Date: Thursdays at 4:00pm.
Room: Mathematics Building 2300.
Organized
by: T. Darvas, Y.A. Rubinstein.
The aim of this seminar is
to attract graduate students to Geometric Analysis, through learning and
research talks. All talks should be accessible to beginning graduate students
who might have background either in PDE or in geometry, but not necessarily
in both.
Previous years: 2012-2013, 2013-2014, 2014-2015.
- September 17, Jim
Isenberg (University of Oregon),
Some Open Problems in Mathematical
Relativity
Einstein’s theory of General Relativity (GR), which celebrates its centenary
this year, provides a beautiful mathematical model of large scale
gravitational phenomena in astrophysics and cosmology. It also provides
mathematicians with wonderfully challenging mathematical questions to
explore. After introducing GR and its initial value problem, I discuss a
few of the outstanding questions in mathematical relativity which are
currently being studied. Depending on time, I will discuss
1) Parametrization and construction of initial data sets which satisfy
the Einstein constraint equations
2) Strong cosmic censorship and the long time behavior of spacetime solutions of Einstein’s equations
3) Stability of the Kerr black hole solution.
- September 24, Jim
Isenberg (University of Oregon),
Some Open Problems in Mathematical
Relativity: Parametrization and construction of initial data sets
- October 1, Jacob
Bernstein (Johns Hopkins University)
Title: Properties of hypersurfaces
of low entropy
Abstract: The entropy is a quantity introduced by Colding and Minicozzi and
may be thought of as a rough measure of the geometric complexity of a
hypersurface of Euclidean space.
It is closely related to the mean curvature flow. On the one hand, the entropy controls
the dynamics of the flow. On the
other hand, the mean curvature flow may be used to study the
entropy. In this talk I will
survey some recent results with Lu Wang that show that hypersurfaces of
low entropy really are simple.
- October 8, Amitai Yuval (Hebrew University),
Title: Geodesics of positive Lagrangians and
almost Calabi-Yau reduced spaces
Abstract: The space of
positive Lagrangians in an almost Calabi-Yau manifold is an open set in the space of
all Lagrangian submanifolds.
A Hamiltonian isotopy class of positive Lagrangians
admits a natural Riemannian metric, which gives rise to a notion of
geodesics. The geodesic equation is a fully non-linear degenerate
elliptic PDE, and it is not known yet whether the initial value problem
and boundary problem have solutions in general.
We will talk about Hamiltonian classes of positive Lagrangians
which are invariant under a Lie group Hamiltonian action. Such a
Hamiltonian class is isometric to the corresponding class in the symplectic reduced space, which has a natural almost
Calabi-Yau structure. We will show that when
the symplectic reduced space is of real
dimension 2, both the initial value problem and boundary problem have
unique solutions. As examples, we will discuss Hamiltonian classes of
symmetric positive Lagrangians in toric Calabi-Yau manifolds
and Milnor fibers. As time permits, we will show as an application that
in these cases, the Riemannian metric induces a metric space structure
on every Hamiltonian isotopy class, and that the obtained metric spaces
can be embedded isometrically in L^2 spaces.
- October 15, Amitai Yuval (Hebrew University),
Title: Geodesics of positive Lagrangians and
almost Calabi-Yau reduced spaces (continued)
- October 22, Hans Joachim Hein (UMD)
Title: Calabi-Yau cones
A Riemannian cone is a warped product space C = (0,infty) x L with
metric g_C = dr^2 + r^2*g_L,
where r denotes the standard coordinate on (0,infty) and (L, g_L) is some given closed Riemannian manifold called
the link or cross-section of the cone. We say that (C, g_C) is a Calabi-Yau cone
if the metric g_C is Ricci-flat Kahler. I will try to explain why people care about
such cones and what you can do with them.
- October 29, Martin Li (Hong Kong)
Title: Simons’ identity and its consequences
Abstract: In a celebrated work of J. Simons in 1968, he discovered a
fundamental identity about the Laplacian of the second fundamental form
of a minimal submanifold. The identity (and
its inequality form) gives curvature estimates for stable minimal
hypersurfaces, which is closed related to the classical Bernstein
theorem and regularity theory of minimal hypersurfaces. On the other
hand, when the ambient space is homogeneous like the round sphere, the
identity gives nice rigidity results about its minimal submanifolds. We will discuss some old and new
results in this aspect and also indicate how this could be related to
the study of free boundary minimal surfaces.
- November 5, David
Hoffman (Stanford)
Title: Limiting behavior of
sequences of embedded minimal disks
Abstract: we prove that it is possible to get families of catenoids as limit leaves of a limit lamination of
embedded minimal disks. We can
also produce sequences whose curvature blows up on any specified closed
subset of the real line. Our method allows us to give another
counterexample to the general Calabi-Yau
conjecture for hyperbolic space, producing a complete and embedded---but
not properly embedded---simply connected minimal surface on either side
of any area-minimizing catenoid in hyperbolic
space. This is joint work with Brian White.
- November 12, John Loftin
(Rutgers),
Title: "Affine Spheres and
the Real Monge-Ampere Equation"
Abstract: Affine differential geometry is the study of differential
invariants of hypersurfaces in R^{n+1} which are invariant under
volume-preserving affine actions on R^{n+1}. We'll define and discuss some of the
basic objects in the theory (affine spheres, affine maximal
hypersurfaces), and their relation to real Monge-Ampere
equations. We'll focus on the
case of hyperbolic affine spheres, and discuss some issues in existence
and regularity of solutions due to Cheng-Yau.
- November 19, Xin Dong (UMD)
Title: Korn's
Inequality and a Theorem on Geometric Rigidity
- December 3, Yakov Shlapentokh-Rothman
(Princeton)
Title: Introduction to the
Black-Hole "No Hair Conjecture"
Abstract: We will introduce and motivate the notion of a black-hole
in general relativity and explain the famous "no hair
conjecture." Next, we will present the classic Carter-Robinson
theory which establishes the conjecture for asymptotically flat,
axisymmetric black-holes with no matter. If time permits, we will end
with a discussion of recent work (joint with Otis Chodosh)
that shows that this conjecture dramatically fails when one adds in even
the very simple matter model of a massive scalar field.
- December 8, Vamsi Pingali (JHU)
Title: A generalised
Monge-Ampere equation
Abstract: A fully nonlinear PDE of the Monge-Ampere
type will be introduced in this talk. Places where it pops up (both
locally and globally) will be mentioned. A few results (existence and a
priori estimates) - both existing and new will be discussed
- March 4, Dmitry
Jacobson (McGill) (note special time)
Title: Nodal sets in conformal geometry.
Abstract: we study conformal invariants that arise from nodal sets
and negative eigenvalues of conformally
covariant operators, which include the Yamabe
and Paneitz operators. We give several
applications to curvature prescription problems. We establish a version
in conformal geometry of Courant's Nodal Domain Theorem. We also show
that on any manifold of dimension at least 3, there exist many metrics
for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number
of negative eigenvalues on any manifold of dimension n>=3. We obtain
similar results for some higher order GJMS operators on some Einstein
and Heisenberg manifolds. This is joint work with Yaiza
Canzani, Rod Gover
and Raphael Ponge. If time permits, we shall
discuss related results for operators on graphs.
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