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27th Annual
Geometry Festival
Duke University, Durham, NC
April 27-29, 2012
ABSTRACTS
OF TALKS
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| Simon Brendle,
Rotational symmetry of self-similar solutions to the Ricci flow VIDEO |
In
this talk I will prove Perelman's conjecture that a 3 dimensional
steady gradient Ricci soliton which is nonflat and kappa noncollapsed
is spherically symmetric and hence is isometric to the Bryant soliton.
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Carla Cederbaum,
From Newton to Einstein: A guided tour through space and time
(General Audience Lecture) VIDEO |
The cosmos and its laws have fascinated people since the
ancient times. Many scientists and philosophers have tried to describe
and explain what they saw in the sky. And almost all of them have used
mathematics to formulate their ideas and compute predictions for the
future. Today, we have made huge progress in understanding and
predicting how planets, stars, and galaxies behave. But still, the
mysteries of our universe are formulated and resolved in mathematical
language and always with new mathematical methods and ideas.
In this lecture, you will hear about two of the most famous physicists
of all times, Isaac Newton (1643-1727) and Albert Einstein (1879-1955),
and about their theories of the universe. You will learn about common
features and central differences in their viewpoints and in the
mathematics they used to formulate their theories. In passing, you will
also encounter the famous mathematician Carl Friedrich Gauss (1777-1855)
and his beautiful ideas about curvature.
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| John Etnyre,
Surgery and tight contact structures VIDEO
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One
of the fundamental problems in 3-dimensional contact geometry is the
construction of tight contact structures on closed manifolds. Two
obvious ways to try to construct such structures are via Legendrian
surgery and admissible transverse surgery. It was long thought that
when performed on a closed tight contact manifold these operations
would yield a tight contact manifold. We show that this is not true for
admissible transverse surgery. Along the way we discuss the relations
between these two surgery operations and construct some contact
structures with interesting properties.
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| Fernando Coda Marques,
Min-max theory and the Willmore conjecture VIDEO
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In
1965, T. J. Willmore conjectured that the integral of the square of the
mean curvature of any torus immersed in Euclidean three-space is at
least 2\pi^2. In this talk we will describe a solution to the Willmore
conjecture based on the min-max theory of minimal surfaces.
This is joint work with Andre Neves (Imperial College, UK).
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| Gordana Matic,
Contact invariant in sutured Floer homology and fillability VIDEO
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In the 70's Thurston and Winkelnkemper
showed how an open book decomposition of a 3-manifold can be used to
construct a contact structure. In 2000 Giroux showed that every
contact structure on a 3-manifold can be obtained from that
process. Ozsvath and Szabo used this fact to define an invariant
of contact structures in their Heegaard Floer homology, providing an
important new tool to study contact 3-manifolds.
In joint work with Ko Honda and Will Kazez we describe a simple way to
visualize this contact invariant and provide a generalization and some
applications. When the contact manifold has boundary, we define
an invariant of contact structure living in sutured Floer homology, a
variant of Heegaard Floer homology for a manifold with boundary due to
Andras Juhasz. We describe a natural gluing map on sutured Floer
homology and show how it produces a (1+1)-dimensional TQFT leading to
new obstructions to fillability.
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Jan Metzger,
On isoperimetric surfaces in asymptotically flat manifolds
VIDEO (good audio after 2 minutes) |
I
will present joint work with Michael Eichmair on the existence of large
isoperimetric regions in complete asymptotically flat manifolds of
arbitrary dimension with metric asymptotic to Schwarzschild. The key
idea is an effective isopermetric inequality that forces nearly optimal
regions to center in the manifold.
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| Yanir Rubinstein,
Einstein metrics on Kahler manifolds VIDEO
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The Uniformization Theorem implies that any compact Riemann surface has a
constant curvature metric. Kahler-Einstein (KE) metrics are a natural
generalization of such metrics, and the search for them has a long and
rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin,
Yau (70's) and Tian (90's), among others. Yet, despite much progress, a
complete picture is available only in complex dimension 2.
In contrast to such smooth KE metrics, in the mid 90's Tian conjectured
the existence of KE metrics with conical singularities along a divisor
(i.e., for which the manifold is `bent' at some angle along a complex
hypersurface), motivated by applications to algebraic geometry and
Calabi-Yau manifolds. More recently, Donaldson suggested a program for
constructing smooth KE metrics of positive curvature out of such singular
ones, and put forward several influential conjectures.
In this talk I will try to give an introduction to Kahler-Einstein
geometry and briefly describe some recent work mostly joint with R. Mazzeo
that resolves some of these conjectures. One key ingredient is a new
C^{2,\alpha} a priori estimate and continuity method for the complex
Monge-Ampere equation. It follows that many algebraic varieties that may
not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless
admit KE metrics bent along a simple normal crossing divisor.
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| Valentino Tosatti,
The evolution of a Hermitian metric by its Chern-Ricci curvature VIDEO
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I
will discuss the evolution of a Hermitian metric on a compact complex
manifold by its Chern-Ricci curvature. This is an evolution equation
which coincides with the Ricci flow if the initial metric is Kahler,
and was first studied by M.Gill. I will describe the maximal existence
time for the flow in terms of the initial data, and thendiscuss the
behavior of the flow on complex surfaces and on some higher-dimensional
manifolds. This is joint work with Ben Weinkove.
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| Mu-Tao Wang, A variational problem for isometric embeddings and its applications in general relativity VIDEO
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I shall discuss a variational problem arising from the study of quasilocal energy in general relativity. Given a spacelike 2-surface in spacetime, the Euler-Lagrange equation for the quasilocal energy is the isometric embedding equation into the Minkowski space coupled with a fourth order nonlinear elliptic equation for the time function. This equation is important in that it gives the ground configuration in GR. In joint work with PoNing Chen and Shing-Tung Yau, we solved this system in the cases of large and small sphere limits.
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