Vladimir Chernov (Tchernov)

Curriculum Vitae
 
 
 

Department of Mathematics

6188 Kemeny Hall

Tel: +1-(603) 646 24 21

Fax: +1-(603) 646 13 12

E-mail: Vladimir.Chernov@dartmouth.edu

Dartmouth College

Hanover, NH 03755-3551, USA

 

 

 

 

 

 

 

 

 

 

Personal Information

Previous spelling of the last name: Tchernov
Birth date: January 9, 1972
Citizenship: Russian citizen, US permanent resident

Research Interests

·        Contact and Symplectic Topology, specifically:  Legendrian and transverse knots (57R17) and their relations to wave front propagation (78A05);

·        Bennequin-type inequalities; Legendrian and Lagrangian immersions and embeddings.

·        Geometric and Low-Dimensional Topology, specifically: knots and links in manifolds (57M27); Vassiliev invariants of knots; invariants of immersed curves and surfaces; knots transverse to a vector field on a $3$-manifold; higher dimensional link theory  (57Q45); generalized linking numbers for nonzero homologous submanifolds.

·        Algebraic Topology and Global Analysis, specifically: spaces of imbeddings and immersions (58D10) and their homotopy groups; manifolds of mappings (58D15);  graded Poisson (17B63) algebra structures on bordism and homology groups of the space of mappings (55N22, 55N45); algebras a la Chas-Sullivan.

·        Interactions between link theory, Lorentz manifolds (53C50), contact and symplectic topology, and mathematical physics. Specifically: relations between causality and linking in strongly causal and globally hyperbolic spacetimes  (83C75); nonrefocussing spacetimes; Legendrian and Lagrangian submanifolds in manifolds of geodesics for semi-Riemannian manifolds; Arnold invariants of wave fronts.

Degrees

(Because my scientific advisor held positions at both Universities, I had to enroll into Graduate Programs at both places and to defend Ph.D.s at both places. The theses for the two Ph.D.s are based on different material.)

·         Licentiate in Mathematics. Uppsala University, 1996

·         Diploma in Mathematics. St. Petersburg State University, Russia, 1996

·         M.Sci. in Mathematics. University of California at Riverside (UCR), 1994

Employment and Teaching Experience

  1. Dartmouth College, Mathematics Department. Fall 2001 - Now. Assistant Professor. Teaching the following courses:

·  Math 8 Calculus of one and several variables Spring 2002;

·  Math 9 Multivariable Calculus (Honors) Fall 2001, Fall 2002, Fall 2003;

·  Math 11 Calculus for First-Year Students with two terms of AP credit, Fall 2004;

·  Math 13 Calculus of Vector Valued Functions Winter 2004;

·  Math 14  Calculus of Vector Valued Functions Winter 2002, Winter 2003, Winter 2004;

·  Math 23 Differential Equations Spring 2002, Winter 2006;

·  Math 31 Topics in Algebra Winter 2004;

·  Math 73 Multi-Variable Analysis Winter 2003;

·  Math 103, Topics in Analysis, Fall 2004, Fall 2005;

·  Math 124 Current Problems in Topology Fall 2002, Fall 2003, Fall 2005.

  1. Zurich University, Mathematics Department. Fall 2000 - Summer 2001. Postdoc. Teaching a course in 3-dimensional Topology and discussion sections for Algebra 3 and 4 (groups, ring, modules)
  2. Max-Planck-Institut fur Mathematik, Bonn. Fall 1999 - Summer 2000. Scientific Visitor for one year.
  3. ETH Zurich. Fall 1998 - Summer 1999. Postdoc.
  4. University of California at Riverside (UCR), Department of Mathematics. Fall 1993 - Spring 1994, Winter 1995, Winter 1996, Fall 1996 - Spring 1997. Teaching Assistant for Introduction to College Math., Calculus for Business, First Year Calculus, Applied Matrix Applied Linear Algebra; grader for Game Theory, Complex Analysis, graduate level Topology, graduate level Algebra.
  5. Uppsala University, Department of Mathematics. Fall 1994, Spring 1995, Fall 1995, Spring 1996. Research Assistant. Spring 1998 Teaching assistant for Transformation methods course.

Graduate Students: 

Allison Henrich

Publications

1.     V. Chernov (Tchernov),Yu. Rudyak: “Toward a General Theory of Linking Invariants”, Geometry and Topology Vol. 9 (2005) Paper no. 42, pages 1881-1913

2.     V Chernov (Tchernov): “Framed Knots in $3$-manifolds and affine self-linking numbers” published at J. Knot Theory Ramifications  14 (2005) no. 6  791-819

3.      V Chernov (Tchernov): “Relative Framing of Transverse knots”,  published in Int. Math. Res. Not. Issue 52 (2004) 2773-2797

4.      V Chernov (Tchernov): The universal order one invariant of framed knots in most S1-bundles over orientable surfaces”,  Algebraic and Geometric Topology Vol. 3 (2003) pp. 89-101

5.     V. Tchernov (Chernov): “Isomorphism of the groups of Vassiliev Invariants of Legendrian and of pseudo Legendrian knots” Compositio Math. 135 (2003), no. 1, 103-122.

6.     V. Tchernov (Chernov): “Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds”. Topology 42 (2003), no. 1, 1-33.

7.      A. Stoimenow, V. Tchernov (Chernov), A. Vdovina: “The canonical genus of a classical and virtual knot”, MPI preprint 108 (2000) Geom. Dedicata 95 (2002), 215-225.

8.      V. Tchernov (Chernov): “Arnold-type invariants of wave fronts on surfaces” Topology 41 (2002), no. 1, 1-45

9.     V. Tchernov (Chernov): “Homotopy groups of the space of curves on a surface”. Math. Scand. 86 (2000), no. 1, 36-44.

10.  V. Tchernov (Chernov): “Arnold-type Invariants of Curves on Surfaces”, J. Knot Theory Ramifications 8 (1999), no. 1, 71-97. preliminary version first appeared as a preprint  “Strangeness- and J±-type invariants of immersed oriented curves”, Preprint, Dept. of Math., Uppsala Univ., U.U.D.M. report 1997:14 

11. V. Tchernov (Chernov): “Shadows of wave fronts and Arnold-Bennequin type invariants of fronts on surfaces and orbifolds.” Differential and symplectic topology of knots and curves, 153-184, Amer. Math. Soc. Transl. Ser. 2, 190, Amer. Math. Soc., Providence, RI, (1999).

12. V. Tchernov (Chernov): “The most refined Vassiliev invariant of degree one of knots and links in R1-fibrations over a surface.”  J. Knot Theory Ramifications 7 (1998), no. 2, 257-266.

Recent Preprints

  1. V. Chernov (Tchernov): “Graded Poisson algebras on bordism groups of garlands and their applications”, preprint math.GT/0608153 at http://www.arxiv.org, 49 pages (2006). Currently under revision. Expected to be submitted for publication in January-February 2007
  2. V. Chernov (Tchernov), Yu. Rudyak: “Linking and causality in globally hyperbolic spacetimes”, preprint math.GT/0611111 at http://www.arxiv.org 41 pages (2006) A part of the paper (several results of sections 3,4,5,8,9) is an extension and development of our work math.GT/0207219 in the context of Lorentzian geometry. The results of sections 6,7,10 and Appendix B are new. Expected to be submitted for publication in November 2006
  3. V. Chernov (Tchernov), Yu. Rudyak: “On generalized winding numbers”,  preprint math.GT/0301117 at http://www.arxiv.org (2003), greatly revised in 2006 to be re-submitted for publication in November 2006.
  4. V. Chernov (Tchernov), Yu. Rudyak: “Algebraic structures on generalized strings”, preprint math.GT.0306140 at http://www.arxiv.org (2003) currently under major revision. Target date for submission - end of 2007.

Works in Progress

1.     V. Chernov (Tchernov): “Towards classification of knots transverse to a given vector field on a manifold.” (tentative title), the preprint is expected to be completed by the Summer 2007.

Talks and Participation in Conferences (talk titles are provided when available)

  1. A talk "Some applications of the Graded Poisson algebras on bordism groups of garlands" at the Topology Seminar, Mathematics Department, Ohio State University, October 31, 2006
  2. An invited talk "Poisson and other algebras on bordism groups of garlands." on October 29 (2006) at the “Special Session on Homotopy Theory of Compactified Moduli Spaces” at the AMS 2006 Fall Eastern Section #1021 Meeting Storrs, CT, held October 28-29, 2006
  3. A talk "Graded Poisson algebras, bordism groups, affine linking numbers and applications." at the Gauge Theory and Topology Seminar, Department of Mathematics, Harvard University, October 20, 2006
  4. A talk "Graded Poisson algebras on bordism groups of garlands and minimizing intersection of loops on a surface." at the Topology Seminar, Mathematics Department, Brandeis University, October 17, 2006
  5. An invited talk "Graded Poisson algebras on bordism groups of odd-dimensional garlands and minimizing the intersection of loops on a surface." on October 8 (2006) at the "Special Session on Floer Methods in Low-dimensional Topology" of the AMS  2006 Fall Western Section Meeting #1019, Salt Lake City, UT, held October 7-8, 2006
  6. A talk "Graded Poisson algebras on bordism groups of garlands."at the “Rokhlin Topology Seminar”, the St. Petersburg Department of the Steklov Institute of Mathematics, St Petersburg, Russia, September 11, 2006
  7. A talk "Applications of the Ω*-algebra of the space of garlands." on June 9 2006 at the Lehigh University Geometry and Topology Conference, held June 8-10, 2006
  8. A talk "Generalized strings algebra." at the Geometry and Topology seminar, Department of Mathematics and Statistics, University of Calgary, Canada, May 10, 2006
  9. Colloquium talk "Affine linking numbers and their applications to the study of wave front propagation." Department of Mathematics and Statistics, University of Calgary, Canada, May 5, 2006
  10. A talk "Relative framing of transverse knots." at the Symplectic Topology Seminar, Department of Mathematics and Statistics, University of Calgary, Canada, May 1, 2006
  11. Two talks "Affine linking number and causality on globally hyperbolic Lorentz manifolds and a  roundtrip journey from singular null surfaces to Legendrian mappings, part I.” and "A roundtrip journey from singular null surfaces to Legendrian mappings, part II."  at the Geometry and Topology Seminar, Department of Mathematics, Dartmouth College, April 20 and May 18, 2006
  12. A talk “Knot Theory and the Study of Wave Front Propagation.” at the XIII Hudson River Undergraduate Mathematics Conference, Westfield, MA, April 8, 2006
  13. A talk "New algebraic structures on the generalized string algebra." on December 4 2005 at the “Algebraic Topology” section of the Union College Mathematics Conference, Union College, Schenectady NY, held December 3-4, 2005
  14. A talk "Framed knots in 3-manifolds and affine self-linking numbers." at the “Rokhlin Topology Seminar”, Steklov Institute of Mathematics at St Petersburg, Russia, September 12, 2005
  15. A talk "Relative framings of transverse knots." at the “Moscow-Petersburg Seminar on Low-Dimensional Mathematics”, Steklov Institute of Mathematics at St Petersburg, Russia, September 9, 2005
  16. A talk "Generalizations of Chas-Sullivan string homology and applications to link and coincidence theories." on July 26 2005 at the 5 International Siegen Topology Symposium “Manifolds and Their mappings” held at Siegen University, Germany, on July 25-29, 2005
  17. A talk "Relative framing of transverse knots and Vassiliev invariants of transverse and pseudo-Legendrian knots." on June 5 2005 at the AMS-IMS-SIAM Summer Research Conference "Quantum Topology-Contemporary Issues and Perspectives" held at Snowbird Utah, June 5-9, 2005
  18. two talks "Applications of Vassiliev invariants to distinguishing pseudo-Legendrian knots, parts I and II." at the Geometry and Topology seminar, Department of Mathematics, Dartmouth College, May 26 and June 2, 2005
  19. A talk "Relative framing of transverse knots." on November 7 2004 at the “Special Session on Invariants of knots and 3-manifolds” of the 2004 Fall Eastern Section Meeting of the AMS +1042, Pittsburgh, PA, November 6-7, 2004
  20. A talk "Algebraic structures on generalized string and applications." at the Differential Geometry and Topology Seminar of the Universite du Quebec a Montreal, Montreal, Canada, September 17, 2004
  21. A talk "Relative framings of transverse knots." on August 26 2004 at the “L.V. Keldysh Centennial Conference in Geometric Topology, Discrete Geometry and Set Theory”  held August 24-28 at the Steklov Mathematics Institute of the Russian Academy of Science and at the Moscow State University, Moscow, Russia
  22. A talk “Affine linking numbers, their applications, and operations on generalized strings.” at the Geometry and Topology seminar, Department of Mathematics, University of Pennsylvania, April 8, 2004
  23. An invited talk “Algebraic structures on generalized strings.” on April 18 2004 at the Special “Homotopical Physics” Section of the Spring Eastern AMS Section Meeting #997, Lawrenceville NJ, held on April 17-18, 2004
  24. A talk "Generalizations of the complex analysis Cauchy integral formula and related algebraic structures on generalized string homology." at the Geometry and Topology seminar, Department of Mathematics, Dartmouth College, March 9, 2004
  25. A talk "Affine linking numbers and generalized string homology.” on December 5 2003 at the “Borders in 3-dimensional topology” conference held at the Ohio State University, Columbus, December 5-7, 2003
  26. A talk "Vassiliev invariants of Legendrian, pseudo-Legendrian, transverse, and framed knots in contact 3-manifolds." at the Geometry and Topology seminar, Mathematics Department, SUNY Buffalo, November 14, 2003
  27. A talk ``Affine linking numbers, their applications and algebraic structures on generalized strings.” on November 8 2003 at the “Algebraic Topology” Section of the Union College Mathematics Conference, Union College, Schenectady, NY, held on November 8-9, 2003
  28. A talk “Affine linking and winding numbers and the study of front propagation.” on October 25 2003 at the “Homological Physics” Section of the Fall AMS Southeastern Section Meeting #991, at the University of North Carolina, Chapel Hill, NC, October 24-25, 2003
  29. A talk “Vassiliev invariant of Legendrian, pseudo-Legendrian and framed knots in contact 3-manifolds.” at the “Gauge Theory and Topology Seminar”, Department of Mathematics, Harvard University, October 10, 2003
  30. A talk “Vassiliev invariants of Legendrian, pseudo-Legendrian, transverse and framed knots in contact 3-manifolds.” at the Geometric Topology Seminar, Department of Mathematics, Columbia University,  September 19, 2003
  31. A talk “Affine linking and winding numbers and their applications to the study of causality and of front propagations.” at the “Arnold’s Seminar on Singularity Theory”, Mathematics Department of Moscow State University, Moscow, Russia, September 9, 2003
  32. A talk "Affine linking numbers and their application to the study of front propagations." at the “Moscow-Petersburg Seminar on Low-Dimensional Mathematics”, Steklov Institute of Mathematics at St Petersburg, Russia, September 5, 2003
  33. A talk “Algebraic structures on generalized strings and the general theory of affine linking numbers." at the “Rokhlin Topology Seminar”, Steklov Institute of Mathematics at St Petersburg, Russia, September 1, 2003
  34. A talk “Affine linking and winding numbers and the study of front propagation.” on July 18 2003 at the “XII Oporto meeting in Geometry Topology and Physics”, Universidade do Porto, Portugal, held July 17-20, 2003
  35. A talk “Affine linking and winding numbers and the study of front propagation.” at the Mathematical Physics Seminar,  Departamento de Matemática Instituto Superior Técnico Lisboa, Portugal, July 17, 2003
  36. A talk “Affine linking and winding numbers and the study of front propagation.” on July 12 2003 at the “Topological Dynamics and Ergodic Theory” session of the “2003 Summer Conference on Topology and its Applications”, Howard University, held on July 9-12 2003
  37. A talk “Affine linking and winding numbers and the study of front propagation.” on June 13 2003  at the Lehigh University Geometry and Topology Conference, held on June 12-14, 2003
  38.  A talk "Vassiliev invariants of Legendrian, Pseudo-Legendrian and framed knots in contact 3-manifolds." on June 6 2003 at the “NSF-NSA Conference on Interactions between Representation Theories, Knot Theory, Topology, Quantum Field Theory, Category Theory and Mathematical Physics”, held at SUNY Potsdam on June 2 - June 6, 2003
  39. A talk "Affine linking numbers and their applications to the study of wave front propagation and causality." on June 4 2003 at the “NSF-NSA conference on Interactions between Representation Theories, Knot Theory, Topology, Quantum Field Theory, Category Theory and Mathematical Physics”, held at SUNY Potsdam, NY, on June 2-6, 2003
  40. Colloquium talk "Affine Linking Numbers and the Causality Relation for Wave Fronts." at the Department of Mathematics, Boise State University, Boise, ID, May 13, 2003
  41. A talk "Vassiliev-Goussarov invariants of Legendrian pseudo-Legendrian and framed knots." at the Topology Seminar, Department of Mathematics, Boise State University, May 12, 2003
  42. A talk "Affine linking numbers and causality relations for wave fronts." at the Topology Seminar, Mathematics Department, Yale University, March 27, 2003
  43. Two talks “Towards the general theory of affine linking numbers part 1.” and “Towards the general theory of affine linking numbers part 2: Relations with winding number and front propagation.” On February 20 and March 13, 2003
  44. A talk “Vassiliev invariants of Legendrian, Pseudo-Legendrian and framed knots in contact 3-manifolds.” at the Topology Seminar, Mathematics Department, Brandeis University, February 12, 2003
  45. A talk "Affine linking numbers and their applications to the causality relation.” at the Geometry and Topology, Department of Mathematics, Dartmouth College, November 21, 2003
  46. A talk "Universal order one Vassiliev invariants of framed knots in the total spaces of S1-bundles over surfaces." at the Geometry and Topology seminar, Department of Mathematics, Dartmouth College, November 14, 2003
  47.  A talk “The universal order one invariant of framed knots in the total spaces of S1 bundles over orientable surfaces F≠ S2, S1 × S1.” on November 10 at the special session on “Invariants of knots and low-dimensional manifolds” of the AMS Southeastrn Section meeting # 982 at Orlando Florida, held on November 9-10, 2002
  48.  An invited talk “Affine linking Numbers and Causality relations for wave fronts.” on November 10 2002 at the special session on “Homotopy Theory and Geometric Topology” at the AMS Southeastrn Section meeting meeting #982 at Orlando Florida, held  on November 9-10, 2002
  49. A talk "Universal order one Vassiliev invariants of framed knots in the total spaces of S1-bundles over surfaces." at the “Rokhlin Topology Seminar”, Steklov Institute of Mathematics at St. Petersburg, Russia, September 9, 2002
  50. A talk "The causality relation and affine linking numbers." at the “Rokhlin Topology Seminar”, Steklov Institute of Mathematics at St. Petersburg, Russia, July 8, 2002
  51. Two talks “Affine linking number and the Causality Relation invariant, parts 1 and 2.” at the Geometry and Topology seminar, Department of  Mathematics, Dartmouth College on May 14 and 21, 2002
  52. A talk “Framed knots in 3-manifolds and the natural framings of transverse knots.” on March 8 2002 at the “Workshop in Geometric Topology and Geometric Group Theory”, University of Florida, Gainesville held on March 3-10, 2002
  53. A talk “Vassiliev invariants of Framed, Pseudo-Legendrian and Legendrian knots in contact 3-manifolds.”, on March 4 2002 at the “Workshop in Geometric Topology and Geometric Group Theory”, University of Florida, Gainesville, held on March 3-10, 2002
  54. A talk “Vassiliev invariants of Framed, Pseudo-Legendrian and Legendrian knots in contact 3-manifolds.” at the Geometry-Topology Seminar, Department of Mathematical Sciences, Binghamton University, Binghampton, NY, February 14 2002
  55. A talk "Legendrian and Pseudo-Legendrian knots and Vassiliev invariants for them." at the Rokhlin Topology Seminar, St Petersburg Department of Steklov Institute of Mathematics, St Petersburg, Russia, December 17 2001
  56.  Two talks "Framed knots in 3-manifolds and some new invariants of transverse knots, parts 1 and 2.”at the Geometry and Topology Seminar, Department of Mathematics, Dartmouth College,  October 4 and 11, 2001
  57. Two talks “Finite order invariants of Legendrian, Transverse and framed knots in contact 3-manifolds, parts I and II.” at the “Rokhlin Topology Seminar”, Steklov Institute of Mathematics at St. Petersburg, Russia, September 3 and 10, 2001
  58. .A talk at the Mathematical Physics seminar at the ETH Zurich, Switzerland 2001
  59.  Invited speaker at the Contact Geometry Workshop at Stanford University, December 13-17 2000
  60.  Talk at the Oberseminar in Complex Geometry, University of Zurich, Zurich, Switzerland, 2000
  61. Talk at the Topology Seminar at the Max-Planck-Institut fur Mathematik, Bonn,  Germany, 2000
  62.  Colloquium Talk, at the Mathematics Department, University of Siegen, Siegen, Germany, 2000
  63.  Talk at the Geometry Seminar at the Institute of Mathematics of the University of Basel, Switzerland, 2000
  64.  Talk at the Geometry seminar at the Ruhr-Universitat, Bochum, Germany, 2000
  65.  Talk at the Geometry seminar at the Albret-Ludwigs-Universitat, Freiburg, Germany, 2000
  66.  Talk at the Geometry Seminar at the Bonn University, Bonn, Germany, 2000
  67.  Talk at the Geometry and Topology Seminar, University Paul Sabatier, Toulouse, France,2000
  68.  Talk at the Topology seminar, Strasbourg University, Strasbourg, France, 2000
  69.  Talk at the Oberseminar, at the Max-Planck-Institut fur Mathematik, Bonn,  Germany, 1999
  70.  Invited speaker at the "Geometric and symplectic topology" program, Pisa University, Pisa, Italy, 1999
  71.  Talk at the ``Sur les singularites'' seminar at the l'Ecole Normale Superieure in Paris, France, 1999
  72.  Speaker at the “VIII OPORTO Meeting in Geometry Topology and Physics”, Porto, Portugal, 1999
  73.  Two talks at the Symplectic Geometry Seminar at the ETH Zurich, Switzerland, 1998-1999
  74.  Talk at the Geometry Seminar at the Institute  of Mathematics of the University of Basel, Switzerland, 1999
  75.  Talk at the V.I. Arnold's seminar on Singularity Theory, Moscow State University, Russia,1999
  76.  Talk at the Colloquium at the Deptartment of Mathematics at the University of California Riverside, CA, 1997
  77.  Talk at the AMS meeting #920 at the University of Maryland, College Park, MD, 1997
  78.   Principal speaker at the MSRI workshop ``Combinatorial Problems Arising in Knots and 3-manifolds'', Berkeley, CA, 1997.

Service and Synergistic Activities

  1. The Faculty Advisor to the “Dartmouth Undergraduate Mathematics Society” DUMS 2004-2006. In 2002-2004, before DUMS was officially registered with the Councilon Student Organizations, I conducted regular unofficial meetings, where invited faculty, graduate and undergraduate speakers were giving Mathematics Talk for Dartmouth Undergraduates students.
  2. Dartmouth college Putnam Mathematics Exam coach in 2001, 2002, 2003, 2004, 2005, 2006; and Mathematics fun problem solving session organizer in 2002, 2003, 2004, 2005, 2006. (During the 2001-2005 period Dartmouth Putnam Exam team greatly improved its performance.
  3. A Chair of the Thayer Prize Mathematical Competition committee in 2004 and in 2006. A member of the committee  in 2002 and 2003.
  4. Mathematics Department Colloquium Director in the Fall of 2003 and in the Spring of 2002.
  5. Member of the Graduate Admissions Committee in the 2002—2003, 2004-2005 academic years, and in Winter-Spring of 2007.
  6. Member of the Teaching Evaluation Committee in the 2002-2003 academic year.
  7. Member of the Undergraduate Program Committee in the 2001-2002 academic year.
  8. Member of the ad-hoc committee for the Math 17 “An Introduction to Mathematics Beyond Calculus” curriculum development.
  9. Course supervisor for the Math 104 class “Topics in Topology” in Spring 2004.
  10. Reviewer for Mathematical Reviews and Zentralblatt.
  11. External referee for the Pacific Mathematics Journal and Springer.

PC Experience

  • Administrating WeBWorK  program that allows students to do the homework online.
  • Teaching practice sessions for the Matlab based course in Applied Matrix Algebra at the UC Riverside.
  • Software development for the Biology Dept. at the St. Lawrence University, 1993.
  • I completed a Maple based research project in Topological Quantum Field Theory in 1994.
  • Knowledge of C, Turbo Pascal, Fortran, UNIX, Windows 2000, XP, NT, MS DOS.