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Fall
Quarter
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26 September
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Speaker: TBA
Title:
Abstract:
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3 October
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Speaker: TBA
Title:
Abstract:
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10 October
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Speaker: Xin Zhou (Stanford)
Title: Min-max minimal hypersurface in (M^{n+1},g) with Ric>0 and 2≤n≤6
Abstract: In this talk, we will discuss the shape of the min-max minimal
hypersurface produced by Almgren-Pitts corresponding to the fundamental
class of a Riemannian manifold (M^{n+1},g) of positive Ricci curvature with
2 ≤ n ≤ 6. We characterize the Morse index, volume and multiplicity of this
min-max hypersurface. In particular, we show that the min-max hypersurface
is either orientable and of index one, or is a double cover of a
non-orientable minimal hypersurface with least area among all embedded
minimal hypersurfaces.
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17 October 3pm--4pm (special time but the same location)
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Speaker: Yi Wang (Stanford)
Title: A subelliptic Bourgain-Brezis inequality
Abstract: In this talk, I will present an approximation result for functions in a Sobolev space with critical power on stratified homogeneous groups, generalizing a result of Bourgain-Brezis on the Euclidean space. I will also talk about its application to solve a divergence-curl inequality for ¯∂b operator on Heisenberg groups. Higher order approximation result is also derived. This is a joint work with Po-Lam Yung.
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17 October
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Speaker: Lashi Bandara (Australian National
University)
Title: L^\infty coefficient operators and non-smooth Riemannian metrics.
Abstract: The Kato square root problem on smooth Riemannian manifolds is
the study of L^\infty coefficient uniformly elliptic divergence form
operators. The stability given by the L^\infty coefficients enable us to
reduce the study of such problems on non-smooth metrics to the smooth
case by absorbing this lack of regularity in a corresponding equivalent
operator on a smooth geometry. I will give a brief account of the
solution to the smooth problem and discuss my current research on
non-smooth metrics and possible connections to geometric flows.
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24 October
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Speaker: Eva Silverstein (Stanford,
Physics)
Title: Inflation, de Sitter spacetime and the origin of structure (PART 1)
Abstract: Inflation is a (deceptively) simple idea in cosmology which ties directly
to observational data on the one hand (real experiments), and to difficult problems in
formulating quantum gravity on the other hand (thought experiments). These talks will start
by explaining the theory of the origin of structure in the universe
as being seeded by quantum fluctuations generated during a period of accelerated expansion. I will go on to describe
the current observational and theoretical state of the subject and its connection to interesting mathematical structures
such as de Sitter spacetime and string compactifications.
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25 October 4:15pm, 380-W (MRC Distinguished Lecture Series)
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Speaker: Michael Struwe (ETH, Zurich)
Title: Conformal metrics of prescribed Gauss curvature
Abstract:
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29 October 4pm (special time but the same location 383N)
(MRC Distinguished Lecture Series)
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Speaker: Michael Struwe (ETH, Zurich)
Title: Well-posedness for a critical nonlinear wave equation in 2 space dimensions (Joint with MRC)
Abstract: We show that the Cauchy problem for the equation
\begin{equation*}
u_{tt}-\Delta u+ ue^{u^2}=0 \hbox{ on } {\mathbb R}\times {\mathbb R}^2
\end{equation*}
related to the Trudinger-Moser embedding on ${\mathbb R}^2$ admits a global
smooth solution for arbitrary smooth initial data.
Previously, Ibrahim, Majdoub, and Masmoudi had obtained well-posedness of
this equation for smooth Cauchy data with small
energy and had conjectured that the problem for large initial data might
exhibit supercritical behavior. However, in 2009 I had
been able to show global well-posedness in the case of radial symmetric
data. The resolution of the general case
requires a new approach, which also involves a subtle improvement of the
Trudinger-Moser inequality.
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31 October 3pm--4pm (special time but the same location 383N)
(MRC Distinguished Lecture Series)
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Speaker: Michael Struwe (ETH, Zurich)
Title: Towards the Rabinowitz Conjecture (Joint with MRC)
Abstract: As a model problem for the study of supercritical partial
differential equations, on a smoothly bounded domain $\Omega\subset {\mathbb
R}^n$, $n\ge 3$, for an exponent $p>2^*=\frac{2n}{n-2}$, the critical
exponent for Sobolev's embedding, we consider the boundary value problem
\begin{equation}\label{1}
-\Delta u=|u|^{p-2}u \hbox{ in } \Omega,\ u = 0 \hbox{ on } \partial\Omega,
\end{equation}
and the associated heat flow
\begin{equation}\label{2}
u_t-\Delta u=|u|^{p-2}u \hbox{ in } \Omega,\ u = 0 \hbox{ on }
\partial\Omega,\ u = u_0 \hbox{ at } t=0
\end{equation}
for given smooth initial data $u_0$. Similar to standard variational
approaches to the existence of positive solutions to \eqref{1}, the flow
\eqref{2} may be regarded as a (negative) gradient flow for the energy
functionall associated with equation \eqref{1}, whose rest points exactly
correspond to the solutions of \eqref{1}. However, solutions to the flow
\eqref{2} may blow up in finite time. We partially extend the pioneering
analysis of the possible blow up profiles of \eqref{2} by Giga-Kohn to the
supercritical range $p>2^*$ and discuss possible consequences for the
existence of positive solutions to \eqref{1} on topologically non-nontrivial
domains.
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31 October 4pm--5pm
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Speaker: Eva Silverstein (Stanford, Physics)
Title: Inflation, de Sitter spacetime and the origin of structure (PART 2)
Abstract:see part 1
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7 November
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Speaker: Kristen Moore (Max Planck, Potsdam)
Title: Evolving hypersurfaces by their inverse null mean curvature
Abstract: We introduce a new second order parabolic evolution equation where the speed is given by the reciprocal of the null mean curvature. This flow is a generalisation of inverse mean curvature flow and it is motivated by the study of black holes and mass/energy inequalities in general relativity. We present a theory of weak solutions using level-set methods and an appropriate variational principle, and outline a natural application of the flow as a variational approach to constructing marginally outer trapped surfaces (MOTS), which play
the role of quasi-local black hole boundaries in general relativity.
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14 November
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Speaker: Bijan Sahamie (University of
Munich)
Title: Engel Structures on Circle Bundles
Abstract: Engel structures are maximally non-integrable 2-plane distributions on 4-manifolds. We will present a brief introduction to Engel structures and then discuss a classification result on circle bundles over closed oriented 3-manifolds.
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28 November
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Speaker: Semyon Dyatlov (Berkeley)
Title: Resonances for normally hyperbolic trapped sets
Abstract: Resonances are complex analogs of eigenvalues for Laplacians on noncompact manifolds, arising in long time resonance expansions of linear waves. We prove a Weyl type asymptotic formula for the number of resonances in a strip, provided that the set of trapped geodesics is r-normally hyperbolic for large r and satisfies a pinching condition. Our dynamical assumptions are stable under small smooth perturbations and motivated by applications to black holes. We also establish a high frequency analog of resonance expansions.
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5 December
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Speaker: Gabriele Di Cerbo (Princeton)
Title: Positivity questions in Kahler-Einstein theory.
Abstract: I will show how some questions motivated by the theory of Kahler-Einstein metrics with singularities can be solved using classical theorems in the minimal model program. For example, I will characterize the pairs which admit Kahler-Einstein metrics with negative scalar curvature and small cone-edge singularities
along a simple normal crossing divisor. If time permits, I will explain how these results can be used to bound the
number of cusps of certain complex hyperbolic manifolds in terms of their volume.
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13 December
Note special day (Thursday)
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Speaker: Keti Tenenblat (University of Brasilia)
Title: On Ribaucour Transformations for surfaces
Abstract: We consider Ribaucour transformations for linear Weingarten surfaces in space forms. We show that such a transformation is a Darboux transformation if, and only if, the surfaces have the same constant mean curvature. We show that such transformations for minimal surfaces in IR^3 produce embedded planar ends while for cmc1 surfaces immersed in IH^3 they produce embedded ends of horosphere type. We prove that the Lawson correspondence commutes with the Darboux transformations. We show that the property of completeness is preserved by this commutativity. Aplications and examples of these results give families of explicitly parametrized complete surfaces with any finite or infinite number of planar ends, bubbles, "segments" or embedded ends of horosphere type. Other applications will be given for flat surfaces in IH^3 and for a class of PDEs associated to the linear Weingarten surfaces.
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