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Spring Quarter
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3 April
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Speaker: Clifford Taubes (Harvard)
Title: How many monopoles can fit in your refrigerator?
Abstract: Bolognese conjectured that a large number, N, of non-Abelian SU(2) monopoles on R3
(solutions to the time independent self-dual equations) can fit in a ball of radius R only if R is no smaller than N. I explain the sense in which Bolognese’s bound is true and describe a construction of N-monopole solutions that come close to realizing Bolognese’s minimal radius bound.
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10 April
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Speaker: Martin Li (UBC)
Title: Minimal surfaces in the unit ball
Abstract: Minimal surfaces in the unit ball arise as solutions to an extremal
eigenvalue problem for surfaces with boundary, due to the work of A. Fraser and R.
Schoen. In this talk, we will discuss some recent results concerning the geometry
and the variational properties of these minimal surfaces. For the embedded ones, we
give an eigenvalue estimate and a smooth compactness result. As an application, we
construct new examples of embedded Willmore surfaces of arbitrary genus. (Part of
this is joint work with A. Fraser.)
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12 April, 2pm (NOTE: SPECIAL DAY AND TIME)
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Speaker: Artem Pulemotov (Queensland)
Title: The Dirichlet problem for the prescribed Ricci curvature equation
Abstract: We will discuss the following question: is it possible to find a
Riemannian metric whose Ricci curvature is equal to a given tensor
on a manifold $M$? To answer this question, one must analyze a
weakly elliptic second-order geometric PDE. In the first part of the
talk, we will review the history of the subject and state several
classical theorems. After that, our focus will be on new results
concerning the case where $M$ has nonempty boundary.
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17 April
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Speaker: TBA
Title: TBA
Abstract:
TBA
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24 April
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Speaker: Nick Edelen (Stanford)
Title: Constant mean curvature, flux conservation, and symmetry
Abstract: As first observed by Korevaar, Kusner and Solomon, constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields. I will discuss joint work with Bruce Solomon, in which we generalize this law by relaxing the topological restrictions assumed by KKS, and by allowing a weighted mean curvature functional. We also prove a partial converse. Roughly, it states: when flux is conserved along enough Killing fields, the hypersurface splits into a region with constant mean curvature, and a region fixed by the given Killing fields. I will demonstrate our theory by using it to derive a first integral for twizzlers, i.e. helicoidal surfaces of constant mean curvature in R3.
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1 May
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Speaker: TBA
Title: TBA
Abstract:
TBA
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8 May
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Speaker: Qi Zhang (Riverside)
Title: Volume non-inflating under Ricci flow and a convergence result
Abstract: First we present a proof of the so called volume non-inflating
property of Ricci flows, which can be regarded as the opposite statement of
Perelman's volume non-collapsing property.
This implies, combining with Perelman's result, that the Kaehler Ricci flow in the positive Chern class case, converges
to a metric space in the Gromov-Hausdorff topology.
Second we discuss a gradient bound for the heat equation which is independent of a
lower bound of the Ricci curvature. Applying this together with Cheeger's result, one can
show that the limiting metric space has certain regularity.
The talk is based on a paper by the speaker and on a joint paper with Gang Tian.
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14 May, 4pm-5pm (NOTE: SPECIAL DAY and SPECIAL ROOM GESB 131)
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Speaker: Panagiota Daskalopoulos (Columbia)
Title: Ancient Solutions to the Yamabe Flow
Abstract: We will discuss the existence of new type II ancient compact solutions to the Yamabe flow. These solutions are rotationally symmetric and converge, as $t \to -\infty$, to a tower of spheres.
We will also discuss complete non-compact Yamabe shrinking solitons and the singularity formation of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have cylindrical behavior at in?nity.
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15 May
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Speaker: Curtis McMullen (Harvard)
Title: The Gauss-Bonnet theorem for cone manifolds and moduli spaces
Abstract: TBA
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22 May
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Speaker: Otis Chodosh (Stanford)
Title: Ricci solitons asymptotic to cones
Abstract:
We'll discuss the relationship between expanding Ricci
solitons and metric cones, in particular the works by Schulze--Simon
and Cabezas-Rivas–Wilking constructing Ricci flows with certain cones
as initial conditions. We will then discuss two recent uniqueness
results for expanding solitons asymptotic to "round" cones (one of
these results is joint work with Frederick Tsz-Ho Fong).
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29 May
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Speaker: Karl-Theodor Sturm (Bonn)
Title: The space of mm spaces is an Alexandrov space
Abstract:
The space X of all metric measure spaces (X,d,m) plays an important rôle in image analysis, in the investigation of limits of Riemannnian manifolds and metric graphs as well as in the study of geometric flows that develop singularities. We show that the space X – equipped with the L2-distortion distance ∆ – is a challenging object of geometric interest in its own. In particular, we show that it has nonnegative curvature in the sense of Alexandrov. Geodesics and tangent spaces are characterized in detail. Moreover, classes of semiconvex functionals and their gradient flows on X are presented.
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