Jean-Pierre Bourguignon

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.

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.

Simon Brendle

Minimal Tori in S^{3} 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 S^{3} of genus 0 is totally geodesic, hence congruent to the equator. In 1970, Blaine Lawson discovered a large class of embedded minimal surfaces in S^{3}, 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 S^{3} of genus 1.

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.

Robert Bryant

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]*.

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.

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.

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

*sup*. Here

_{Br(x)}|Rm|≤ r^{-2}*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.

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.

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.

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.

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 c_{1}(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.

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.

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.

Mark Haskins

Recent progress in *G _{2}* 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

*G*; in particular they exist on any Riemannian manifold with holonomy group contained in

_{2}*G*. Finding compact associative 3-folds in compact manifolds with

_{2}*G*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

_{2}*G*manifolds that contained rigid associative 3-folds and some of the other advances we made in the process.

_{2}
Nigel Hitchin

The Dirac operator for Higgs bundles.

Spiro Karigiannis

A survey of results about G_{2} conifolds

The exceptional properties of the octonion algebra allow us to define the notion of a G_{2} 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 G_{2} 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 G_{2} *conifolds*, which have a finite number of isolated "cone-like" singularities.

After some background on G_{2} manifolds and their moduli, we will present (an admittedly biased) survey
of some results on G_{2} conifolds, and the closely related asymptotically conical G_{2} manifolds,
including:

- a theorem (K., 2009) on desingularization of G
_{2}conifolds by glueing, and its implications about the G_{2}moduli space - a new result (K.-Lotay, 2012) on the deformation theory of G
_{2}conifolds, including several important applications - a brief discussion about a potential method of constructing G
_{2}conifolds, generalizing an existing construction of smooth compact G_{2}manifolds (K.-Joyce, 2013)

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.

Conan Leung

Instantons in G_{2} geometry

M-theory on G_{2} 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 G_{2}
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.

Aaron Naber

Classification of Tangent Cones and Lower Ricci Curvature

We consider limit spaces (*M _{i},g_{i},p_{i}) → (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.

Louis Nirenberg

Singular Solutions of Nonlinear Elliptic And Parabolic Equations

Various forms of of strong maximum principle are presented with application to symmetry and monotonicity, and viscosity solutions. A number of small results are presented.

Slides

Sema Salur

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
G_{2} 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.

In this talk, I will first give brief introductions to G_{2} manifolds, and then discuss relations between G_{2} and
contact structures.

If time permits, I will also show that techniques from symplectic geometry can be adapted to the G_{2} setting.
These are joint projects with Hyunjoo Cho, Firat Arikan and Albert Todd.

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.

I. M. Singer

Beyond the string genus

Gang Tian

Conic Kähler-Einstein metrics

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.

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,b _{2}-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(b*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

_{2}-3,3)/SO(2)× SO(b_{2}-3,1)*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).

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).

John Wermer

Function Algebras and Boundaries of Complex Varieties

Let M be a smooth, compact, oriented manifold in C^{n}, 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.