Welcome to my research page

I am interested in geometry and dynamical systems. Some of my papers use category theory.

Preprints, papers, and books:

[52] Lee DeVille,Eugene Lerman, James Schmidt,
Runge-Kutta and Networks

[51] Eugene Lerman,James Schmidt
Networks of hybrid open systems
J. Geometry and Physics, v 149, March 2020

[50] Eugene Lerman, Left-invariant vector fields on a Lie 2-group
Theory and Applications of Categories, vol 34, 2019, No 21, pp 604-634

[49] Eugene Lerman, Networks of open systems,
The paper supersedes arXiv.org/abs/1602.01017

[48] Eugene Lerman, A category of hybrid systems,
Preliminary version. Comments welcome.

[47] Daniel Berwick-Evans, Eugene Lerman,
Lie 2-algebras of vector fields

[46] Eugene Lerman, David I. Spivak,
An algebra of open continuous time dynamical systems and networks

[45] Brian Collier, Eugene Lerman and Seth Wolbert,
Parallel Transport on Principal Bundles over Stacks
J. of Geometry and Physics, Vol. 107, September 2016, pp 187--213
DOI: 10.1016/j.geomphys.2016.05.010

[44] Dmitry Vagner, David I. Spivak, Eugene Lerman,
Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams
Theory and Applications of Categories, Vol. 30, 2015, No. 51, pp 1793-1822.
Published 2015-12-03. http://www.tac.mta.ca/tac/volumes/30/51/30-51.pdf

[43] Eugene Lerman, Invariant vector fields and groupoids
Int Math Res Notices (2015) 2015 (16): 7394-7416. doi: 10.1093/imrn/rnu170

[42] Lee DeVille and Eugene Lerman, Modular dynamical systems on networks
arXiv.org/abs/1303.3907 A major revision of [39] below.
Journal of the European Mathematical Society Volume 17, Issue 12, 2015, pp. 2977-3013, DOI: 10.4171/JEMS/577

[41] Lee DeVille and Eugene Lerman, Dynamics on networks of manifolds
www.arXiv.org/abs/1208.1513 (first version of the paper was posted under the title "Tinker toy dynamics.")
SIGMA 11 (2015), 022, 21 pages DOI: 10.3842/SIGMA.2015.022

[40] Eugene Lerman, Geometric quantization; a crash course
Appeared in Mathematical Aspects of Quantization, Contemporary Math. v 583, Evens et al, eds., AMS, 2012.

[39] R. E. Lee DeVille and Eugene Lerman, Dynamics on networks I. Combinatorial categories of modular continuous-time systems

[38] Eugene Lerman, Categories of symplectic toric manifolds as Picard stack torsors

[37] Eugene Lerman and Anton Malkin, Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds
Advances in Mathematics. Volume 229, Issue 2, 30 January 2012, Pages 984-1000 DOI: 10.1016/j.aim.2011.10.013

[36] Yael Karshon and Eugene Lerman, Non-compact symplectic toric manifolds.
SIGMA 11 (2015), 055, 37 pages, DOI: 10.3842/SIGMA.2015.055
Hand-written notes of a talk on the paper.

[35] Eugene Lerman and Anton Malkin, Equivariant differential characters and symplectic reduction
Comm. Math. Phys. 289 (2009), no. 2, 777--801.

[34] Eugene Lerman, Orbifolds as stacks?
L'Enseign. Math. (2) 56 (2010), no. 3-4, 315--363

[33] Eugene Lerman and Anton Malkin, Differential characters as stacks and prequantization
J. Gokova Geom. Topol. GGT 2 (2008), 14--39.

[32] D. Burns, V. Guillemin and E. Lerman, Kaehler metrics on singular toric varieties,
Pacific J. Math. 238 (2008), no. 1, 27--40.

[31] D. Burns, V. Guillemin and E. Lerman, Toric symplectic singular spaces I: isolated singularities
Conference on Symplectic Topology. J. Symplectic Geom. 3 (2005), no. 4, 531--543.

[30] Eugene Lerman, Gradient flow of the norm squared of a moment map
L'Enseign. Math. (2) 51 (2005), no. 1-2, 117--127.

[29] Viktor Ginzburg and Eugene Lerman, Existence of relative periodic orbits near relative equilibria,
Math. Res. Lett. 11 (2004), no. 2-3, 397--412.

[28] V. Guillemin and E. Lerman, Melrose--Uhlmann projectors, the metaplectic representation and symplectic cuts
J. Differential Geom. 61 (2002), no. 3, 365--396.

[27] Eugene Lerman, Contact fiber bundles
J. Geom. Phys. 49 (2004), no. 1, 52--66

[26] D. Burns, V. Guillemin and E. Lerman, Kaehler cuts.

[25] Eugene Lerman, On maximal tori in the contactomorphism groups of regular contact manifolds.

[24] Eugene Lerman, Maximal tori in the contactomorphism groups of circle bundles over Hirzebruch surfaces
Math. Res. Lett. 10 (2003), no. 1, 133--144.

[23] Eugene Lerman, Homotopy groups of K-contact toric manifold
Trans. Amer. Math. Soc. 356 (2004), no. 10, 4075--4083 ,
www.arXiv.org/abs/math/0204064 .

[22] Eugene Lerman, Geodesic flows and contact toric manifolds,
Symplectic geometry of integrable Hamiltonian systems (Barcelona, 2001), 175--225, Adv. Courses Math. CRM Barcelona, Birkhauser, Basel, 2003. www.arXiv.org/abs/math/0201230 .

These are notes for a course on contact manifolds and torus actions delivered at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems at Centre de Recerca Matematica in Barcelona in July 2001.

[21] Lerman, E., Contact toric manifolds,
J. Symplectic Geom. 1 (2003), no. 4, 785--828. www.arXiv.org/abs/

[20] Lerman, E., Shirokova, N., Completely integrable torus actions on symplectic cones
Math Research Letters, 9 (2002), no. 1, 105--115.
preprint: Toric integrable geodesic flows www.arXiv.org/abs/math.DG/0011139 .

[19] Lerman, E., A convexity theorem for torus actions on contact manifolds
Illinois J. Math , 46 (2002), no. 1, 171--184.

[18] Lerman, E., Contact Cuts, Israel J. Math , 124 (2001), 77--92; www.arXiv.org/abs/math.SG/000204 .

[17] 2001k:53163 Lerman, E.; Tolman, S. Intersection cohomology of S^1 symplectic quotients and small resolutions. Duke Math. J. 103 (2000), no. 1, 79--99.

[16] 2001j:53112 Lerman, Eugene; Willett, Christopher, The topological structure of contact and symplectic quotients, Internat. Math. Res. Notices 2001, no. 1, 33--52.

[15] 2000b:37066 Lerman, Eugene; Tokieda, Tadashi, On relative normal modes C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 413--418.

[14] 99j:58077 Lerman, E.; Singer, S.F., Stability and persistence of relative equilibria at singular values of the moment map, Nonlinearity 11 (1998), no. 6, 1637--1649. [original article]
(access may require subscription to Nonlinearity ).
Early version of this paper was posted as dg-ga/9706009 (see below)

[13] 99a:58069 Lerman, Eugene; Meinrenken, Eckhard; Tolman, Sue; Woodward, Chris, Nonabelian convexity by symplectic cuts. Topology 37 (1998), no. 2, 245--259. postscript file

[12] 98i:58085 Bates, L.; Lerman, E., Proper group actions and symplectic stratified spaces. Pacific J. Math. 181 (1997), no. 2, 201--229. [original article]

[11] 98e:58077 Karshon, Yael; Lerman, Eugene, The centralizer of invariant functions and division properties of the moment map. Illinois J. Math. 41 (1997), no. 3, 462--487. 58F05 (57S15 57S25) gzip'ed dvi file

[10] 98d:58074 Guillemin, Victor; Lerman, Eugene; Sternberg, Shlomo, Symplectic fibrations and multiplicity diagrams. Cambridge University Press, Cambridge, 1996. xiv+222 pp. ISBN: 0-521-44323-7 58F06 (17B99 22E45 58F05 81S10)

[9] 98a:57043 Lerman, Eugene; Tolman, Susan, Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Amer. Math. Soc. 349 (1997), no. 10, 4201--4230. [original article]

[8] 97g:57058 Lerman, Eugene, A compact symmetric symplectic non-Kaehler manifold. Math. Res. Lett. 3 (1996), no. 5, 587--590. 57S25 (53C15 57R15 57S15)

[7]97c:32045 Lerman, Eugene; Sjamaar, Reyer, Reductive group actions on Kähler manifolds. Conservative systems and quantum chaos (Waterloo, ON, 1992), 85--92, Fields Inst. Commun., 8, Amer. Math. Soc., Providence, RI, 1996. 32M05 (58F05)

[6] 96f:58062 Lerman, Eugene Symplectic cuts. Math. Res. Lett. 2 (1995), no. 3, 247--258. 58F05 (57S25)
dvi , LaTeX

[5] 95h:58054 Lerman, Eugene; Montgomery, Richard; Sjamaar, Reyer, Examples of singular reduction. Symplectic geometry, 127--155, London Math. Soc. Lecture Note Ser., 192, Cambridge Univ. Press, Cambridge, 1993. 58F05 (58A35)

[4] 92g:58036Sjamaar, Reyer; Lerman, Eugene; Stratified symplectic spaces and reduction. Ann. of Math. (2) 134 (1991), no. 2, 375--422. 58F05 (57R15)

[3] 92f:58058 Guillemin, V.; Lerman, E.; Sternberg, S., On the Kostant multiplicity formula. J. Geom. Phys. 5 (1988), no. 4, 721--750 (1989). 58F05 (22E46 22E60 58F06 58G10)

[2] 90k:58070 Lerman, Eugene, On the centralizer of invariant functions on a Hamiltonian $G$-space. J. Differential Geom. 30 (1989), no. 3, 805--815.

[1] 89k:53033 Lerman, Eugene, How fat is a fat bundle? Lett. Math. Phys. 15 (1988), no. 4, 335--339.

old preprints/:

dg-ga/9706009 [abs, src, ps, other] :

Title: Relative equilibria at singular points of the moment map
Author: Eugene Lerman (University of Illinois at Urbana-Champaign)
Comments: 10 pages, LaTeX2e
Subj-class: Differential Geometry
MSC-class: 58F; 70H
A later version is item [14] above

dg-ga/9608010 [abs, src, ps, other] :

Title: Stability of symmetric tops via one variable calculus
Author: Eugene Lerman
Comments: 10 pages, LaTeX, uses epic and eepic style files
Subj-class: Differential Geometry
MSC-class: 58F
It got rejected once by Physica D . I never got around to publishing it.

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last changed January 17, 2017