The speakers are listed below. Click on the name of a speaker to find their title and abstract.
The schedule of talks, together with locations is below the list of abstracts.
Each plenary speaker will give a series of four one-hour lectures, introducing a research topic in geometric analysis. The plenary speakers are:
Along side the lecture series, we will also have a collection of research talks on a variety of subjects in geometric analysis. These talks will be given by:
- Siqi He (Chinese Academy of Sciences)
- Mirela Kohr (Babeş-Bolyai University)
- Greg Parker (Stanford)
- Marco Usula (Université libre de Bruxelles)
- Max Zimet (Stanford)
Iterated edges and applications
My goal is to review some methods of geometric microlocal analysis with special attention to degenerate operators with edge and iterated edge singularities. These arise in many geometric problems, for example in gauge theory, index theory, variational problems, geometric flows of noncompact and singular spaces, etc. I will discuss specific linear and nonlinear problems where various aspects of this theory are illuminating and useful.
- Examples of edges in geometry and analysis, including incomplete conic and edge singularities, elliptic boundary problems, and asymptotically hyperbolic manifolds. Deficiencies of classical theory in this setting, and a review of the new analytic phenomena which appear in elliptic and parabolic edge problems.
- The edge calculus and parametrix constructions for elliptic and parabolic problems with edges. Sharp regularity theory, exhibited as generalizations of standard classical results, but with some new twists.
- Some examples: conic surfaces with constant Gauss curvature; geometric flows of conic and edge spaces; coalescing singularities (and perhaps more).
- Iterated edges and stratified spaces, with attention to two particular problems: knot singularities in gauge theory and multi-valued harmonic spinors and 1-forms which branch along a graph.
Analysis on manifolds with nice ends, pseudodifferential operators, and applications
The main goal is to extend to non-compact manifolds and singular spaces some
of the basic results on Euclidean spaces and on closed manifolds. As a main example, we will examine the Fredholm property of differential (and pseudodifferential) operators on certain non-compact manifolds with nice ends. I will assume some knowledge of manifold theory (tangent and cotangent space, metric, connection, differential forms and integration, vector bundles) and some knowledge of basic analysis (L^p and Sobolev spaces, Banach and Hilbert spaces, continuous/bounded operators on normed spaces). Here is a tentative plan of the lectures:
- Pseudodifferential operators on manifolds
- Differential and regularizing operators, Schwartz’s kernel theorem
- Hormander and classical symbol classes and the local form of pseudodifferential operators
- Pseudodifferential operators on manifolds and their main properties
- Manifolds with cylindrical ends and adapted pseudodifferential
operators (the “inv-calculus” and its relation to the “b-calculus of
Melrose; properties in preparation of Kohr’s talk)
- Manifolds with nice ends (Melrose’s program and Lie algebroids)
- Review of results in the compact case and b. Some “paradoxes” on non-smooth domains and on non-compact manifolds. (Issues: loss of regularity, loss of Fredholmness
- Examples of manifolds with nice ends: manifolds with cylindrical ends,
manifolds with fibered boundaries (also edges; Mazzeo, Grushin, … ), …
- Lie algebroids
- Lie manifolds
- Main geometric properties of Lie manifolds.
- Differential operators and Sobolev spaces.
- Background on functional analysis:
- Fredholm operators and Atkinson’s theorem
- Concrete and abstract C^*-algebras
- *-representations and invertibility (morphisms of C^*-algebras are spectrum preserving, the injective ones are also isometric).
- Pseudodifferential operators on Lie manifolds and characterization of Fredholm operators on “nice” Lie manifolds
- Lie groupoids (definition and examples)
- Pseudodifferential operators on groupoids
- Groupoid C^*-algebras and their regular representaions
- Applications to Fredholm conditions
Approximating the area functional in codimension two: Yang-Mills-Higgs energy on U(1) bundles
onstructing minimal submanifolds in arbitrary Riemannian ambients has always been one of the most popular and important problems in the calculus of variations and geometric analysis, giving birth in the 60s to the vast field of geometric measure theory. For low dimensional submanifolds (curves and surfaces), a fruitful approach is to parametrize the submanifold, viewing it as the image of a map, typically relaxing area with the Dirichlet energy. In a dual way, in low codimension (one or two), it is useful to view the submanifold as the zero set of a map, and to seek an energy which resembles the area of the level set from a variational standpoint. In codimension one, this was successfully achieved using the Allen-Cahn energy, which models phase transitions between two pure phases. In codimension two, together with Daniel Stern, we showed that a successful candidate is the Yang-Mills-Higgs energy on U(1) bundles, coming from the Ginzburg-Landau model of superconductivity; additional evidence for this was obtained in later works with Davide Parise and Daniel Stern. The aim of this minicourse is to survey this “level set approach” and some of the functionals which have been proposed so far, with particular emphasis on Yang-Mills-Higgs and the relevant techniques.
Tentative outline of the lectures:
- Survey of known ways to approximate area for variational purposes. Brief review of basic geometric measure theory (Hausdorff measures, rectifiable sets, currents, varifolds).
- Allen-Cahn energy. Sharp monotonicity and the role of Modica’s bound on the “discrepancy” function.
- Ginzburg-Landau energy with no induced magnetic field (in the Bethuel-Brezis-Hélein simplified version). Good and bad features, positive results and counterexamples. Ambrosio-Soner rectifiability result for generalized varifolds.
- Yang-Mills-Higgs energy, i.e., the previous one with induced magnetic field, specialized to the self-dual regime. Analogue of Modica’s inequality. Quantization and integrality of the limit varifold. Further results: Gamma-convergence to area and gradient flow analysis.
(Of course, this plan of time allocation will not be respected in a strict way; in particular, the contents of lecture 4 will actually occupy more than one hour.)
Exploring Z2 Harmonic 1-forms on Projective Varieties
Z2 harmonic 1-forms, a concept introduced by Taubes, play a significant role in gauge theory and differential geometry. In this talk, we’ll dive into the world of Z2 harmonic 1-forms over projective varieties. We’ll explore how Z2 harmonic 1-forms are connected to Higgs bundles and a recent proposal by Chen-Ngo about the surjectivity of the Hitchin morphism. We’ll also look at several applications, including how Z2 harmonic 1-forms help us construct the Hitchin section and establishing the generalized Milnor-Wood inequality. Furthermore, we will explain how Z2 harmonic 1-forms could play a role in the Simpson integral conjecture. We will discuss a new proof using Finsler metric rigidity to prove characteristic rigidity and integrality for arithematic varieties with rank bigger than one. This talk is based on collaborated work with J.Liu and N.Mok.
On some elliptic operators on manifolds with cylindrical ends
We study some elliptic operators (Laplace, Stokes) on a manifold with cylindrical ends. To this purpose, we obtain useful Fredholm, regularity, and invertibility results. An important role is played by an adapted pseudodifferential calculus on manifolds with straight cylindrical ends which contains the inverses of its $L^2$-invertible, elliptic operators of non-negative order. We also obtain the well-posedness of the corresponding Dirichlet problem. Joint work with Victor Nistor (Metz) and Wolfgang L. Wendland (Stuttgart).
Deformations of Z_2 Harmonic Spinors
Z_2-harmonic spinors are singular generalizations of classical harmonic spinors that allow topological twisting around a submanifold of codimension 2 called the singular set. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and are expected to contribute to new gauge-theoretic invariants via wall-crossing formulas. In this talk, I will focus on a recent result showing that the universal moduli space of Z_2-harmonic spinors at sufficiently generic points is locally a codimension 1 submanifold, i.e. a “wall” in the space of parameters. The key ingredients in the proof are differentiating the relevant Dirac operator with respect to deformations of the singular set, and an application of the Nash-Moser Implicit Function Theorem.
0-elliptic boundary value problems
Given a compact manifold with boundary, there is a natural ring of “0-differential operators”, locally given by compositions of vector fields vanishing along the boundary. Typical examples are the natural geometric operators (Laplacians, Dirac operators etc.) associated to asymptotically negatively curved metrics in the interior. These operators act on natural weighted (Sobolev or Holder) spaces, but because of their degeneracy, they are typically not Fredholm. For sufficiently negative weights, however, it is often possible to supplement the operators with appropriate boundary conditions to ensure Fredholmness. In this talk, we will discuss the formulation of these boundary value problems, focusing on the parametrix construction.
K3 Surfaces as Gauge-Theoretic Moduli Spaces
I will explain a novel construction (in progress) of K3 surfaces as moduli spaces of singular equivariant instantons on a 4-torus. This yields explicit formulae for K3 metrics near torus orbifold limits. I will also describe a novel Fredholm theory — and its development using interesting ideas from microlocal analysis — for Laplacians and Dirac operators constructed out of these very singular connections which should enable the construction. I will next introduce a variant of the Donaldson-Uhlenbeck-Yau theorem that operates in this setting, some of its consequences (such as non-emptiness of our moduli spaces), and the novel notion of stability for these singular connections that enters into the theorem. This theorem is proved by studying the gradient flow equation for the Yang-Mills functional, and I will describe the Fredholm theory for the heat operator that undergirds the proof of the short- and long-time existence and regularity of the flow. While these results are all proved for singular connections on a 4-torus, they are expected to generalize to enable the study of vast new classes of gauge-theoretic moduli spaces consisting of objects with severe singularities in codimension at least three. Based on joint work with Andras Vasy.
All the talks will be in the Salle Solvay, on the 5th floor of building NO except for Tuesday when the talks will be in Forum G. You can find these rooms via this map of the Campus de la Plaine.
The lunches are provided to all the registered participants. They are at the “Maison des anciens étudiants”, which is also on campus, southwest of the Forum, in the building marked 21 (which is visible if you zoom in on the map).