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Facade of the New Academic Building (author: Rafal Komendarczyk).

Nodal Sets and Contact Structures

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2 ROBERT GHRIST AND RAFAL KOMENDARCZYK Question 1.1

Three lectures by young topologists: Professors Eli Grigsby (Boston College), Matt Hedden (Michigan State University), and Rafal Komendarczyk (Tulane University). In addition to their mathematical content will have the effect of showing examples of success stories starting out as graduate students through obtaining a tenure track position.

Time: Thursday, 05/03, 3-4pm
Speaker: John Etnyre (Georgia Tech)
Title: Curvature and (contact) topology
Abstract: Contact geometry is a beautiful subject that has important interactions with topology in dimension three. In this talk I will give a brief introduction to contact geometry and discuss its interactions with Riemannian geometry. In particular I will discuss a contact geometry analog of the famous sphere theorem and more generally indicate how the curvature of a Riemannian metric can influence properties of a contact structure adapted to it. This is joint work with Rafal Komendarczyk and Patrick Massot.

ETNYRE, RAFAL KOMENDARCZYK, AND PATRICK MASSOT ABSTRACT

  • 8:00 a.m.
    Helicity and Energy Bounds For Vector Fields.
    Jason H Cantarella*, University of Georgia
    Jason Parsley, Wake Forest University
    (1067-57-2065)
  • 8:30 a.m.
    New perspectives on helicity.
    Jason Cantarella, University of Georgia
    Jason Parsley*, Wake Forest University
    (1067-57-2260)
  • 9:00 a.m.
    The Search for Higher Helicities.
    Clayton Shonkwiler*, Haverford College
    Dennis DeTurck, University of Pennsylvania
    Herman Gluck, University of Pennsylvania
    Rafal Komendarczyk, Tulane University
    Paul Melvin, Bryn Mawr College
    David Shea Vela-Vick, Columbia University
    (1067-57-1502)
  • 9:30 a.m.
    Optimally Immersed Planar Curves under Möbius Energy.
    Ryan P Dunning*, St. Mary's University
    (1067-49-937)
  • 10:00 a.m.
    Self-organization resulting from conservation of magnetic helicity, a distributed form of linkages; applications to lab and solar phenomena.
    Paul M Bellan*, Caltech
    (1067-78-1149)

Rafal Komendarczyk, Paul Melvin, Haggai Nuchi and Clayton Shonkwiler.

Rafal Komendarczyk
Associate Professor

Tulane University
6823 St. Charles Ave.
Gibson Hall
New Orleans, LA 70118

Abstract: Knots are tangled closed loops in 3-dimensional space which have been studied mathematically at least since the 19th century. I will first discuss the idea of knot invariants, which can be used to prove that a knot can’t be untangled. Links are like knots, but with multiple strings instead of just one. I will discuss the Gauss linking number, which is an invariant of 2-component links and is arguably simpler than any knot invariant. It detects linking of the two components, but it ignores knotting of either component. I will describe this invariant in elementary terms by counting crossings. I will also describe it as the degree of a map of surfaces. If time permits, I will further describe it in terms of homotopy classes of maps of configuration spaces. This latter description generalizes to an invariant of n-strand links which ignores knotting but conjecturally detects all linking phenomena. This is the subject of recent joint work with Fred Cohen, Rafal Komendarczyk, and Clay Shonkwiler, where we proved a certain analogue of this conjecture.