math 4263 Spring 2015,
Math 2163 Fall 2014,
Math 4013 Fall 2014,
Math 6283 (Several Complex Variables) Spring 2014,
Math 2144 Fall 2013 (used D2L),
UW Math 222 Spring 2013,
UW Math 322 Fall 2012,
UW Math 621 Fall 2012,
UW Math 522 Spring 2012,
UW Math/CS 240 Spring 2012,
UW Math 521 Fall 2011,
UW Math 213 Fall 2011,
UCSD Math 140B Spring 2011,
UCSD Math 20B Spring 2011,
UCSD Math 10C Winter 2011,
UCSD Math 140A Winter 2011,
UCSD Math 20C Fall 2010,
UCSD Math 20D Fall 2010,
UIUC Math 285 Spring 2010,
UIUC Math 595 Spring 2010,
UIUC Math 444 Fall 2009,
UIUC Math 225 Spring 2009,
UIUC Math 286 Spring 2009,
UIUC Math 286 Fall 2008,
UIUC Math 124 Spring 2008,
UIUC Math 380 Fall 2007.

Anne-Katrin Gallagher, Jiří Lebl, and Koushik Ramachandran,
*Convexity of level lines of Martin functions and applications*,
preprint arXiv:1710.00280,

Jiří Lebl, Alan Noell, and Sivaguru Ravisankar,
*On Lewy extension for smooth hypersurfaces in ${\mathbb C}^n \times {\mathbb R}$*,
preprint arXiv:1704.08662,

Jiří Lebl, Asif Shakeel, and Nolan Wallach,
*Unentangled Measurements and Frame Functions*,
preprint arXiv:1701.06069.

Jiří Lebl,
*An interesting example of a compact non-${\mathbb C}$-analytic real
subvariety of ${\mathbb R}^3$*,
preprint arXiv:1412.4838.

Jiří Lebl, Alan Noell, and Sivaguru Ravisankar,
*Codimension two CR singular submanifolds and extensions of CR functions*,
Journal of Geometric Analysis,
**27** (2017), no. 3, 2453-2471,
MR3667437,
DOI:10.1007/s12220-017-9767-6,
arXiv:1604.02073.

Jiří Lebl, Alan Noell, and Sivaguru Ravisankar,
*Extension of CR functions from boundaries in ${\mathbb C}^n \times {\mathbb R}$*,
Indiana University Mathematics Journal,
**66** (2017), no. 3, 901-925,
MR3663330,
DOI:10.1512/iumj.2017.66.6067,
arXiv:1505.05255.

Jiří Lebl, Asif Shakeel, and Nolan Wallach,
*Local Distinguishability of Generic Unentangled Orthonormal Bases*,
Physical Review A,
**93** (2016), no. 1, 012330,
DOI:10.1103/PhysRevA.93.012330,
arXiv:1502.06639.

Dusty Grundmeier and Jiří Lebl,
*Initial monomial invariants of holomorphic maps*,
Mathematische Zeitschrift,
**282** (2016), no. 1, 371-387,
MR3448385,
DOI:10.1007/s00209-015-1543-3,
arXiv:1502.03434.

John P. D'Angelo and Jiří Lebl,
*Homotopy equivalence for proper holomorphic mappings*,
Advances in Mathematics,
**286** (2016), 160-180,
MR3415683,
DOI:10.1016/j.aim.2015.09.007,
arXiv:1408.1104.

Jiří Lebl,
*Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian*,
International Journal of Mathematics,
**26** (2015), no. 5, 1550036 (17 pages),
MR3345513,
DOI:10.1142/S0129167X15500366,
arXiv:1407.5913.

Xianghong Gong and Jiří Lebl,
*Normal forms for CR singular codimension two Levi-flat submanifolds*,
Pacific Journal of Mathematics,
**275** (2015), no. 1, 115-165,
MR3336931,
DOI:10.2140/pjm.2015.275.115,
arXiv:1403.0558.

Jiří Lebl, André Minor, Ravi Shroff, Duong Son, and Yuan Zhang,
*CR singular images of generic submanifolds under holomorphic maps*,
Arkiv För Matematik,
**52** (2014), no. 2, 301-327,
MR3255142,
DOI:10.1007/s11512-013-0193-0,
arXiv:1205.5309.

Dusty Grundmeier, Jiří Lebl, and Liz Vivas,
*Bounding the rank of Hermitian forms and rigidity for CR mappings
of hyperquadrics*,
Mathematische Annalen,
**358** (2014), no. 3-4, 1059-1089,
MR3175150,
DOI:10.1007/s00208-013-0989-z,
arXiv:1110.4082.

Orest Bucicovschi and Jiří Lebl,
*On the continuity and regularity of convex extensions*,
Journal of Convex Analysis,
**20** (2013), no. 4, 1113-1126,
MR3184299,
arXiv:1012.5796.

Jennifer Halfpap and Jiří Lebl,
*Signature pairs of positive polynomials*,
Bulletin of the Institute of Mathematics, Academia Sinica, New Series,
**8** (2013), no. 2, 169-192,
MR3098535,
arXiv:1211.0997.

Jiří Lebl,
*Singular set of a Levi-flat hypersurface is Levi-flat*,
Mathematische Annalen,
**355** (2013), no. 3, 1177-1199,
MR3020158,
DOI:10.1007/s00208-012-0821-1,
arXiv:1012.5993.

Jiří Lebl and Han Peters,
*Polynomials constant on a hyperplane and CR maps of spheres*,
Illinois Journal of Mathematics,
**56** (2012), no. 1, 155-175,
MR3117023,
arXiv:1105.2343.

Jiří Lebl,
*Algebraic Levi-flat hypervarieties in complex projective space*,
Journal of Geometric Analysis,
**22** (2012), 410-432,
MR2891732,
DOI:10.1007/s12220-010-9201-9,
arXiv:0805.1763.

John P. D'Angelo and Jiří Lebl,
*Pfister's theorem fails in the Hermitian case*,
Proceedings of the American Mathematical Society,
**140** (2012), 1151-1157,
MR2869101,
DOI:10.1090/S0002-9939-2011-10841-4,
arXiv:1010.3215.

Jiří Lebl,
*Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics*,
Michigan Mathematical Journal,
**60** (2011), no. 3, 603-628,
MR2861091,
DOI:10.1307/mmj/1320763051,
arXiv:0906.0325.

John P. D'Angelo and Jiří Lebl,
*Hermitian symmetric polynomials and CR complexity*,
Journal of Geometric Analysis,
**21** (2011), no. 3, 599-619,
MR2810845,
DOI:10.1007/s12220-010-9160-1,
arXiv:1003.0126.

Jiří Lebl and Han Peters,
*Polynomials constant on a hyperplane and CR maps of hyperquadrics*,
Moscow Mathematical Journal,
**11** (2011), no. 2, 287-317,
MR2859238,
arXiv:0910.2673.

Jiří Lebl and Daniel Lichtblau,
*Uniqueness of certain polynomials constant on a line*,
Linear Algebra and its Applications,
**433** (2010), no. 4, 824-837,
MR2654111,
DOI:10.1016/j.laa.2010.04.020,
arXiv:0808.0284.
See also the addendum
arXiv:1302.1441.

John P. D'Angelo and Jiří Lebl,
*On the complexity of proper holomorphic mappings between balls*,
Complex Variables and Elliptic Equations,
**54** (2009), nos. 3-4, 187-204,
MR2513534,
DOI:10.1080/17476930902759403,
arXiv:0802.1739.

Jiří Lebl,
*Levi-flat hypersurfaces with real analytic boundary*,
Transactions of the American Mathematical Society,
**362** (2010), no. 12, 6367-6380,
MR2678978,
DOI:10.1090/S0002-9947-2010-04887-1,
arXiv:0710.3801.

John P. D'Angelo and Jiří Lebl,
*Complexity results for CR mappings between spheres*,
International Journal of Mathematics,
**20** (2009), no. 2, 149-166,
MR2493357,
DOI:10.1142/S0129167X09005248,
arXiv:0708.3232.

Jiří Lebl,
*Extension of Levi-flat hypersurfaces past CR boundaries*,
Indiana University Mathematical Journal,
**57** (2008), no. 2, 699-716,
MR2414332,
DOI:10.1512/iumj.2008.57.3203,
arXiv:math.CV/0612071.
Further notes / additions

John P. D'Angelo, Jiří Lebl, and Han Peters,
*Degree estimates for polynomials constant on a hyperplane*,
Michigan Mathematical Journal,
**55** (2007), no. 3, 693-713,
MR2372622,
DOI:10.1307/mmj/1197056463,
arXiv:math.CV/0609713.
Further notes / additions

Jiří Lebl,
*Nowhere minimal CR submanifolds and Levi-flat hypersurfaces*,
Journal of Geometric Analysis, **17** (2007), no. 2, 321-342,
MR2320166,
DOI:10.1007/BF02930726,
arXiv:math.CV/0606141.

Jiří Lebl,
*Addendum to Uniqueness of certain polynomials constant on a line*,
arXiv:1302.1441.

Jiří Lebl,
*Pullback of varieties by finite maps*,
arXiv:0812.2498.

Jiří Lebl,
*Singularities and complexity in CR geometry*,
PhD thesis, University of California, San Diego, Spring 2007.

Jiří Lebl,
*Quasiconformal extensions of quasisymmetric mappings*,
Masters thesis, SDSU, Spring 2003.

[List of my papers on arXiv] [Recent papers in Complex Variables on arXiv]

Notes on Diffy Qs: Differential Equations for Engineers. A textbook for an undergraduate level course on differential equations broadly aimed at engineers or other technical fields. It covers first and second order ODE, systems of linear ODEs, Fourier series methods and PDEs, boundary and eigenvalue problems, Laplace transform, power series methods, and nonlinear systems. It grew out of my notes for Math 286 at UIUC. It has been and is being used at a number of different schools from community college level, to top research schools. For example, it has been adopted as a standard book at UC-Irvine. 351 pages, available as a PDF download (free) or get it as cheap paperback. It is also browsable as web pages. | |

Basic Analysis: Introduction to Real Analysis. A textbook for a proof-based undergraduate real analysis course. Covers the real number system, sequences and series, continuous functions, the derivative, the Riemann integral, sequences of functions, and metric spaces. It started its life as my notes for Math 444 at UIUC. It has been and is being used at a number of different schools and even been adopted as a standard book at a few. 259 pages, available as a PDF download, or get it as cheap paperback. | |

Basic Analysis II: Introduction to Real Analysis, Volume II. Second volume for the Basic Analysis book. This volume covers multivariable topics, starting with differential calculus, differentiation under the untegral, path integrals, and the multivariable Riemann integral. The idea is that both volumes together can be used for a two-semester sequence, where Volume I covers a bit more than one semester, and then Volume II covers the rest. 114 pages, available as a PDF download, or get it as cheap paperback. | |

Tasty Bits of Several Complex Variables. Introduction to Several Complex Variables. These were my class notes for the SCV graduate course at OSU given in Spring 2014 and Spring 2016. It aims to be self contained and from a more traditional viewpoint (as compared to the course below). 140 pages, available as a PDF download, or a cheap paperback. | |

Hermitian Forms Meet Several Complex Variables: Minicourse on CR Geometry Using Hermitian Forms. A set of notes for a half-semester graduate mini-course on Hermitian forms and CR geometry given at UIUC in the spring of 2010. Should serve as a quick introduction to the subject of CR geometry from a somewhat different viewpoint than is usual. 62 pages, available as a PDF download. |

*Genius* - A free software (GPL) mathematics package for Unix/Linux (also works on MacOS X). See
the website for more
information and the latest release. Some sort of release is usually
included in most Linux software repositories (e.g. universe in Ubuntu).
I have been working on this since about '97 and it is quite capable nowadays.

*On Lewy extension for smooth hypersurfaces in ${\mathbb C}^n \times
{\mathbb R}$*, Complex Geometry and PDEs, May 2017, Beirut, Lebanon.
(slides)

*On Lewy extension for smooth hypersurfaces in ${\mathbb C}^n \times
{\mathbb R}$*, AMS special session, April 2017, Pullman, WA.
(slides)

*Codimension two CR singular submanifolds and extensions of CR
functions*, Midwest SCV, May 2016, Toledo, OH.
(slides)

*Extensions of CR functions in ${\mathbb C}^n \times {\mathbb R}$*,
AMS special session, April 2016, Salt Lake City, UT.
(slides)

*Local and Global Aspects of Singular Levi-flat Hypersurfaces*,
Advanced course at 30^{o} Colóquio Brasileiro de Matemática,
26-31 of July 2015,
IMPA, Rio de Janeiro, Brazil.

*Homotopy equivalence for proper holomorphic mappings*,
AMS special session, March 2015, Michigan State University, MI.
(slides)

*Normal forms for CR singular codimension two Levi-flat submanifolds*,
Workshop on Complex Analysis and Geometry in Wuhan University,
May 2014, Wuhan, China.
(slides)

*Signature pairs of positive polynomials*,
Joint Mathematical Meetings, AMS special session,
January 2013, San Diego, CA. (slides)

*Bounding the rank of Hermitian forms and rigidity for CR mappings of
hyperquadrics*,
International Workshop on Several Complex Variables and Complex Geometry,
July 2012, Taipei, Taiwan. (slides with typos corrected)

*Polynomials constant on a hyperplane and CR maps of spheres*,
RTG Workshop on Complex Analysis,
October 2011, Ann Arbor, MI. (slides)

Poster *Singular set of a Levi-flat hypersurface is Levi-flat*,
VI Workshop on Geometric Analysis of PDE and Several Complex Variables,
August 2011, Serra Negra, Brazil. (the poster in
pieces)

*Hermitian forms and rational maps of hyperquadrics*,
CIRM - CR-Geometry and PDE's - IV, June 2010, Levico Terme, Italy.

*Polynomials constant on a hyperplane and CR maps of spheres*,
Special session AMS meeting, March 2010, Lexington, KY.

*Hermitian forms and rational maps of hyperquadrics*,
RTG Workshop on Holomorphic Maps and Iterations, March 2010, Ann Arbor, MI.
(slides)

*Uniqueness of certain polynomials constant on a hyperplane*,
Applications of Computer Algebra 2009, Montréal, Canada.

*Singular Levi-flat hypersurfaces in complex projective space*,
Conference on Complex and CR Geometry, Partial Differential Equations and
Invariant Theory in honor of Joseph J. Kohn, July 2008, Prague, Czech Rep.

*Singular Levi-flat hypersurfaces in complex projective space*,
CIRM - CR-Geometry and PDE's - III, June 2008, Levico Terme, Italy.

*Levi-flat hypersurfaces with real analytic boundary*,
Special session CMS meeting, December 2007, London, ON.

*Extensions of Levi-flat hypersurfaces past CR boundaries*,
Special session AMS meeting, October 2007, Chicago, IL.

*Singularities of Levi-Flat Hypersurfaces*, International Conference in
PDE, Complex Analysis, and Differential Geometry, June 2006, Notre Dame, IN.

MGSS Seminar notes November 2013: Several Complex Variables are Better than One (PDF). Large font was for me, if you want smaller font get the tex file. This is an updated version of the talk I gave in the Food for Thought seminar from '06 at UCSD.

After my masters thesis defense I had some free time and dabbled into cryptography for a bit (I was taking a crypto class at SDSU) and there was a final project, so I wrote up a short paper where I came up with two new toy public key crypto systems (not useful, but perhaps the ideas could be refined). Anyway this is from May 2003 "Toy Public Key Cryptography" (PDF).

I went on a holiday in Hawaii and so obviously I brought the MAA Monthly and solved a few problems. Here's 10998 as postscript or latex (appeared in the December 2004 issue), and 11002 as postscript or latex (both done circa spring 2003).

Some very old unfinished (and never will be finished) research on counting b-nary fraction sequences summing to 1. Here as postscript file (circa summer 2002). After I got the main counting result I found it was already published in the 70's (MR 50:1767) (concentrating on b=2 and done with simpler formulas). I haven't actually found anything really new beyond that.

Also I've written quite a few of the encyclopedia entries on Planetmath. Unfortunately it has been overrun by cranks and there doesn't seem to be the will to kick them off. I was thinking of forking planetmath to save the good parts of the encyclopedia, but have not had (and likely will not have) the time to do so. 90% is good I would say, but the presence of inaccurate, nonsensical, or sometimes just plain wrong entries undermines credibility of the website.

Here are some wxMaxima scripts I used in math 10c and 20c (multivariable calculus) at UCSD.

I always used to write a blog, before it was called a blog. I tend to write things very sporadically. I've recently opened a blog at wordpress instead of advogato since they allow latex. So go read my blog.

If you are interested in a hardcopy version of some of the notes above, my thesis, some of my strange czech poetry, or some photobooks, or whatever else I've put on lulu ... see my lulu page.

Here are some pictures from the MSRI, CR Geometry workshop in 2005. Not really pictures from the workshop, but more from walking around Berkeley. And here are some pictures from the AIM workshop on CR complexity in September 2006. In similar spirit. There are some other pictures from conferences and random other things at our galleries: old gallery, new gallery

If you ever wondered what a five-legged goat looks like, see Goat Love, Inc.

See my non-math home page for other boring info including some more math stuff.