Nov 18, 2024
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Geometric analysis on singular complex spaces
Jian Song (Rutgers University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We establish a uniform Sobolev inequality and diameter bound for Kahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular Kahler metrics on Kahler spaces with normal singularities. This allows us to build a general theory of global geometric analysis on singular Kahler spaces including the spectral theorem, heat kernel estimates, eigenvalue estimates and diameter estimates. Such estimates were only known previously in very special cases such as Bergman metrics. As a consequence, we derive various geometric estimates, such as the diameter estimate and the Sobolev inequality, for Kahler-Einstein currents on projective varieties with definite or vanishing first Chern class.
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Analysis and degenerations of ALH* gravitational instantons
Xuwen Zhu (Northeastern University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will focus on the ALH*type which has a non-compact end with inhomogeneous collapsing near infinity. I will talk about a joint project with Rafe Mazzeo on the Fredholm mapping property of the Laplacian and the Dirac operator, where the geometric microlocal analysis of fibered metrics plays a central role. Application of this Fredholm theory includes the L^2 Hodge theory, polyhomogeneous expansion and local perturbation theory. I will also discuss a joint project with Yu-Shen Lin and Sidharth Soundararajan on the degeneration of such metrics which gives a partial compactification of their moduli space.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Self-Expander of Mean Curvature Flow and Applications
Lu Wang (Yale University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Self-expanders are a special class of solutions to the mean curvature flow, in which a later time slice is a scale-up copy of an earlier one. They are also critical points for a suitable weighted area functional. Self-expanders model the asymptotic behavior of a mean curvature flow when it emerges from a cone singularity. The nonuniqueness of self-expanders presents challenges in the study of cone-like singularities in the flow. In this talk, I will discuss some recent development on a variational theory for self-expanders and an application to the question on lower density bounds for minimal cones
- Supplements
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03:00 PM - 03:00 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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The (spherical) Mahler measure of the X-discriminant
Sean Paul (University of Wisconsin-Madison)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Let P be a homogeneous polynomial in N+1 complex variables of degree d. The logarithmic Mahler Measure of P (denoted by m(P) ) is the integral of log|P| over the sphere in C^{N+1} with respect to the usual Hermitian metric and measure on the sphere. Now let X be a smooth variety embedded in CP^N by a high power of an ample line bundle and let $\Delta$ denote a generalized discriminant of X wrt the given embedding , then $\Delta$ is an irreducible homogeneous polynomial in the appropriate space of variables.
In this talk I will discuss work in progress whose aim is to find an asymptotic expansion of m(\Delta) in terms of the degree of the embedding. The technical machinery required for this task was developed by Jean-Michel Bismut in several articles in the 1990's
- Supplements
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Nov 19, 2024
Tuesday
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09:30 AM - 10:30 AM
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(Remote) Closed 3-forms in five dimensions and SL(3,C) structures in dimension six
Simon Donaldson (Imperial College, London)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
The talk is based on joint work with Fabian Lehmann. The main thrust of that work was to develop a deformation theory for torsion-free SL(3,C) structures on 6-manifolds with boundary, partially extending the well-known Torelli theory in the closed case. We will outline this and a number of related constructions and questions which arise. These include a local differential geometric invariant of closed 3-forms on 5-manifolds, an obstruction space related to CR theory and a possible generalisation of the affine isoperimetric inequality.
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Special Lagrangian pair of pants
Yang Li (University of Cambridge)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Special Lagrangian submanifolds are a distinguished class of minimal surfaces inside Calabi-Yau manifolds. I will discuss the construction of the special Lagrangian analogue of the pair of pants surfaces in all dimensions, using a combination of PDE and geometric measure theory methods.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Torelli parameters for Hitchin moduli spaces on the four-punctured sphere
Laura Fredrickson (University of Oregon)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Asymptotic geometry at infinity of quiver varieties
Frederic Rochon (Université du Québec à Montréal)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Using an approach developed by Melrose, we will explain how to show that quiver varieties are quasi-asymptotically conical (QAC) provided some genericity assumption holds. Relying on this fine description of the geometry at infinity and a spectral gap for singular 3-Sasakian manifolds, we will then explain how the reduced L^2 cohomology of a quiver variety can be computed, confirming a prediction made by Vafa and Witten in 1994. This is a joint work with Panagiotis Dimakis.
- Supplements
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04:30 PM - 06:00 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Nov 20, 2024
Wednesday
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09:30 AM - 10:30 AM
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Seiberg-Witten equations in all dimensions
Joel Fine (Université Libre de Bruxelles)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
I will describe a generalisation of the Seiberg-Witten equations to a Spin-c manifold of any dimension. The equations are for a U(1) connection A and spinor \phi and also an odd-degree differential form b (of inhomogeneous degree). Clifford action of the form is used to perturb the Dirac operator D_A. The first equation says that (D_A+b)(\phi)=0. The second equation involves the Weitzenböck remainder for D_A+b, setting it equal to q(\phi), where q(\phi) is the same quadratic term which appears in the usual Seiberg-Witten equations. This system is elliptic modulo gauge in dimensions congruent to 0,1 or 3 mod 4. In dimensions congruent to 2 mod 4 one needs to take two copies of the system, coupled via b. I will also describe a variant of these equations which make sense on manifolds with a Spin(7) structure, or a G_2 structure.
The most important difference with the familiar 3 and 4 dimensional stories is that compactness of the space of solutions is, for now at least, unclear. I will outline a very speculative suggestion that if this problem can be overcome and the equations can be used to define an invariant then it might be related to the existence of certain geometric structures on the underlying manifold (much as in dimension 4, vanishing of the Seiberg-Witten invariant obstructs the existence of a symplectic structure). This is joint work with Partha Ghosh and, in the Spin(7) and G_2 setting, Ragini Singhal.
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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A variational theory for the area of Legendrian surfaces
Alessandro Pigati (Università Commerciale ``Luigi Bocconi'')
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Nov 21, 2024
Thursday
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09:30 AM - 10:30 AM
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Chambered invariants of real Cauchy–Riemann operators
Thomas Walpuski (Humboldt-Universität)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 10:40 AM
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Group Photo
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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The role of flat SL(2,C) connections in counting special Lagrangians
Gregory Parker (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
There are numerous speculative proposals for constructing enumerative invariants of Calabi-Yau 3-folds by counting special Lagrangian submanifolds. One of the many difficulties in defining such invariants is that special Lagrangians may collapse to multiple covers as parameters vary. For two-fold covers specifically, work of Siqi He shows that the local deformation theory is governed by the existence of Z_2-harmonic 1-forms. On the other hand, Taubes originally defined Z_2-harmonic 1-forms to describe limits of diverging sequences of flat SL(2,C) connections. In this talk I will discuss the connection between these two theories, and discuss progress toward and obstructions to gluing results providing the obverse of He's result in the realm of flat SL(2,C) connections.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Stationary Integral Varifolds near Multiplicity 2 Planes
Paul Minter (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Allard’s regularity theorem provides regularity for stationary integral varifolds near a plane of multiplicity one: they must be smooth minimal graphs. What can be said when the plane instead has (integer) multiplicity larger than 1 has remained an open question and is crucial to the global regularity of stationary integral varifolds, as it currently prevents us concluding almost everywhere regularity. Examples such as the catenoid and complex varieties illustrate the complication. Even uniqueness of a multiplicity 2 plane as a tangent cone is not known in general, and those cases where there are results have additional variational assumptions, such as being area minimising.
In this talk, I will discuss a new regularity result for stationary integral varifolds near planes of multiplicity 2. This holds in arbitrary dimension and codimension, providing full regularity as a Lipschitz 2-valued graph (with certain improved estimates). The only additional assumption one needs is of a topological nature on the support of the varifold: no other variational assumptions are needed other than stationarity. This assumption can be directly checked in several cases of interest. I will also discuss hopes for the future. This is all based on joint work with Spencer Becker-Kahn and Neshan Wickramasekera.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Analysis of singularities of area minimizing currents
Brian Krummel (The University of Melbourne)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In his monumental work in the 1980s, Almgren showed that for $n$-dimensional locally area minimizing rectifiable currents of codimension $> 1$, the singular set has Hausdorff dimension at most $n-2$. In contrast with codimension one, it is well-known that area minimizers of higher codimension can admit branch point singularities, at which at least one tangent cone is an $n$-dimensional plane with integer multiplicity $\geq 2$. Almgren’s lengthy proof consisted of first using an elementary argument based on tangent cone type to show that the set of non-branch point singularities has Hausdorff dimension at most $n-2$, and then using a powerful array of new ideas to bound the dimension of the set of all branch point singularities. In this strategy, the exceeding complexity of the argument stems largely from the lack of an estimate giving decay of $T$ towards a unique tangent plane at a branch point.
We will discuss a new approach (joint work with Neshan Wickramasekera) in which we first focus on showing uniqueness of tangent cones at $\mathcal{H}^{n-2}$-a.e. singular point, and then we study the size and fine structure of the singular set. As part of our method, we introduce an approximate monotonicity formula for a new intrinsic planar frequency function which quantifies the rate at which $T$ decays towards an $n$-dimensional plane. We decompose the singular set of $T$ into the set of branch points $\mathcal{B}$ at which $T$ decays to a unique tangent plane faster than a fixed rate (of radius to a power) and the set $\mathcal{S}$ of all remaining singular points, which a priori might contain a large set of branch points. Using the monotonicity of planar frequency functions together with the relatively elementary parts of Almgren’s proof (namely Lipschitz and harmonic approximation), we obtain locally uniform estimates indicating that near each point of $\mathcal{S}$ and at each sufficiently small scale, $T$ is significantly closer to some non-planar cone than it is to any plane. An analysis of singularities using the planar frequency function and locally uniform estimates recovers Almgren’s dimension bound for the singular set of $T$ in a simpler way. Using a blow-up methods of L. Simon and Wickramaskera and the locally uniform estimates for $\mathcal{S}$, we show that $T$ has a unique non-planar tangent cone at $\mathcal{H}^{n-2}$-a.e. point of $\mathcal{S}$, and thus $T$ has a unique tangent cone at $\mathcal{H}^{n-2}$-a.e. singular point. Again using the blow-up method together with the approximate monotonicity of the planar frequency function, we show that the entire singular set, including $\mathcal{B}$, is countably $(n-2)$-rectifiable and that $T$ has a unique homogeneous harmonic blow-up relative to its tangent plane at each point of $\mathcal{B}$.
- Supplements
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Nov 22, 2024
Friday
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09:30 AM - 10:30 AM
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Bernstein-type and sheeting-type results for stable minimal hypersurfaces
Costante Bellettini (University College London)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties with an extrinsic area growth assumption). For n \geq 7 we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively the analogous Schoen-Simon-Yau estimates (obtained for n \leq 5) and the Schoen-Simon sheeting theorem (valid for embeddings).
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Uniform estimates for weighted cscK metrics
Eleonora Di Nezza (Institut de Mathématiques de Jussieu; École Normale Supérieure)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
A central theme in Kähler geometry is the search for canonical Kähler metrics. The concept of constant weighted scalar curvature Kähler metrics (weighted cscK for short), introduced by Lahdili, provides a unification of various geometric settings (such as cscK metrics, Kähler-Ricci solitons or Calabi extremal metrics). In this talk I will present uniform estimates for weighted cscK potentials. This is a joint work with S. Jubert and A. Lahdili.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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