When people ask me what kind of math I do, I have a hard time answering this question. I started my career working on Hamiltonian systems with symmetry. I did some work on singular reduction. Then I moved to contact geometry and did some work on contact toric manifolds. After that I wanted to figure out if it is useful to think of orbifolds as stacks and what happens when you do. Thinking of orbifolds as stacks turned out to be useful when one is interested in their quantization. Understanding geometric stacks required coming to terms with Lie groupoids. This led to work on vector fields on stacks (which form a Lie 2-algebra). It also, accidentally, led to work on vector fields with groupoid symmetries on coupled cell networks. And developed into further work on various kinds of networks including networks of hybrid open systems. Right before the pandemic started I was at a small meeting and complained about the way people were handling singular spaces. When the pandemic hit, I had nothing better to do, so I decided to write a paper on vector fields, differential forms and Cartan calculus on differential spaces and, more generally, on C∞-ringed spaces. I ended up with three papers in this subject.
[56] Yael Karshon and Eugene Lerman, Vector fields and flows on subcartesian spaces, arXiv:2307.10959
SIGMA 19 (2023), 093, 17 pages https://doi.org/10.3842/SIGMA.2023.093
[55] Eugene Lerman, Cartan calculus for C∞-ringed spaces, arXiv:2307.05604
[54] Eugene Lerman, Differential forms on C∞-ringed spaces arXiv:2212.11163
Journal of Geometry and Physics, Volume 196, February 2024, 105062 https://doi.org/10.1016/j.geomphys.2023.105062
[53] Eugene Lerman, Stability and bifurcations of symmetric tops https://arxiv.org/abs/2111.04855
(an update of https://arxiv.org/abs/dg-ga/9608010). Published version: Pure and Applied Mathematics Quarterly
Volume 19, Number 4, 2037–2065, 2023
[52] Lee DeVille,Eugene Lerman, James Schmidt,
Runge-Kutta and Networks
arxiv.org/abs/1908.11453. (Turns out the results were known, so the paper will not be submitted for publication.)
[51] Eugene Lerman,James Schmidt
Networks of hybrid open systems
arxiv.org/abs/1908.10447
J. Geometry and Physics, v 149, March 2020
[50] Eugene Lerman, Left-invariant vector fields on a Lie 2-group
arxiv.org/abs/1808.02920
Theory and Applications of Categories, vol 34, 2019, No 21, pp 604-634
[49] Eugene Lerman, Networks of open systems,
arXiv.org/abs/1705.04814
The paper supersedes arXiv.org/abs/1602.01017
10.1016/j.geomphys.2018.03.020
[48] Eugene Lerman, A category of hybrid systems,
arXiv.org/abs/1612.01950
Preliminary version. Comments welcome.
[47] Daniel Berwick-Evans, Eugene Lerman,
Lie 2-algebras of vector fields
arXiv.org/abs/1609.03944
[46] Eugene Lerman, David I. Spivak,
An algebra of open continuous time dynamical systems and networks
arXiv.org/abs/1602.01017
[45] Brian Collier, Eugene Lerman and Seth Wolbert,
Parallel Transport on Principal Bundles over Stacks
arXiv.org/abs/1509.05000
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
arXiv.org/abs/1408.1598
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
arXiv.org/abs/1307.7733
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
www.arXiv.org/abs/1206.2334
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
www.arXiv.org/abs/1008.5359
[38] Eugene Lerman, Categories of symplectic toric manifolds as Picard stack torsors
www.arXiv.org/abs/0908.2783
[37] Eugene Lerman and Anton Malkin, Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds
www.arXiv.org/abs/0908.0903
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.
www.arXiv.org/abs/0907.2891
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.
www.arXiv.org/abs/0807.0058
[34] Eugene Lerman, Orbifolds as stacks?
L'Enseign. Math. (2) 56 (2010), no. 3-4, 315--363
www.arXiv.org/abs/0806.4160
[33] Eugene Lerman and Anton Malkin, Differential characters as stacks and prequantization
J. Gokova Geom. Topol. GGT 2 (2008), 14--39.
www.arXiv.org/abs/0710.4340
[32] D. Burns, V. Guillemin and E. Lerman, Kaehler metrics on singular toric varieties,
Pacific J. Math. 238 (2008), no. 1, 27--40.
www.arXiv.org/abs/math/0501311
[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.
www.arXiv.org/abs/math/0501310
[30] Eugene Lerman, Gradient flow of the norm squared of a moment map
L'Enseign. Math. (2) 51 (2005), no. 1-2, 117--127.
www.arXiv.org/abs/math/0410568
[29] Viktor Ginzburg and Eugene Lerman, Existence of relative periodic orbits near relative equilibria,
Math. Res. Lett. 11 (2004), no. 2-3, 397--412.
www.arXiv.org/abs/math/0402264
[28] V. Guillemin and E. Lerman, Melrose--Uhlmann projectors, the metaplectic representation and symplectic cuts
J. Differential Geom. 61 (2002), no. 3, 365--396.
www.arXiv.org/abs/math/0302150
[27] Eugene Lerman, Contact fiber bundles
J. Geom. Phys. 49 (2004), no. 1, 52--66
www.arXiv.org/abs/math/0301137
[26] D. Burns, V. Guillemin and E. Lerman, Kaehler cuts.
www.arXiv.org/abs/math/0212062
[25] Eugene Lerman, On maximal tori in the contactomorphism groups of regular contact manifolds.
www.arXiv.org/abs/math/0212043
[24] Eugene Lerman, Maximal tori in the contactomorphism groups of circle bundles over Hirzebruch surfaces
Math. Res. Lett. 10 (2003), no. 1, 133--144.
www.arXiv.org/abs/math/0204334/
[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. arXiv:math/0107201
[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.
www.arXiv.org/abs/math.SG/0012017
[18] Eugene Lerman, Contact Cuts, Israel J. Math , 124 (2001), 77--92; arXiv:math/0002041
[17] Eugene Lerman and Susan Tolman, Intersection cohomology of S^1 symplectic quotients and small resolutions. Duke Math. J. 103 (2000), no. 1, 79--99.
[16] Lerman, Eugene; Willett, Christopher, The topological structure of contact and symplectic quotients, Internat. Math. Res. Notices 2001, no. 1, 33--52. arXiv:math/0008178
[15] Lerman, Eugene; Tokieda, Tadashi, On relative normal modes C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 413--418.
[14] 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]
Early version of this paper was posted as dg-ga/9706009
[13] Lerman, Eugene; Meinrenken, Eckhard; Tolman, Sue; Woodward, Chris, Nonabelian convexity by symplectic cuts. Topology 37 (1998), no. 2, 245--259. arXiv:dg-ga/9603015
[12] Bates, L.; Lerman, E., Proper group actions and symplectic stratified spaces. Pacific J. Math. 181 (1997), no. 2, 201--229. [original article] arXiv:dg-ga/9407003
[11] 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.
arXiv:dg-ga/9506008
[10] 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] 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] Eugene Lerman, A compact symmetric symplectic non-Kaehler manifold. Math. Res. Lett. 3 (1996), no. 5, 587--590.
[7] Eugene Lerman and Reyer Sjamaar, 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] Eugene Lerman, Symplectic cuts. Math. Res. Lett. 2 (1995), no. 3, 247--258.
[5] 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] Reyer Sjamaar and Eugene Lerman, Stratified symplectic spaces and reduction. Ann. of Math. (2) 134 (1991), no. 2, 375--422.
[3] 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] Lerman, Eugene, On the centralizer of invariant functions on a Hamiltonian G-space. J. Differential Geom. 30 (1989), no. 3, 805--815.
[1] Lerman, Eugene, How fat is a fat bundle? Lett. Math. Phys. 15 (1988), no. 4, 335--339
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