Abstracts for A conference to describe the published scientific
work of Jim Simons on the occasion of his 75th birthday
A conference to describe the published scientific
work of Jim Simons on the occasion of his 75th birthday
Abstracts
Tuesday May 28, 2013
Edward Witten
The Chern-Simons Functional In Condensed Matter Physics
Beginning with materials in ordinary three-dimensional space, I will explain in general how one understands the interaction of light with an insulating material. Then I will explain unusual phenomena that can occur in the case of a thin insulating film and which are linked theoretically to the existence of the Chern-Simons invariant of a connection on a U(1) bundle over a closed three-manifold M.
In general, a thin insulating film is characterized by an integer k, which one can view as the coefficient with which the Chern-Simons invariant appears in the appropriate effective action. For k nonzero, there are ``edge currents'' at the boundary of such a film; these are associated to the mathematical properties of the Chern-Simons invariant for the case of a three-manifold with boundary. To date, our experimental knowledge of materials with nonzero k is mainly associated to studies of the edge currents, though there are theoretical predictions of unusual behavior of light interacting with these materials.
Pierre Deligne
An analogue in complex algebraic geometry
A complex vector bundle V on a manifold M has Chern classes cp(v) in H2p(M,Z).
Story 1: A connection on V lifts this class to a Cheeger-Simons' differential character.
Story 2: If M is a projective complex algebraic variety, giving the structure of an analytic (equivalently: algebraic) vector bundle ensures that cp(v) is of type (p,p) and lifts cp(v) to a class in an extension of Hp,p(M,Z) by the "intermediate jacobian" related to H2p-1(M).
We will explain how the two stories are parallel.
Blaine Lawson
Jim Simons' Work on the Theory of Minimal Varieties
In 1967 Jim Simons wrote a remarkable paper in which he established
the interior regularity, up to dimension 7, of Federer-Fleming solutions to
the Plateau problem in codimension-one. He also extended the Bernstein
Theorem for minimal graphs up to dimension 8. He then produced an example
which was subsequently used to show that both results were sharp.
At the same time Simons laid the foundations for the (ultimately very
effective) use of minimal submanifolds as a tool in riemannian geometry.
I will discuss this work, its background and its impact in geometry and
analysis.
Structure Theory (Berger, Chern, Alexseevski, and Brown-Gray)
First Global Results (Calabi, Calabi-Yau)
The first lecture will be an historical and introductory overview of the concept of holonomy and its ubiquity in geometry.
Beginning with Schouten and Cartan, early in the 20th century, the importance of understanding the global effects of parallel transport in Riemannian geometry was recognized and studied. The connection with non-holonomic mechanics led E. Cartan to describe the dependence of parallel translation on the path of translation in terms of the so-called holonomy group. Later work of de Rham, Borel-Lichnerowicz, and Ambrose-Singer clarified the basic concepts and the relationship between curvature and holonomy setting the stage for M. Bergers famous 1954 classification of the possible Riemannian holonomy groups.
In 1962, J. Simons gave an important conceptual explanation for this classification, which laid the groundwork for future advances in the subject. (The search for more conceptual proofs of this fundamental result has continued, with C. Olmos contributing new insights and simplifications in 2004.)
The importance of metrics with reduced holonomy has grown over the years, starting with the recognition that Kähler geometry is one particular case of reduced holonomy. Many of the leading researchers in geometry contributed to its general development, including M. Berger and S.-s. Chern among others. The challenge of understanding global and compact examples, both for theoretical and applied reasons, has garnered much attention and driven the application of nonlinear PDE to geometry. One of the more spectacular cases was the resolution by S.-T. Yau of the Calabi Conjecture about metrics with holonomy SU(n). The cases n=2 and n=3 are particularly interesting because of their applications in algebraic geometry and string theory.
Wednesday May 29, 2013
Robbert Dijkgraaf
The Many Incarnations of Chern-Simons Theory
Chern-Simons gauge theory was the first topological field theory and it has guided the development of this class of quantum field theories ever since. It has been a "unreasonable" rich theme leading to many variations, both in physics and mathematics. I will review the various generalizations and applications,in particular in string theory.
Mike Hopkins
Generalized differential homology, I
As the use of Chern-Simons invariants in string theory became more elaborate, a demand for a more sophisticated apparatus involving the Cheeger-Simons theory of differential characters developed. In this lecture I will describe some of these issues and how they eventually led to the construction of "generalized" Cheeger-Simons cohomology theories.
Brian White
Stable Minimal Surfaces
In 1968, Jim Simons published pioneering work
on stable minimal varieties, including the
theorem that any stable minimal hypercone
of dimension less than 7 is a plane.
He also gave an example (now known as the
"Simons cone") of a 7 dimensional nonplanar
stable minimal hypercone.
In this talk I will discuss some of the
subsequent developments
about stable minimal surfaces.
Robert Bryant
Special Holonomy (Basic Theory)
HyperKähler (Fujiki, Mukai, Beauville, HKLR)
Quaternion Kähler (Berger, Wolf, Alexseevski, and later)
G2 (Bonan, Bryant, Bryant-Salamon, Hitchin)
Spin(7) (Bonan, Bryant, Bryant-Salamon, Hitchin)
Appearance of holonomy in theoretical physics
The second lecture will consider the various special cases of reduced holonomy in turn. (All of these terms will be defined and explained fully in the lecture.)
HyperKähler geometry (once thought not to exist for global reasons) has turned out to be fundamental in several different areas. From the constructions using algebraic geometry and the Calabi-Yau theorem by Fujiki, Mukai, and Beauville to the HKLR construction generalizing symplectic reduction, these special metrics make several appearances as moduli spaces of solutions of important nonlinear equations in geometry and integrable systems.
Less is known about the global Quaternionic Kähler case, and one of the enduring mysteries is whether non-symmetric examples even exist in the case of positive scalar curvature.
The two 'exceptional cases' of G2 (for 7-manifolds) and Spin(7) (for 8-manifolds) resisted understanding for a long time, with no examples being known, even local ones. Their local existence and generality were determined by Bryant in 1984, and complete examples were constructed by Bryant and Salamon in 1987. The geometry and topology of such spaces is rather constrained, and the sorts of results that one can derive from standard methods will be explained and set in context.
The fact that these 'exceptional' cases of reduced holonomy corresponds to the existence of parallel spinor fields makes a direct link with supersymmetric theories in high-energy physics and accounts for the interest in these manifolds in theoretical physics. Unfortunately, compact examples were lacking (and were not even known to exist) until the work of Joyce, which will be explored in the following lectures by Mark Haskins.
Thursday May 30, 2013
Robbert Dijkgraaf
The Many Incarnations of Chern-Simons Theory
Chern-Simons gauge theory was the first topological field theory and it has guided the development of this class of quantum field theories ever since. It has been a "unreasonable" rich theme leading to many variations, both in physics and mathematics. I will review the various generalizations and applications,in particular in string theory.
Blaine Lawson
Analytic and Homological Aspects of Differential Cohomology
I will discuss how differential characters naturally arise in analysis as
mediators between smooth and "integral" objects. I will then show how,
with a natural topology, differential cohomology obeys Poincaré-Pontrjagin
duality. Finally I will present a homological apparatus which establishes the
equivalence of different structures which yield differential cohomology without
establishing the full axioms (in particular, without addressing the multiplicative structure).
Brian White
Singularities of minimal varieties
Jim Simons' 1968 theorem about stable
minimal hypercones implied, by earlier results,
that area minimizing hypersurfaces of dimension
less than seven must be smooth. Shortly afterwards,
Bombieri, De Giorgi, and Giusti showed that there are
singular examples in higher dimensions.
(Specifically, they proved that the Simons cone
is area minimizing.)
In this talk, I will discuss some of the subsequent developments
and outstanding open questions about singularities of minimal
varieties.
Mark Haskins
Compact manifolds with G2 holonomy and noncompact Calabi-Yau manifolds
This pair of lectures picks up from where Robert Bryant's lectures finish. In the mid 90s Dominic Joyce revolutionised the field of exceptional holonomy by proving the existence of compact 7-manifolds with holonomy G2 and compact 8-manifolds with holonomy Spin(7) using geometric PDE 'gluing' methods. I will describe Joyce's approach and more recent developments in the field, concentrating on results for compact manifolds with holonomy G2. I will explain how all the currently known constructions of compact G2 manifolds rely heavily on the powerful existence results now known for various types of complete noncompact Calabi-Yau manifolds by solving complex Monge-Ampere equations. I will discuss some of the recent advances in our understanding of the topology of certain compact G2 manifolds and how to find interesting calibrated submanifolds in compact G2 manifolds. Finally I will discuss some of the many current open problems in the field.
Friday May 31, 2013
Isadore Singer
Chern-Simons Perturbation Theory
We show this theory is finite using Chern's derivation
of the Chern-Simons invariant by a choice of framing of the three manifold.
He proves that a different choice of framing changes the invariant by an integer.
[S.S. Chern "Complex Manifolds without Potential Theory", section 7,
Springer-Verlag, 2nd Addition]
A more extensive exposition of the Chern-Simons invariant and the action of the
Gauge Group can be found in "Lecture notes on Chern-Simons-Witten Theory"
by Sen Hu, World Scientific.
Mike Hopkins
Generalized differential homology, II
Using generalized Cheeger-Simons chomology I will explain Witten's realization of the Kervaire invariant of a manifold of dimension six, and the generalization to all dimensions of the form (4k+2). I will also describe some open related problems.
Aaron Naber
Quantitative Stratification and Estimates for Minimizing Hypersurfaces
In this joint work with Jeff Cheeger we discuss a new technique for taking noneffective local behavior, e.g. the tangent cone behavior classified by Simons, and deriving from this effective estimates on regularity. This involves two key steps: the quantitative stratification, which generalizes the standard notion of a stratification and gives proofs for the Hausdorff estimates on singular sets, combined with new epsilon regularity theorems. For minimizing hypersurfaces we prove Lp estimates for p<7 for the second fundamental form, and we prove estimates for the regularity scale. These results are more powerful and they are sharp.
Mark Haskins
Compact manifolds with G2 holonomy and noncompact Calabi-Yau manifolds
This pair of lectures picks up from where Robert Bryant's lectures finish. In the mid 90s Dominic Joyce revolutionised the field of exceptional holonomy by proving the existence of compact 7-manifolds with holonomy G2 and compact 8-manifolds with holonomy Spin(7) using geometric PDE 'gluing' methods. I will describe Joyce's approach and more recent developments in the field, concentrating on results for compact manifolds with holonomy G2. I will explain how all the currently known constructions of compact G2 manifolds rely heavily on the powerful existence results now known for various types of complete noncompact Calabi-Yau manifolds by solving complex Monge-Ampere equations. I will discuss some of the recent advances in our understanding of the topology of certain compact G2 manifolds and how to find interesting calibrated submanifolds in compact G2 manifolds. Finally I will discuss some of the many current open problems in the field.