Organized by: T. Darvas, H.-J. Hein, V.P. Pingali, Y.A. Rubinstein, B. Shiffman, R. Wentworth, S. Wolpert.
The seminar is a combination of a learning and a research seminar. The first 15 minutes or so of each talk are a "trivial notions" talk defining all basic notions, giving examples and intuition to the subject, and should be accessible to a beginning graduate student. The next 50 minutes are a regular seminar talk.
Previous years: 2012-2013, 2013-2014.
September
9, 3 PM, room TBA
(UMD) (note special time!)
Jake
Solomon (Hebrew University, Jerusalem)
Title:
Geometry of the space of positive Lagrangians
Abstract:
A Lagrangian submanifold of a Calabi-Yau manifold is called positive
if the real part of the holomorphic volume form restricted to it is
positive. A Hamiltonian isotopy class of positive Lagrangian
submanifolds admits a Riemannian metric with non-positive curvature.
Its universal cover admits a functional, with critical points
special Lagrangians, that is strictly convex with respect to the
metric. Solutions of the geodesic equation, both smooth (with A.
Yuval) and viscosity (with Y. Rubinstein), will be discussed. Mirror
symmetry relates these phenomena with analogous phenomena for the
space of Hermitian metrics on a holomorphic vector bundle and the
space of Kahler metrics.
October
7, (UMD)
Jesus
Martinez-Garcia (JHU)
Title: Singular Kahler-Einstein metrics
of small angles
Abstract:
The existence of a Kahler-Einstein metric on a Fano variety is
equivalent to the algebro-geometric concept of K-stability. However
K-stability is very difficult to test. For those Fano varieties
which are not K-stable, we can define a singular Kahler-Einstein
metric known as Kahler-Einstein metric with edge singularities,
depending on a parameter \beta\in (0,1]. These metrics also have a
reformulation in terms of log K-stability. It is well known that a
smooth del Pezzo surface admits a Kahler-Einstein metric if and only
if it is not the blow-up of \mathbb P^2 in one or two points.
However they always admit a Kahler-Einstein edge metric. In this
talk, after introducing all these topics, I explain how we can use
birational geometry and log canonical thresholds to find
Kahler-Einstein edge metrics on all del Pezzo surfaces. This is
joint work with Ivan Cheltsov.
October
21, (JHU)
Chengjian
Yao (SUNY, Stony Brook)
Title:
Kahler-Einstein
metrics on singular Fano varieties.
Abstract:
The existence of smooth Kahler-Einstein metrics is equivalent to
being polystable for smooth Fano manifolds. The generalization of
this equivalence to the situation of singular Fano varieties is an
interesting question since it might be used to construct
compactified moduli space for K-polystable/Kahler-Einstein Fano
manifolds as conjectured by Odaka-Spotti-Sun. In this talk, I will
discuss some recent progress about the existence of Kahler-Einstein
metrics on singular Fano varieties which is smoothable and
K-polystable. This is a joint work with Cristiano Spotti and Song
Sun.
October
28, (UMD)
Connor
Mooney(Columbia)
Title: Singular solutions to the Monge-Ampere
equation
Abstract:
Strictly convex solutions to the Monge-Ampere equation \det D^2u = 1
are smooth. However, there are examples of singular solutions, due
to Pogorelov and Caffarelli, that degenerate along line segments. I
will discuss recent optimal estimates for the Hausdorff dimension of
the singular set and applications to the regularity theory for
singular solutions.
November
11, (JHU)
Valentino
Tosatti
(Northwestern
University)
Title: The Kahler-Ricci flow and its
singularities
Abstract:
I will give an introduction to the study of Ricci flow on compact
Kahler manifolds, and explain how its behavior reflects the
structure of the complex manifold. I will then describe a result
(joint with T.Collins) which gives a geometric description of the
set where finite-time singularities occur, answering a conjecture of
Feldman-Ilmanen-Knopf and Campana.
December
2, (UMD)
Heather
Macbeth (Princeton)
Title:
Kaehler-Einstein metrics and higher alpha-invariants
Abstract:
In the most delicate cases, the proof of existence of
Kaehler-Einstein metrics on Fano surfaces M uses the
"alpha-invariants" \alpha_{m,1}(M) and \alpha_{m,2}(M). I
will give a survey of that proof (from 1990 and due to Tian), then
discuss recent work on what can be learned in higher dimensions from
the higher alpha-invariants \alpha_{m,k}(M).
December
9, (UMD)
Richard Wentworth (UMD)
Title:
The Yang-Mills flow on Kaehler manifolds
Abstract:
The fundamental work of Donaldson and Uhlenbeck-Yau proves the the
smooth convergence of the Yang-Mills flow of stable integrable
unitary connections on hermitian vector bundles over Kaehler
manifolds. This was generalized by Bando and Siu to incorporate
certain (singular) hermitian structures on reflexive sheaves.
Bando-Siu also conjectured what happens when the initial sheaf is
unstable; namely, that the limiting behavior should be controlled by
the Harder-Narasimhan filtration of the sheaf. In this talk I will
describe the solution to this question, which draws
on
the work of several authors.
February
3, (UMD)
Bingyuan Liu, (Washington University)
Title:
Recent progresses in automorphism groups
Abstract:
Let O be
a bounded domain of C^n. By a 1935 theorem of Cartan, all
biholomorphisms from O
onto
O form
a (real) finite dimensional Lie group, which is denoted by Aut(O).
When O is
in complex space of one dimension, the study of Aut(O)
is
classical. However, as one considers domains with higher
dimensions,Aut(O)
shows
both similarity and dissimilarity in terms of algebraic and
topological properties comparing with those in one dimension. In
this talk, I will give a short introduction and exhibit several
recent progresses in the geometry of complex domains with
non-compact automorphism groups.
February
11, 4:30 PM, (JHU, Shaffer 100) Note special time and place!
Claire
Voisin, (Jussieu/IAS)
Title:
Coniveau and algebraic cycles on very general complete intersections
Abstract:
There are two notions of coniveau (Hodge and geometric) for smooth
projective varieties, which should be equivalent according to
Grothendieck-Hodge conjecture. On the other hand, the Bloch
conjecture is a prediction that Chow groups of algebraic varieties
are trivial in dimension smaller than the coniveau. We proved this
conjecture for very general complete intersections. The talk will be
mainly devoted to explaining the notions and motivating the
conjecture mentioned above.
March
24, (JHU, Shaffer 302)
Michael Lock (University of
Texas)
Title:
Scalar-flat Kahler ALE metrics on minimal resolution
Abstract:
Scalar-flat Kahler ALE surfaces have been studied in a variety of
settings since the late 1970s. All previously known examples have
group at infinity either cyclic or contained in SU(2). I will
describe an existence result for scalar-flat Kahler ALE metrics with
group at infinity G, where the underlying space is the minimal
resolution of C^2/G, for all finite subgroups G of U(2) which act
freely on S^3. I will also discuss a non-existence result for
Ricci-flat metrics on certain spaces, which is related to a
conjecture of Bando-Kasue-Nakajima.
March
31, (JHU)
Dror Varolin (Stony Brook)
Title:
Two Bergman-type interpolation problems on finite Riemann
surfaces
Abstract:
Let X be an open Riemann surface with a Hermitian metric and a
weight function (i.e. non-trivial metric for the trivial line
bundle). Given a closed discrete subset G in X, the above data
defines a Bergman space on X and a Hilbert space on G (in a standard
way). We say that G is an interpolation set if the restriction map
from the Bergman space on X to the Hilbert space on G is surjective.
The interpolation problem consists in characterizing all
interpolation sets. When X is the complement of a finite set in a
compact Riemann surface (i.e., a compact Riemann surface with some
finite number of punctures), the metric g is flat outside some
compact subset of X, and the curvature of the weight satisfies
certain positivity and boundedness conditions, we give a complete
solution to the interpolation problem. We then turn our attention to
more general bordered Riemann surfaces with finitely many punctures.
We equip these with the unique metric of constant negative curvature
-4, and point out that the same Bergman interpolation problem
discussed above does not have a reasonable solution. We therefore
modify the problem so that it doesn't change in the asymptotically
flat case, but has a reasonable solution in the hyperbolic case.
Finally, we give a complete characterization of interpolation sets
for this modified problem.
April
7, (4:30 PM, JHU)
Ved Datar, (Notre Dame)
Title:
Connecting toric manifolds by conical Kahler-Einstein metrics.
Abstract:
Conical Kahler-Einstein metrics have played an important role in the
recent breakthrough on the existence of smooth Kahler-Einstein
metrics on Fano manifolds. In this talk, I will first show that a
log-Fano toric pair has conical KE metric if and only if the
barycenter of the corresponding moment polytope is zero. This
confirms the log version of the Yau-Tian-Donaldson conjecture for
toric pairs, and extends a fundamental result of Wang and Zhu. I
will then show that any two toric manifolds of the same dimension
can be connected by a continuous family of toric manifolds paired
with conical Kahler-Einstein metrics in the Gromov-Hausdorff
topology. This is joint work with Bin Guo, Jian Song and Xiaowei
Wang.
April
7, (5:30 PM, JHU)
Long Li, (McMaster)
Title:
On the convexity of the Mabuchi energy functional along
geodesics
Abstract:
It is conjectured by X.X. Chen that the Mabuchi energy functional is
convex along the geodesic connecting two Kaehler metrics, during his
study in uniqueness of csck metrics. Now we can give an affirmative
answer to this question in the joint work with X.X. Chen and Mihai
Paun. The first breakthrough in this subject is the work by Berman
and Berndtsson last year, where they proved the weak convexity of
the Mabuchi energy functional based on the log-subharmonicity of
Bergman kernels. Our work is somewhat a "global version"
of Bergman kernels approximation, and also completes the conjecture
by proving the continuity of the Mabuchi energy functional along the
geodesic. Finally, we also calculated the almost convexity of the
Mabuchi energy functional along the \ep-approximation geodesics.
April
21, (JHU)
Leon
Takhtajan
(Stony
Brook)
Title:
On Bott-Chern forms of Hermitian holomorphic vector
bundles
Abstract:
I will discuss a method for explicit computation of Bott-Chern forms
of holomorphic Hermitian vector bundles over a complex manifold.
After reviewing the basic properties of Chern and Bott-Chern forms,
I will describe descent and ascent equations for explicitly
computing these differential forms.
Driving
directions to JHU. Park in South Garage (see map)
on any level (except the reserved spaces). Take a ticket when
entering. The Department will provide a visitor parking pass to use
when exiting.
Driving
and parking directions to UMD. Park in Paint Branch Drive Visitor
Lot (highlighted in yellow in the lower right corner of the second
map in the previous link), or in Regents Drive Garage (highlighted in
the upper right corner). If you arrive after 4pm you do not need to
pay: see the instructions in the previous link.