Jan 18, 2016
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
- --
- Supplements
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09:30 AM - 10:30 AM
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Manifolds with lower sectional curvature bounds and Alexandrov geometry
Karsten Grove (University of Notre Dame)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The aim of the talk is to provide a survey of the main tools, results and open problems concerning manifolds with a lower (sectional) curvature bound. It is well known that local bounds on sectional curvature can be described geometrically via local distance comparison to constant curvature spaces. For lower curvature bounds this comparison is global, as expressed in the Toponogov Comparison Theorem. This together with critical point theory for distance functions paved the way for studying manifolds with only a lower sectional curvature bound, resulting in Finiteness, Structure, and Recognition Theorems. There are several equivalent versions of Toponogov’s Comparison Theorem, some of which make sense in a general metric space. Moreover, such a metrically expressed lower curvature bound is preserved by the process of taking a Gromov-Hausdor limit. An Alexandrov space is a finite Hausdor dimensional, inner metric space with a lower curvature bound. It turns out that, despite their general definition, Alexandrov spaces have a surprisingly rich structure and are natural objects in their own rite. Their applications and significance to Riemanian geometry stems from the fact that there are several natural geometric operations that are closed in Alexandrov geometry, but not in Riemanian geometry. These include taking Gromov-Hausdor limits, taking quotients and taking joins of positively curved spaces. All concepts alluded to above will be explained and discussed, as will examples, some of the main results, and fundamental open problems.
- 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
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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Comparison geometry for Ricci curvature I
Guofang Wei (University of California, Santa Barbara)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Ricci curvature occurs in the Einstein equation, Ricci flow, optimal transport, and is important both in mathematics and physics. Comparison method is one of the key tools in studying the Ricci curvature. We will start with Bochner formula and derive Laplacian comparison, Bishop-Gromov volume comparison, first eigenvalue and heat kernel comparison and some application. Then we will discuss some of its generalizations to Bakry-Emery Ricci curvature and integral Ricci curvature
- 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
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Ricci flat spaces and metrics with special or exceptional holonomy I
Mark Haskins (Duke University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
To this day the only compact irreducible complete Ricci-flat Riemannian metrics arise as manifolds with special or exceptional holonomy. This pair of talks will give an introduction to special and exceptional holonomy metrics and the resulting constructions of both compact and noncompact Ricci-flat manifolds. We will indicate how ideas from Riemannian convergence theory have provided motivation for the currently known (and possibly for future) constructions of metrics with exceptional holonomy
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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03:30 PM - 04:30 PM
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Aspects of Einstein metrics on 4-manifolds I
Michael Anderson (State University of New York, Stony Brook)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
These lectures will discuss classical and more recent research on selected topics in the area of Einstein metrics on 4-dimensional manifolds. The basic questions concern existence and uniqueness of such metrics. Broadly speaking, the first lecture will discuss uniqueness issues; more precisely the apriori structure of the (moduli) space of Einstein metrics on a given manifold. The second lecture will turn to the more difficult existence questions (on manifolds with boundary)
- Supplements
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Jan 19, 2016
Tuesday
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09:30 AM - 10:30 AM
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Mean curvature flow
Tobias Colding (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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--
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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Positively and non-negatively curved manifolds and (torus) symmetries
Catherine Searle (Wichita State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The classification of Riemannian manifolds with positive or non-negative sectional curvature is a long-standing problem in Riemannian Geometry. This talk will give a survey of tools and techniques, results and open problems concerning this class of manifolds with an emphasis on how (torus) symmetries play an important role in obtaining classification results
- 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
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Metric measure spaces satisfying Ricci curvature lower bounds I
Andrea Mondino (University of Warwick)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80s and was pushed by Cheeger and Colding in the '90s who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, Levy-Gromov isoperimetric inequality, etc.), permitting to give further insights in the theory. The goal of the lectures is to give an introduction to the subject:
-In the first lecture I will introduce Ricci curvature lower bounds for metric measure spaces and discuss basic properties.
- The second lecture will be devoted to the study of analytic and geometric properties of such spaces, in particular I will present the proof of the Levy-Gromov isoperimetric inequality (joint work with Fabio Cavalletti)
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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03:30 PM - 04:30 PM
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Semiconcave functions in Alexandrov geometry
Vitali Kapovitch (University of Toronto)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will talk about semiconcave functions on Alexandrov spaces with curvature bounded below. Semiconcave functions and their gradient flows provide an important technical tool for studying Alexandrov spaces with curvature bounded below. I will discuss basic properties and constructions of semi-concave functions and will talk about applications such as Morse theory on Alexandrov spaces which shows that a finite dimensional Alexandrov spaces are stratified manifolds. I will also discuss other applications such as various nilpotency theorems as well as some open problems.
- Supplements
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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Jan 20, 2016
Wednesday
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09:30 AM - 10:30 AM
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A panoramic glimpse of nonnegative curvature
Karsten Grove (University of Notre Dame)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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Comparison geometry for Ricci curvature II
Guofang Wei (University of California, Santa Barbara)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
Ricci curvature occurs in the Einstein equation, Ricci flow, optimal transport, and is important both in mathematics and physics. Comparison method is one of the key tools in studying the Ricci curvature. We will start with Bochner formula and derive Laplacian comparison, Bishop-Gromov volume comparison, first eigenvalue and heat kernel comparison and some application. Then we will discuss some of its generalizations to Bakry-Emery Ricci curvature and integral Ricci curvature
- Supplements
-
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Jan 21, 2016
Thursday
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09:30 AM - 10:30 AM
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Ricci flat spaces and metrics with special or exceptional holonomy II
Mark Haskins (Duke University)
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- Location
- SLMath:
- Video
-
- Abstract
To this day the only compact irreducible complete Ricci-flat Riemannian metrics arise as manifolds with special or exceptional holonomy. This pair of talks will give an introduction to special and exceptional holonomy metrics and the resulting constructions of both compact and noncompact Ricci-flat manifolds. We will indicate how ideas from Riemannian convergence theory have provided motivation for the currently known (and possibly for future) constructions of metrics with exceptional holonomy
- Supplements
-
--
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10:30 AM - 11:00 AM
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|
Break
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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11:00 AM - 12:00 PM
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Aspects of Einstein metrics on 4-manifolds II
Michael Anderson (State University of New York, Stony Brook)
|
- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
These lectures will discuss classical and more recent research on selected topics in the area of Einstein metrics on 4-dimensional manifolds. The basic questions concern existence and uniqueness of such metrics. Broadly speaking, the first lecture will discuss uniqueness issues; more precisely the apriori structure of the (moduli) space of Einstein metrics on a given manifold. The second lecture will turn to the more difficult existence questions (on manifolds with boundary).
- Supplements
-
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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The maximal symmetry rank conjecture for nonnegative curvature
Catherine Searle (Wichita State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- --
- Supplements
-
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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03:30 PM - 04:30 PM
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Minimal submanifolds and lower curvature bounds I
Richard Schoen (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In these lectures we will discuss the role that minimal submanifolds play in the study of lower curvature bounds. The first lecture will be an introduction to the theory including existence theory and the geometry of the second variation. The second lecture will be a survey of applications especially to lower scalar and isotropic curvature bounds
- Supplements
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--
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Jan 22, 2016
Friday
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09:30 AM - 10:30 AM
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Minimal submanifolds and lower curvature bounds II
Richard Schoen (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In these lectures we will discuss the role that minimal submanifolds play in the study of lower curvature bounds. The first lecture will be an introduction to the theory including existence theory and the geometry of the second variation. The second lecture will be a survey of applications especially to lower scalar and isotropic curvature bounds
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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11:00 AM - 12:00 PM
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Mean curvature flow
Tobias Colding (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- --
- Supplements
-
--
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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02:00 PM - 03:00 PM
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Metric measure spaces satisfying Ricci curvature lower bounds II
Andrea Mondino (University of Warwick)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80s and was pushed by Cheeger and Colding in the '90s who investigated the structure of the spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago; with this approach one can give a precise meaning of what means for a non smooth space to have Ricci curvature bounded from below by a constant. This approach has been refined in the last years by a number of authors and a number of fundamental tools have now been established (for instance the Bochner inequality, the splitting theorem, Levy-Gromov isoperimetric inequality, etc.), permitting to give further insights in the theory. The goal of the lectures is to give an introduction to the subject:
-In the first lecture I will introduce Ricci curvature lower bounds for metric measure spaces and discuss basic properties.
- The second lecture will be devoted to the study of analytic and geometric properties of such spaces, in particular I will present the proof of the Levy-Gromov isoperimetric inequality (joint work with Fabio Cavalletti).
- Supplements
-
|
03:00 PM - 03:30 PM
|
|
Tea
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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03:30 PM - 04:30 PM
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Measure-metric boundary and Liouville theorem in Alexandrov geometry
Vitali Kapovitch (University of Toronto)
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- Location
- SLMath:
- Video
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- Abstract
I will introduce the notion of measure-metric boundary on measure-metric spaces and discuss various motivational examples. I will then describe some recent joint work on measuremetric boundary with Alexander Lytchak and Anton Petrunin. We show that if an Alexandrov space has a zero measure metric boundary then for almost every point in almost every direction there exists an infinite geodesic and the geodesic flow preserves the Liouville measure. We conjecture that any Alexandrov space without boundary has zero measure-metric boundary and hence the above result should hold for all such spaces. While we can not prove it in general we show that this is true for convex hypersurfaces in smooth manifolds. We also show that finite dimensional Alexandrov spaces have finite measure-metric boundary.
- Supplements
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