• Previous Article
    Branches of positive solutions of subcritical elliptic equations in convex domains
  • PROC Home
  • This Issue
  • Next Article
    Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays
2015, 2015(special): 239-247. doi: 10.3934/proc.2015.0239

Bridges between subriemannian geometry and algebraic geometry: Now and then

1. 

Department of Mathematics, Imperial College, London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

2. 

Mathematics and Computer Science, Santa Clara University, Santa Clara, CA, United States

3. 

Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA

Received  September 2014 Revised  February 2015 Published  November 2015

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.
Citation: Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239
References:
[1]

V. I. Arnold, Simple singularities of curves,, Proc. Steklov Inst. Math., 226 (1999), 20.

[2]

V. I. Arnold, A.N. Varchenko, and S. Guzein-Sade, Singularités des Applicants Différentiables,, Editions MIR, (1986).

[3]

A. Castro, Chains and Monsters,, Ph.D thesis, (2010).

[4]

A. Castro and W. Howard, A Monster tower approach to Goursat multi-flags,, Differential Geom. Appl., 30 (2012), 405.

[5]

A. Castro and W. Howard, A Semple-type approach to a problem of Goursat: the multi-flag case,, C. R. Math. Acad. Sci. Paris, 351 (2013), 921.

[6]

A. Castro and W. Howard, Spelling rules for the Monster/Semple tower,, arXiv:1407.1824., ().

[7]

A. Castro, R. Montgomery, and appendix by W. Howard, Spatial curve singularities and the Monster/Semple tower,, Israel J. Math., 192 (2012), 381.

[8]

S. Colley and G. Kennedy, Triple and quadruple contact of plane curves,, Enumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium, 123 (1991), 31.

[9]

A. Giaro, A. Kumpera, and C. Ruiz, Sur la Lecture correcte d'un résultat d' Élie Cartan,, C.R. Acad. Sci. Paris Sér. A-B, 287 (1978).

[10]

J.C. Hausmann and E. Rodriguez, Holonomy orbits of the snake charmer algorithm,, Geometry and Topology of Manifolds, 76 (2007), 207.

[11]

F. Jean, The car with N trailers: characterisation of the singular configurations,, Control, 1 (1996), 241.

[12]

S. Kuroki and D. Suh, Classification of complex projective towers up to dimension 8 and cohomological rigidity,, arXiv:1203.4403., ().

[13]

M. Lejeune-Jalabert, Chains of points in the Semple tower,, Am. J. Math., 128 (2006), 1283.

[14]

J. Milnor and J. Stasheff, Characteristic Classes,, Princeton University Press, (1974).

[15]

R. Montgomery and M. Zhitomirskii, Geometric appraoch to Goursat flags,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 459.

[16]

R. Montgomery and M. Zhitomirksii, Points and curves in the Monster tower,, Mem. Amer. Math. Soc., 203 (2009).

[17]

P. Mormul, Geometric classes of Goursat flags and their encoding by small growth vectors,, Central European J. Math., 2 (2004), 859.

[18]

R. Murray and S. Sastry, Nonholonomic motion planning: steering using sinusoids,, IEEE Trans. Automat. Control, 38 (1993), 700.

[19]

F. Pelletier and M. Slayman, Configuration of an articulated arm and singularities of special multi-flags,, SIGMA, 10 (2014), 1.

[20]

C. Shanbrom, The Puiseux characteristic of a Goursat germ,, J. Dynamical and Control Systems, 20 (2014), 33.

[21]

K. Shibuya and K. Yamaguchi, Drapeau theorem for differential systems,, Differential Geom. Appl., 27 (2009), 793.

[22]

R. Thom, Quelques propriétés globales des variétés,, Comment. Math. Helv., 28 (1954), 17.

[23]

C. Wall, Singular Points of Plane Curves,, London Mathematical Society Student Texts, (2004).

show all references

References:
[1]

V. I. Arnold, Simple singularities of curves,, Proc. Steklov Inst. Math., 226 (1999), 20.

[2]

V. I. Arnold, A.N. Varchenko, and S. Guzein-Sade, Singularités des Applicants Différentiables,, Editions MIR, (1986).

[3]

A. Castro, Chains and Monsters,, Ph.D thesis, (2010).

[4]

A. Castro and W. Howard, A Monster tower approach to Goursat multi-flags,, Differential Geom. Appl., 30 (2012), 405.

[5]

A. Castro and W. Howard, A Semple-type approach to a problem of Goursat: the multi-flag case,, C. R. Math. Acad. Sci. Paris, 351 (2013), 921.

[6]

A. Castro and W. Howard, Spelling rules for the Monster/Semple tower,, arXiv:1407.1824., ().

[7]

A. Castro, R. Montgomery, and appendix by W. Howard, Spatial curve singularities and the Monster/Semple tower,, Israel J. Math., 192 (2012), 381.

[8]

S. Colley and G. Kennedy, Triple and quadruple contact of plane curves,, Enumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium, 123 (1991), 31.

[9]

A. Giaro, A. Kumpera, and C. Ruiz, Sur la Lecture correcte d'un résultat d' Élie Cartan,, C.R. Acad. Sci. Paris Sér. A-B, 287 (1978).

[10]

J.C. Hausmann and E. Rodriguez, Holonomy orbits of the snake charmer algorithm,, Geometry and Topology of Manifolds, 76 (2007), 207.

[11]

F. Jean, The car with N trailers: characterisation of the singular configurations,, Control, 1 (1996), 241.

[12]

S. Kuroki and D. Suh, Classification of complex projective towers up to dimension 8 and cohomological rigidity,, arXiv:1203.4403., ().

[13]

M. Lejeune-Jalabert, Chains of points in the Semple tower,, Am. J. Math., 128 (2006), 1283.

[14]

J. Milnor and J. Stasheff, Characteristic Classes,, Princeton University Press, (1974).

[15]

R. Montgomery and M. Zhitomirskii, Geometric appraoch to Goursat flags,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 459.

[16]

R. Montgomery and M. Zhitomirksii, Points and curves in the Monster tower,, Mem. Amer. Math. Soc., 203 (2009).

[17]

P. Mormul, Geometric classes of Goursat flags and their encoding by small growth vectors,, Central European J. Math., 2 (2004), 859.

[18]

R. Murray and S. Sastry, Nonholonomic motion planning: steering using sinusoids,, IEEE Trans. Automat. Control, 38 (1993), 700.

[19]

F. Pelletier and M. Slayman, Configuration of an articulated arm and singularities of special multi-flags,, SIGMA, 10 (2014), 1.

[20]

C. Shanbrom, The Puiseux characteristic of a Goursat germ,, J. Dynamical and Control Systems, 20 (2014), 33.

[21]

K. Shibuya and K. Yamaguchi, Drapeau theorem for differential systems,, Differential Geom. Appl., 27 (2009), 793.

[22]

R. Thom, Quelques propriétés globales des variétés,, Comment. Math. Helv., 28 (1954), 17.

[23]

C. Wall, Singular Points of Plane Curves,, London Mathematical Society Student Texts, (2004).

[1]

Penka Georgieva, Aleksey Zinger. Real orientations, real Gromov-Witten theory, and real enumerative geometry. Electronic Research Announcements, 2017, 24: 87-99. doi: 10.3934/era.2017.24.010

[2]

Abbas Bahri. Recent results in contact form geometry. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 21-30. doi: 10.3934/dcds.2004.10.21

[3]

C. Alonso-González, M. I. Camacho, F. Cano. Topological invariants for singularities of real vector fields in dimension three. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 823-847. doi: 10.3934/dcds.2008.20.823

[4]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437

[5]

Janina Kotus, Mariusz Urbański. The dynamics and geometry of the Fatou functions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 291-338. doi: 10.3934/dcds.2005.13.291

[6]

Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437

[7]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291

[8]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407

[9]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249

[10]

Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014

[11]

François Lalonde, Yasha Savelyev. On the injectivity radius in Hofer's geometry. Electronic Research Announcements, 2014, 21: 177-185. doi: 10.3934/era.2014.21.177

[12]

Manuel Gutiérrez. Lorentz geometry technique in nonimaging optics. Conference Publications, 2003, 2003 (Special) : 386-392. doi: 10.3934/proc.2003.2003.386

[13]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

[14]

Weihong Guo, Jing Qin. A geometry guided image denoising scheme. Inverse Problems & Imaging, 2013, 7 (2) : 499-521. doi: 10.3934/ipi.2013.7.499

[15]

Q-Heung Choi, Changbum Chun, Tacksun Jung. The multiplicity of solutions and geometry in a wave equation. Communications on Pure & Applied Analysis, 2003, 2 (2) : 159-170. doi: 10.3934/cpaa.2003.2.159

[16]

Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777

[17]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040

[18]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012

[19]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363

[20]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643

 Impact Factor: 

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]