Sep 03, 2024
Tuesday
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09:15 AM - 09:30 AM
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Welcome to SLMath
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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09:30 AM - 11:00 AM
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Geometric estimates along the Kähler-Ricci flow, part one
Vincent Guedj (Institut de Mathématiques de Toulouse)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green’s functions associated with certain Kähler metrics. In these lectures, we will explain their approach, broaden the scope of their techniques, and apply these results to study the asymptotic behavior of the Kähler-Ricci flow.
- Supplements
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11:00 AM - 11:30 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:30 AM - 12:30 PM
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More Complete Calabi-Yau Metrics of Calabi Type
Yifan Chen (University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In this talk I will construct more complete Calabi-Yau metrics of Calabi type. They are higher-dimensional analogues of ALH* gravitational instantons in two dimensions. This work builds on and generalizes the results of Tian-Yau and Hein-Sun-Viaclovsky-Zhang, creating Calabi-Yau metrics that are only polynomially close to the model space. I will also show the uniqueness of such metrics in a given cohomology class with fixed asymptotic behavior.
- Supplements
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12:30 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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Collapsing spaces in metric Riemannian geometry: old and new pictures
Ruobing Zhang (University of Wisconsin-Madison)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In this talk, we will introduce old and recent developments in the studies of collapsing manifolds with sectional and Ricci curvature bounds.
- 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 - 05:00 PM
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Geometric estimates along the Kähler-Ricci flow, part two
Vincent Guedj (Institut de Mathématiques de Toulouse)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In a remarkable series of works, Guo, Phong, Song, and Sturm have obtained key uniform estimates for the Green’s functions associated with certain Kähler metrics. In these lectures, we will explain their approach, broaden the scope of their techniques, and apply these results to study the asymptotic behavior of the Kähler-Ricci flow.
- Supplements
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05:00 PM - 06:20 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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Sep 04, 2024
Wednesday
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09:00 AM - 10:30 AM
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Hodge cohomology of complete hyperKähler manifolds
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
After reviewing the notion of L^2-cohomology and presenting various conjectures and recent results about the Hodge cohomology (in other words reduced L^2-cohomology) of some complete hyperKähler metrics, I will focus on a result of Hausel-Hunsicker-Mazzeo of 2004 describing the Hodge cohomology of most types of gravitational instantons. In particular, I will explain how this result crucially relies on a decay at infinity of L^2-harmonic forms which will be the object of the next lecture.
- Supplements
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10:30 AM - 11:00 AM
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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 - 12:30 PM
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PDE analysis on 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 will review classical estimates for elliptic PDEs, and the notions of stability and minimality for hypersurfaces. The interplay among these aspects underlies a vast number of geometric results, via the regularity and compactness theory for stable minimal hypersurfaces. We will discuss recent progress and open questions.
- Supplements
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12:30 PM - 02:00 PM
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Lunch
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- Location
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- Video
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- Abstract
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- Supplements
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02:00 PM - 05:00 PM
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Free Afternoon
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- Location
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- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
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- Video
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- Abstract
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- Supplements
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Sep 05, 2024
Thursday
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09:00 AM - 10:30 AM
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Decay of L^2-harmonic forms via microlocal methods
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
I will explain a general and flexible method originally introduced by Richard Melrose for estimating the decay of L^2-harmonic forms at infinity, but focusing on the specific example of L^2-harmonic forms of fibered boundary metrics (e.g. most types of gravitational instantons), in which case the result is due to Hausel-Hunsicker-Mazzeo in 2004 and heavily relies on the parametrix construction of Vaillant. After outlining the method, I will introduce the pseudodifferential calculus needed and describe the important steps of the parametrix construction. I will also indicate other important results that can be obtained with this method.
- Supplements
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10:30 AM - 10:40 AM
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Group Picture
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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10:40 AM - 11:00 AM
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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 - 12:30 PM
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PDE analysis on stable minimal hypersurfaces
Costante Bellettini (University College London)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
-
- Abstract
Zoom Link
We will review classical estimates for elliptic PDEs, and the notions of stability and minimality for hypersurfaces. The interplay among these aspects underlies a vast number of geometric results, via the regularity and compactness theory for stable minimal hypersurfaces. We will discuss recent progress and open questions.
- Supplements
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12:30 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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Geometric flows of G_2 and Spin(7)-structures
Shubham Dwivedi (University of Waterloo)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We will discuss a family of flows of G_2-structures on seven dimensional Riemannian manifolds. These flows are negative gradient flows of natural energy functionals involving various torsion components of G_2-structures. We will prove short-time existence and uniqueness of solutions to the flows and a priori estimates for some specific flows in the family. We will discuss analogous flows of Spin(7)-structures. This talk is based on arXiv:2311.05516 (joint work with P. Gianniotis and S. Karigiannis) and arXiv:2404.00870.
- 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 - 05:00 PM
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Solitons in G_2-Laplacian flow, part one
Anna Fino (Università di Torino; Florida International University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Closed G_2-structures on 7-manifolds are defined by a closed positive 3-form. Although linear, the closed condition for a G_2-structure is very restrictive, and no general results on the existence of closed G_2-structures on compact 7-manifolds are known. In this lecture after an introduction on closed G_2-structures, I will review known results on the G_2-Laplacian flow
- Supplements
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Sep 06, 2024
Friday
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09:00 AM - 10:30 AM
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Solitons in G_2-Laplacian flow, part two
Anna Fino (Università di Torino; Florida International University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Closed G_2-structures on 7-manifolds are defined by a closed positive 3-form. Although linear, the closed condition for a G_2-structure is very restrictive, and no general results on the existence of closed G_2-structures on compact 7-manifolds are known. In this second lecture I will discuss recent results on solitons of the G_2-Laplacian flow.
- Supplements
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10:30 AM - 11:00 AM
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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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Constant scalar curvature metrics and semistable vector bundles
Annamaria Ortu (University of Göteborg)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In this talk, we present a construction of Kaehler metrics with constant scalar curvature on the projectivisation of certain holomorphic vector bundles. When the vector bundle is slope-stable and the base admits a constant scalar curvature metric, it is a classical result of Hong that the total space of the projectivisation admits a constant scalar curvature metric in adiabatic classes. We extend their result to slope-semistable vector bundles: we show that if $E \to B$ is slope-semistable and the total space of the projectivisation is K-polystable then it admits a constant scalar curvature metric in adiabatic classes.
This is joint work with L.M. Sektnan.
- Supplements
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12:00 PM - 01:00 PM
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Structure of singularities for semicalibrated currents
Davide Parise (University of California, San Diego)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Semicalibrated currents are a generalization of calibrated submanifolds and appear naturally in various geometric problems, since the property of being merely semicalibrated is significantly easier to preserve under deformations. Semicalibrated currents are expected to share the regularity properties of area-minimizing currents, and indeed Almgren's celebrated dimension estimate on the interior singular set was shown to also hold for semicalibrated surfaces by Spolaor in 2015. I will talk about joint work with Paul Minter, Anna Skorobogatova, and Luca Spolaor, in which we build on this to establish a sharp structural result (rectifiability) for the interior singular set.
- Supplements
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01:00 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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