Talks (titles and abstracts)

Single talks:

Riccardo Caniato: The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varieties
I will present a joint work with Tristan Rivière in which we prove some new results about uniqueness of tangent maps for general pseudo-holomorphic and strongly approximable maps from an arbitrary almost complex manifold into projective algebraic varieties. As a byproduct of the approach and the techniques developed, in the same work we also show how to obtain the unique tangent cone property for a special class of non-rectifiable positive pseudo-holomorphic cycles.

Guido De Philippis: Non-degenerate minimal surfaces as energy concentration sets: a variational approach
I will show that every non-degenerate minimal sub-manifold of codimension 2 can be obtained as the energy concentration set of a family of critical points of the (rescaled) Ginzburg Landau functional. The proof is purely varia-tional, and follows the strategy laid by Jerrard and Sternberg. in 2009. The same proof applies also to the Yang-Mills-Higgs and to the Allen-Cahn-Hillard energies. This is a joint work with Alessandro Pigati.

Wenshuai Jiang: Nodal set of harmonic functions on manifold with Ricci curvature bounded below
In this talk, we will consider harmonic functions on noncollapsing manifold with lower Ricci curvature bound. By proving an eps-regularity theorem and a bilipschitz estimate for harmonic function, we can get uniform measure estimates of the nodal set. This is a joint work with Jianchun Chu and Huabin Ge.

Daniel Ketover: Higher index minimal surfaces and doublings
I will describe some constructions of minimal surfaces in Riemamnian three-manifolds that have controlled topological type and Morse index greater than 1. As a consequence, I’ll show how one can obtain doublings and triplings, etc, of certain minimal hypersurfaces using variational methods.

Yang Li: Thomas-Yau conjecture
The (wildly open) Thomas-Yau conjecture concerns the existence question of special Lagrangians vs stability conditions in symplectic geometry. I will discuss a variational program based on the minimisation of the Solomon functional. The focus is primarily on the analytic aspects. Time permitting, I hope to mention some open questions.

Jesse Madnick: Compactness and Bubbling of Conformally Calibrated Maps
Let M be a Riemannian n-manifold equipped with a calibration p-form. What is the best way to parametrize the calibrated submanifolds of M? In this talk, we discuss “conformally calibrating maps” (or “Smith maps”) from p-dimensional Riemannian manifolds into M. Smith proved that such maps are weakly conformal, have calibrated submanifolds as their images, and are minimizers of the p-energy functional in their homology class.
A fundamental example of conformally calibrating maps are J-holomorphic maps from Riemann surfaces into almost-Kahler manifolds, the moduli spaces of which play a fundamental role in symplectic geometry. In that direction, several non-trivial analytic properties of J-holomorphic maps have been established in order to compactify the relevant moduli spaces.
In this talk, we explain that many of these analytic properties hold more generally for conformally calibrated maps, at least when the calibration arises from a vector cross product (as happens in the almost-Kahler, G2, and Spin(7) settings). More precisely, we prove an epsilon-regularity theorem, a removable singularity result, an energy gap theorem, and a compactness modulo bubbling result. Further, the resulting “bubble trees” of such maps have no necks, and experience no p-energy loss in the limit. This is joint work with Da Rong Cheng and Spiro Karigiannis.

Andrea Marchese: Geometric and analytic properties of Radon measures through their decompositions along curves
The decomposability bundle, which I introduced with Alberti in [1], is a flexible tool to study the geometry of Radon measures. Roughly speaking, it is a map which captures at almost every point the tangential directions to the Lipschitz curves along which the measure can be decomposed.
I will discuss some recent applications: a characterization of rectifiable measures in terms of Lusin type approximation of Lipschitz functions by functions of class C^1, a classification of those differential operators which are closable, and a characterization of Federer-Fleming flat chains with finite mass as those vector valued measures for which the polar vector belongs to a variant of the decomposability bundle.
References: 1. Alberti, Marchese: On the differentiability of Lipschitz functions with respect to measures in the Euclidean space, GAFA, 2016

Alessandro Pigati: (Non-)quantization phenomena for higher-dimensional Ginzburg-Landau vortices
The Ginzburg-Landau energies for complex-valued maps, initially introduced to model superconductivity, were later proposed in the context of geometric variational problems to renormalize the Dirichlet energy of singular circle-valued maps and, more recently, to approximate the area functional in codimension two.
While the pioneering works of Lin-Rivière and Bethuel-Brezis-Orlandi (2001) showed that, for families of critical maps, energy does concentrate along a codimension-two minimal submanifold, it has been an open question whether this always happens with integer multiplicity. In joint work with Daniel Stern, we give a sharp characterization of the possible multiplicities, showing that they belong to the set {1} U [2,∞). In particular, in contrast to the Allen-Cahn and abelian Yang-Mills-Higgs energies, non-integer densities higher than 2 can arise in ambient dimension at least three.

Antoine Song: The spherical Plateau problem
For any closed oriented manifold with fundamental group G, or more generally any group homology class for a discrete group G, there is a corresponding infinite codimension Plateau problem in a Hilbert classifying space for G. I will discuss some uniqueness and existence results for Plateau solutions. For instance, for a closed oriented 3-manifold M, the intrinsic geometry of any Plateau solution is given by the hyperbolic part of M.

Gang Tian: Yang-Mills fields and minimal surfaces

Thomas Walpuski: The Gopakumar–Vafa finiteness conjecture
The purpose of this talk is to illustrate an application of the powerful machinery of geometric measure theory to a conjecture in Gromov–Witten theory arising from physics. Very roughly speaking, the Gromov–Witten invariants of a symplectic manifold (X,ω) equipped with a tamed almost complex structure J are obtained by counting pseudo-holomorphic maps from mildly singular Riemann surfaces into (X,J). It turns out that Gromov–Witten invariants are quite complicated (or “have a rich internal structure”). This is true especially for if (X,ω) is a symplectic Calabi–Yau 3–fold (that is: dim X = 6, c_1(X,ω) = 0).
In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.
The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that Gromov’s compactness theorem can (and should!) be replaced with the work of Federer–Flemming, Allard, and De Lellis–Spadaro–Spolaor. This upgrade of Ionel and Parker’s cluster formalism proves both the integrality and finiteness conjecture.
This talk is based on joint work with Eleny Ionel and Aleksander Doan.

Lu Wang: Relative expander entropy
In this talk, I will discuss a notion of relative entropy motivated by self-expanding solutions to mean curvature flow. I will also discuss some basic properties of the relative entropy and applications to the study of mean curvature flow coming out of a cone. This is based on joint work with Jacob Bernstein.

Boyu Zhang: On the compactness problem for generalized Seiberg–Witten equations in dimension three
In this talk, I will present a compactness result for a family of generalized Seiberg-Witten equations in dimension 3. This result recovers the compactness theorem for stable flat PSL2(C)-connections by Taubes as well as the compactness theorem for Seiberg–Witten equations with multiple spinors by Haydys-Walpuski. Furthermore, it implies a compactness theorem for the ADHM_{1,2} Seiberg–Witten equations. This is joint work with Thomas Walpuski.

Xin Zhou: Minimal hypersurfaces, min-max theory, and multiplicity
In this talk, we will report some recent developments related to multiplicity in the min-max theory of minimal hypersurfaces. We will first briefly introduce the solution of the Multiplicity One Conjecture which asserts that the multiplicity is always one for bumpy metrics. Then we will discuss a recent joint work with Zhichao Wang (UBC) in which we exhibit a set of non-bumpy metrics on the standard spheres, under which the min-max theory must produce higher multiplicity minimal hypersurfaces.

4-hour minicourses:

Costante Bellettini: Regularity and compactness questions in calibrated geometric analysis

Jake Solomon: The space of positive Lagrangian submanifolds
A Lagrangian submanifold of a Calabi-Yau manifold is called positive if the restriction to it of the real part of the holomorphic volume form is positive. The space of positive Lagrangians admits a Riemannian metric of non-positive curvature. The universal cover of the space admits a convex functional with critical points over special Lagrangians. Understanding the geodesics of the space of positive Lagrangian submanifolds would shed light on questions ranging from the uniqueness and existence of special Lagrangian submanifolds to Arnold's nearby Lagrangian conjecture. The geodesic equation is a nonlinear degenerate elliptic PDE. I will discuss joint work with A. Yuval on the cylindrical transform, which converts the geodesic equation to a family of elliptic boundary value problems. Namely, a geodesic segment is equivalent to a 1-parameter family of special Lagrangian cylinders with boundaries on the positive Lagrangian submanifolds that are the endpoints of the segment. A blowup argument compatible with the cylindrical transform gives tangent geodesics at potentially singular points facilitating analysis of regularity. A priori, a 1-parameter family of special Lagrangian cylinders emanates from any Maslov zero intersection point of positive Lagrangians. This family is expected to undergo surgeries as it passes intersection points of positive index before terminating at intersection points of maximal index. Thus, if one could prove appropriate compactness and gluing results for special Lagrangians cylinders, the existence of geodesic segments with prescribed endpoints would follow.

Luca Spolaor: Almgren’s type regularity for semicalibrated currents