1:30pm Aaron Naber
Classification of Tangent Cones and Lower Ricci Curvature
We consider limit spaces (Mi,gi,pi) → (X,d,p),
where the spaces $M_i$ are noncollapsed and have Ricci curvature uniformly
boundedfrom below. In this case we study the set TC(p) of metric spaces
which consists of the possible tangent cones at p, and give a
classification result which says exactly which subsets of all metric
spaces can arise as TC(p) for some such limit. We use this to build
new examples of limit spaces with particularly degenerate behaviors.
In particular we show limit spaces cannot be stratified based on their
tangent cones, and that there exists a limit space for which there are
even nonhomeomorphic tangent cones at a point. This is joint work with
Toby Colding.
3:30pm Simon Brendle
In this talk, I will describe a proof of this conjecture. The proof involves an application of the maximum principle to a function that depends on a pair of points on the surface.
Minimal Tori in S3 and Lawson's Conjecture
The study of minimal surfaces is one of the oldest pursuits in differential geometry. Of particular interest is the case when the ambient manifold has constant curvature. For example, in 1966, Almgren showed that any immersed minimal surface in S3 of genus 0 is totally geodesic, hence congruent to the equator. In 1970, Blaine Lawson discovered a large class of embedded minimal surfaces in S3, which have genus greater than 1; he also constructed examples of minimal tori, which are immersed but fail to be embedded. Motivated by these results, Lawson conjectured that the Clifford torus is the only embedded minimal surface in S3 of genus 1.
9:30am Phillip Griffiths
Automorphic cohomology and cycle spaces
Automorphic cohomology arose from Hodge theory in the late
1960's. Fairly soon thereafter, many of its representation-theoretic
properties were understood. However, the geometric and arithmetic aspects
of automorphic cohomology remained largely mysterious. Due in significant
part to the work of Carayol, this has begun to change. In this talk we
will explain some of these developments which show that automorphic
cohomology exhibits a rich geometric structure in which cycles on flag
domains play an important role.
11:00am John Wermer
Function Algebras and Boundaries of Complex Varieties
Let M be a smooth, compact, oriented manifold in Cn, dim M = 2p-1.
Question 1: Under what conditions on M does there exist a complex analytic
variety V such that M is the boundary of V? Suppose such a V exists.
Define A to be the algebra of smooth functions on M such that f admits
a holomorphic extension to V, and let Ā be the uniform closure
of A on M. Then Ā is a closed subalgebra of C(M).
Question 2: Describe the elements of Ā.
In the 1950's the case
p = 1 (M is a closed curve) was thoroughly investigated. The answer for Question 1 was given by Reese Harvey and Blaine Lawson in their
fundamental paper "On boundaries of complex analytic varieties,I",
Ann. of Math. 102 (1975). In my talk, I shall discuss these matters.
2:00pm Vincent Guedj
Convergence of the normalized Kähler-Ricci flow on Fano varieties
Let X be a Fano manifold whose Mabuchi functional is proper. A deep result
of Perelman-Tian-Zhu asserts that the normalized Kähler-Ricci flow,
starting from an arbitrary Kähler form in c1(X), smoothly converges towards the unique Kähler-Einstein metric.
We will explain an alternative proof of a weaker convergence result which
applies to the broader context of (log)-Fano varieties.
This is joint work with Berman, Boucksom, Eyssidieux and Zeriahi.
3:30pm Eric Friedlander
Intersection on Singular Varieties
Chow's Moving Lemma justifies the intersection product on rational
equivalences classes of algebraic cycles on smooth varieties.
This talk will discuss work in progress with Joe Ross to define
an intersection product on algebraic cycles on singular (complex)
varieties. Our techniques include the "moving lemma for families"
of algebraic cycles which Blaine and I proved many years ago.
9:30am Cumrun Vafa
Feynman Graphs and Calabi-Yau Threefolds
I discuss how singularities of toric Calabi-Yau threefolds
relate to 5 dimensional superconformal theories. Each such singularity is captured by a Feynman-like diagram with cubic
vertices. The evaluation of the diagram includes integration over
the complexified Kähler moduli of Calabi-Yau and leads to the computation
of the index of the resulting 5d superconformal theory.
11:00am I. M. Singer
Beyond the string genus
2:00pm Jean-Pierre Bourguignon
The purpose of the lecture is to review recent results
in the context of using spinors and Dirac operators as
tools in Riemannian geometry, emphasizing the dependance
of these objects on the metric.
Results related to harmonic and more general special spinors
will be in particular highlighted.
Recent Results on Spinors and Dirac Operators
Spinors and Dirac Operators play for almost a century
a central role in Physics. It took more time for them to
play a similar role in Mathematics.
3:30pm Jeff Cheeger
Volume Estimates on sets of points at which the regularity scale is small
We discuss joint work with Aaron Naber in which Hausdorff
dimension estimates on singular sets for certain elliptic and parabolic
equations are improved to volume estimates on the set of points
at which the "regularity scale" is small. For instance, for Einstein manifolds,
the regularity scale at x is the "curvature radius",
r|Rm|(x) i.e. the supremum of those r ≤ 1, such that
supBr(x) |Rm|≤ r-2. Here Rm denotes the curvature tensor.
After briefly indicating the scope of the applications to date, we
illustrate the method by giving additional details in the Einstein case (which is
typical). The key point is an effective replacement for the iterated blow
up arguments used in proving the earlier Hausorff dimension estimates.
This replacement enables one to work instead on a single scale.
9:30am Reese Harvey
Perspectives on Elliptic PDE's
This will not be a survey of the joint work with Blaine Lawson on nonlinear PDE's listed here, but rather a selection of topics primarily of an elementary nature.
11:00am Conan Leung
Instantons in G2 geometry
M-theory on G2 manifolds is an analog of string theory on
symplectic manifolds. The role of holomorphic curves with Lagrangian
boundary conditions is replaced by associative submanifolds with
coassociative boundary conditions. The work of Fukaya-Oh related
holomorphic disks in cotangent bundles with Morse flow lines in
Lagrangian submanifolds. Wang, Zhu and I generalized this to the G2
setting, namely thin associative submanifolds can be constructed from
regular holomorphic curves in coassiciative submanifolds. This can be
used to construct new examples of associative submanifolds.
2:00pm Mark Haskins
Recent progress in G2 geometry
In their foundational paper Calibrated Geometries, Reese and Blaine discovered two very rich calibrated geometries in 7-dimensional Euclidean space: associative 3-folds and coassociative 4-folds. Both calibrations are intimately linked with the compact exceptional Lie group G2; in particular they exist on any Riemannian manifold with holonomy group contained in G2. Finding compact associative 3-folds in compact manifolds with G2 holonomy has been a particular challenge, in part because their deformation theory is less well-behaved than coassociative 4-folds. We describe recent work joint with Corti, Nordstrom and Pacini in which we construct a plentiful supply of compact G2 manifolds that contained rigid associative 3-folds and some of the other advances we made in the process.
3:30pm Gang Tian
Conic Kähler-Einstein metrics
9:30am Claire Voisin
Unramified cohomology and integral Hodge classes
Unramified cohomology of a complex algebraic variety produces
important birational invariants coming from the comparison between
the Zariski and Euclidean topologies and the associated
Leray spectral sequence, which is the Bloch-Ogus spectral sequence.
The talk will be an introduction to this subject, and we will show
eventually how the non-triviality of unramified cohomology is
related to the defect of the Hodge conjecture for integral Hodge
classes, in adequate degrees (joint work with J.-L. Colliot-Thelene).
11:00am Paulo Lima-Filho
Integral currents, equivariant cohomology and regulators for real varieties
We provide an explicit formula - in the level of complexes - for the regulator map
from the motivic cohomology of real varieties to the integral Deligne cohomology for real
varieties, introduced in joint work with dosSantos. The construction requires a formulation of
ordinary RO(G)-graded equivariant cohomology using complexes of real analytic currents, and some
properties of Milnor K-theory sheaves. Explicit examples are constructed for Voevodsky's
complexes that parallel Totaro's construction for Bloch's higher Chow groups, whose
non-triviality is detected using our regulator maps.
2:00pm Robert Bryant
These structures first arose in connection with generalized geometry and in the construction of certain σ-models in physics. They have been the subject of investigations by Apostolov and Gualtieri (arXiv:math/0605.342) and Apostolov, Calderbank, and Gauduchon (arXiv:math/1010.0992), who classified the examples with a 2-dimensional Lagrangian symmetry group and investigated their relationship with 4-dimensional Einstein orbifolds.
In this talk, after discussing the basic local structure theory, I will provide a complete local classification of these structures by interpreting the integrability conditions as an overdetermined system of PDE that can be treated by the methods of exterior differential systems. In particular, I will provide a complete list of local normal forms, except for a (somewhat mysterious) 6-dimensional family whose existence has been proved but for which the structure equations have (so far) resisted integration.
On the local classification of ambiKähler structures in dimension 4.
An ambiKähler structure on a 4-manifold M is a triple ([g],J+,J-) where [g] is a conformal structure on M
and J+ and J- are [g]-compatible complex structures that commute,
induce opposite orientations on M, and each are Kähler with respect to
some Riemannian metric in the conformal class [g].
3:30pm José Figueroa-O'Farrill
Supersymmetry of hyperbolic monopoles
Hyperbolic monopoles are solutions of the Bogomol'nyi equations on
three-dimensional hyperbolic space. These equations are a natural reduction of
the self-duality equations for Yang-Mills fields in four-dimensional euclidean
space. After some introductory remarks on supersymmetry for mathematicians, I
will present the construction of a supersymmetric Yang-Mills theory on hyperbolic
space, identify hyperbolic monopoles as supersymmetric configurations and will
show how supersymmetry determines the geometry of the moduli space of
hyperbolic monopoles.
9:30am Rick Schoen
Minimal surfaces as extremals of eigenvalue problems
For closed surfaces and for surfaces with boundary there are natural eigenvalue extremal problems whose solutions determine minimal surfaces in the sphere or the ball with a natural boundary condition. We will discuss the geometric properties of extremal metrics and the
difficult problem of existence and regularity. This is joint work with A. Fraser.
11:00am Robert Hardt
Some Homology and Cohomology Theories for a Metric Space
Various classes of chains and cochains may reveal geometric as well as topological properties of
metric spaces. In 1957, Whitney introduced a geometric "flat norm" on polyhedral chains in Euclidean space,
completed to get flat chains, and defined flat cochains as the dual space. Federer and Fleming also
considered these in the sixties and seventies, for homology and cohomology of Euclidean Lipschitz
neighborhood retracts. These include smooth manifolds and polyhedra, but not algebraic varieties or
subspaces of some Banach spaces. In works with Thierry De Pauw and Washek Pfeffer, we find generalizations
and alternate topologies for flat chains and cochains in general metric spaces. With these, we homologically
characterize Lipschitz path connectedness and obtain several facts about spaces that satisfy local linear
isoperimetric inequalities.
2:00pm Sema Salur
In this talk, I will first give brief introductions to G2 manifolds, and then discuss relations between G2 and
contact structures.
If time permits, I will also show that techniques from symplectic geometry can be adapted to the G2 setting.
These are joint projects with Hyunjoo Cho, Firat Arikan and Albert Todd.
Calibrations in Contact and Symplectic Geometry
In a celebrated paper published in 1982, F. Reese Harvey and Blaine Lawson introduced four types of calibrated
geometries. Special Lagrangian submanifolds of Calabi-Yau manifolds, associative and coassociative submanifolds of
G2 manifolds and Cayley submanifolds of Spin(7) manifolds. Calibrated geometries have been of growing interest
over the past few years and represent one of the most mysterious classes of minimal submanifolds.
3:30pm Alice Chang
On a class of non-local conformal invariants on asymptotic hyperbolic manifolds
We will discuss properties of a class of conformal invariants in conformal geometry and their connection to geometric quantities on asymptotically hyperbolic manifolds. Special emphasize will be on the extension theorem of Caffarelli-Silvestre and applications in this setting.
9:00am Spiro Karigiannis
A survey of results about G2 conifolds
The exceptional properties of the octonion algebra allow us to define the notion of a G2 structure on
an oriented spin 7-manifold, which is a certain "nondegenerate" 3-form that induces a Riemannian
metric in a nonlinear way. The manifold is called a G2 manifold if the 3-form is parallel. Such
manifolds are always Ricci-flat, and are of interest in physics. More recently, however, there has been
interest in G2 conifolds, which have a finite number of isolated "cone-like" singularities.
After some background on G2 manifolds and their moduli, we will present (an admittedly biased) survey
of some results on G2 conifolds, and the closely related asymptotically conical G2 manifolds,
including:
10:15am Nigel Hitchin
The Dirac operator for Higgs bundles
11:00am Misha Verbitsky
Global Torelli theorem for hyperkähler manifolds
A mapping class group of an
oriented manifold is a quotient of its diffeomorphism
group by the isotopies. We compute a mapping class group
of a hypekähler manifold M, showing that it is
commensurable to an arithmetic subgroup in SO(3,b2-3) A
Teichmuller space of M is a space of complex structures
on M up to isotopies. We define a birational Teichmuller
space by identifying certain points corresponding to
bimeromorphically equivalent manifolds, and show that the
period map gives an isomorphism of the birational
Teichmuller space and the corresponding period space
SO(b2-3,3)/SO(2)× SO(b2-3,1) We use this
result to obtain a Torelli theorem identifying any
connected component of birational moduli space with a
quotient of a period space by an arithmetic subgroup. When
M is a Hilbert scheme of n points on a K3 surface,
with n-1 a prime power, our Torelli theorem implies the
usual Hodge-theoretic birational Torelli theorem (for
other examples of hyperkähler manifolds the
Hodge-theoretic Torelli theorem is known to be false).