Abstracts
- Teresa Arias-Marco: What is a weakly-Einstein manifold of order
$k$?
Abstract. Weakly Einstein Riemannian manifolds were defined on any
dimension by Y. Euh, J. Park and K. Sekigawa [A curvature identity on a
4-dimensional Riemannian manifold. Results Math. 63 (2013)
107–114]. They did so as an application of a curvature identity obtained
using the generalized Gauss-Bonnet formula for compact and oriented Riemannian
manifolds of dimension four.
The talk will show the known results related to this type of manifolds. In
particular, their relation with Einstein manifolds and its 4-dimensional
classification in the homogeneous case [T. Arias-Marco, O. Kowalski.
Classification of 4-dimensional homogeneous weakly Einstein manifolds.
Czechoslovak Math. J. 65 (2015) 21–59.]
I deeply thank Prof. O. Kowalski for introducing me to this interesting
research line.
- Romina M. Arroyo: The long-time behaviour of the
pluriclosed flow on Lie groups.
Abstract. The pluriclosed flow is a geometric flow that evolves
pluriclosed Hermitian structures (i.e. Hermitian structures for which its
$2$-fundamental form satisfies $\partial \bar \partial \omega =0$) in a given
complex manifold. The aim of this talk is to discuss the asymptotic behaviour
of the pluriclosed flow in the case of left-invariant structures on Lie groups.
More precisely, invariant structures on nilmanifolds and solvmanifolds. We will
analyze the flow and explain how a suitable normalization converges to
pluriclosed solitons, which are self-similar solutions to the flow.
This is a joint work with Ramiro Lafuente (University of Münster).
- Renato Bettiol: Cohomogeneity one Ricci Flow and
Nonnegative Sectional Curvature
Abstract.
We show that some compact 4-manifolds, including $S^4$ and $CP^2$, admit
metrics with nonnegative sectional curvature that immediately lose
this property when evolved via Ricci flow. This behavior, which
indicates certain limitations of Ricci flow beyond dimension 3 (where
nonnegative sectional curvature is preserved), was previously known to
happen only in dimensions $>$ 5 or in noncompact manifolds. Such new
examples stem from studying this evolution equation on manifolds with
isometric cohomogeneity one actions, where it reduces to a system of
PDEs in 2 variables. This is based on joint work with A. Krishnan.
- Jonathan Epstein: Topological Entropy of Left-Invariant
Magnetic Flows on 2-Step Nilmanifolds
Abstract.
We consider magnetic flows on 2-step nilmanifolds $M = \Gamma \backslash G$,
where the Riemannian metric $g$ and the magnetic field $\sigma$ are
left-invariant. These are Hamiltonian flows on the cotangent bundle $T^*M$,
where the standard symplectic structure is twisted by the pullback of the
magnetic field. Such twisted symplectic structure can be realized as the
reduction of the cotangent bundle of an associated nilmanifold of one dimension
more with its canonical symplectic structure. We use this to study the
topological entropy of these magnetic flows. When $\sigma$ represents a
rational cohomology class and its restriction to $\mathfrak{g} = T_eG$ vanishes
on the derived algebra, then the associated magnetic flow has zero topological
entropy. In particular, this is the case when $\sigma$ represents a rational
cohomology class and is exact. In addition, there exist magnetic flows on a
2-step nilmanifolds with positive topological entropy on arbitrarily high
energy levels. Lastly, we discuss the relationship between topological entropy
on different energy levels and the Mañé critical value.
- Anna Fino: $G_2$-structures and Ricci solitons
Abstract. In this talk we present some general results about
$G_2$-structures whose underlying Riemannian metric is Einstein, as well
recent results on the existence of left invariant closed and co-closed $G_2$
forms determining a Ricci soliton metric on nilpotent Lie groups.
For these structures, we will also show some results related to the
Laplacian flow and co-flow.
- Lee Kennard: Torus actions and positive curvature
Abstract. It is an open problem to classify smooth manifolds that admit
Riemannian metrics with positive or non-negative sectional curvature. In fact,
the question of whether these two classes are the same is open for closed,
simply connected manifolds. In this talk, I will discuss work on this problem
in the presence of symmetry. I will focus on simplest case of torus symmetry
and discuss recent classification results in this context.
- Lee Kennard: PUBLIC LECTURE. A beautiful, living
formula
Abstract. In 1750, the famous mathematician Leonard Euler wrote down a
simple yet powerful formula that now bears his name: $V - E + F = 2$. Like
Einstein's famous equation, $E = mc^2$, it involves three letters and the
number two, and it is also surprising. One difference is that Euler's formula
is actually quite easy to prove.
In the centuries since it was first written down, meditations on the formula
have led to vast generalizations and new fields of mathematics. It has also
seen incredible applications, both to our physical world and to the
mathematical sphere that stretches beyond it's borders.
Other applications are less important and yet irresistible in a lecture such as
this. I hope you will join us.
- Ramiro Lafuente: The Ricci flow on solvmanifolds of real
type
Abstract. In this talk we will introduce solvable Lie groups of “real
type”, also known as “almost completely solvable”. After explaining their
geometric relevance, we will give a characterization of such spaces in terms of
the behavior of homogeneous Ricci flows. We will then show that any
appropriately rescaled homogeneous Ricci flow on such a space converges to a
solvsoliton in Cheeger-Gromov topology. Moreover, the limit soliton is unique,
and it does not depend on the initial metric. In the case of an Einstein limit,
the convergence can be improved to $C^infty$.
This is joint work with Christoph Böhm.
- Jorge Lauret: SURVEY LECTURE. Algebraic Solitons
Abstract. The following condition for a left-invariant metric on a
nilpotent Lie group has been studied by many mathematicians since 1998: the
Ricci operator is a multiple of the identity modulo a derivation. The reasons
to be interested in this property were many. On one hand, the condition nicely
combines geometric and algebraic aspects of the metric and holds for some known
distinguished metrics like H-type and naturally reductive. On the other hand,
it clearly produces an Einstein metric on the corresponding solvable extension
and Heber's proof of uniqueness for Einstein metrics on solvable Lie groups
worked perfectly for the metrics satisfying the property. However, nowadays,
perhaps the main feature of such metrics is that they are Ricci solitons.
An algebraic soliton is an invariant metric on a homogeneous space for which
the natural generalization of the above condition holds. The concept goes
beyond Riemannian geometry and includes geometric structures and different
kinds of geometric flows in complex, symplectic and $G_2$ geometry. We will
survey in this talk on the role played by algebraic solitons in providing
canonical geometric structures on Lie groups, in the existence problem of
solitons for different flows as well as in their structure and classification.
- Jorge Lauret: Ricci negative solvmanifolds and the
convexity properties of the moment map
Abstract. We will report on some work in progress on the following
questions:
Which nilpotent Lie algebras have a derivation such that the corresponding
solvable extension admits a Ricci negative left-invariant metric?
Given a nilpotent Lie algebra endowed with a basis, what kind of set is the
cone of all diagonal derivations such that the corresponding solvable extension
admits a Ricci negative left-invariant metric? Is it convex? Is it open in
the space of all diagonal derivations?
This is joint work with Jonas Derè.
- Ricardo Mendes: Minimal hypersurfaces in compact symmetric
spaces
Abstract. A conjecture attributed to R. Schoen states that if (M,g) is
a compact
Riemannian manifold with positive Ricci curvature, then there exists
$C>0$ such that any closed minimal hypersurface $S$ satisfies
index$(S)>C.b_1(S)$. Here $b_1$ denotes the first Betti number, and
index$(S)$ denotes the Morse index of $S$ for the area functional. In
previous work (by Ros, Savo, Ambrozio-Carlotto-Sharp, …) this
conjecture has been established for $(M,g)$ any compact rank-one
symmetric space (CROSS), with $g$ the standard metric. Moreover a weak
version has been proved for (flat) tori, “weak” in the sense the index
is replaced with the index plus nullity.
Our main result is such a weak version of Schoen's conjecture valid
for any compact symmetric space. If the rank is one we also recover
the strong version, albeit with a worse constant $C$ than in the
previous results.
The main new tool used in our proof is a generalization of isometric
immersions of $(M,g)$ into Euclidean space. Namely, we consider
isometric embeddings of the tangent bundle TM into a trivial vector
bundle $MxR^n$, such that the standard connection on $MxR^n$ induces the
Levi-Civita connection of $M$. The fundamentals of the theory of
isometric immersions generalize in similar form, with the important
caveat that the second fundamental form is no longer necessarily
symmetric.
If time allows I will also discuss cases where a robust version of
such index bounds holds, in the sense that there is a constant $C'$, and
a neighbourhood $U$ of the metric $g$, such that for every metric $g'$ in $U$,
and every closed minimal hypersurface $S$ of $(M,g')$, one has
index$(S)>C'.b_1(S)$. These include the CROSS, the Lie groups $Sp(n)$, and
the quaternionic Grassmannians.
- Yuri Nikolayevsky: Two topics on homogeneous geodesics
Abstract. This talk has two parts; we will look at two independent, but
interrelated topics on homogeneous geodesics. In the first part (joint work
with Yurii Nikonorov), we classify all the geodesic orbit metrics on the
Ledger-Obata spaces, the homogeneous spaces $F^m/diag(F)$, where $F$ is a
compact simple group; we will show that almost such metrics are naturally
reductive. In the second part (joint work with Dmitry Alekseevsky), we consider
the stability of homogeneous geodesics in the sense of Arnol'd. The main
question, to which we know only a partial answer is this: any homogeneous space
admits a homogeneous geodesic (some number of them, in many cases), but is it
true that there is always a stable homogeneous geodesic?
- Marcos Origlia: Locally conformal Kähler or symplectic
structures on compact solvmanifolds
Abstract. We consider locally conformal Kähler (LCK) manifolds, that
is, a Hermitian manifold $(M,J,g)$ such that on each point there exists a
neighborhood where the metric is conformal to a Kähler metric. Equivalently,
$(M,J,g)$ is LCK if and only if there exists a closed $1$-form $\theta$ such
that $d\omega=\theta\wedge\omega$, where $\omega$ is the fundamental $2$-form
determined by the Hermitian structure. The $1$-form $\theta$ is called the Lee
form.
On the other hand, the concept of LCK structure can be generalized to the
notion of locally conformal symplectic (LCS) structure, that is a pair
$(\omega, \theta)$ satisfying $d\omega=\theta\wedge\omega$, where $\omega$ is a
non-degenerate $2$-form and $\theta$ is a closed $1$-form.
In this work we study left invariant LCK or LCS structures on solvable Lie
groups and the existence of lattices (co-compact discrete subgroups) on these
Lie groups in order to obtain compact solvmanifolds equipped with these kind of
locally conformal geometric structures.
- Tracy Payne: The Structure of H-like Metric Lie
Algebras
Abstract. We will discuss the structure of H-like metric Lie algebras.
We characterize these algebras in terms of subspaces of cones
over certain $GL_q(\bold{R})$ orbits in $\frak{so}(\bold{R}^q),$ showing
that the classification problem for H-like metric nilpotent Lie
algebras is equivalent to a difficult type of problem in algebraic
geometry that has already been studied. We describe properties of H-like
metric Lie
algebras and present new methods for constructing them. We classify
H-like metric Lie algebras with the property that the associated
$J_Z$-maps have rank two for all nonzero $Z.$ This is joint work with
Cathy Kriloff.
- Benjamin Schmidt: Real projective spaces with all
geodesics closed
Abstract. The study of manifolds with all geodesics closed is classical
in Riemannian geometry. Besides the symmetric metrics on the compact rank one
symmetric spaces and their locally symmetric Riemannian quotients, all
remaining known examples consist of non-symmetric metrics on spheres in each
dimension.
After some historical discussion, I'll describe a new result proved jointly
with Samuel Lin, showing that the only Riemannian metrics with all geodesics
closed on real projective spaces are the constant curvature metrics, except
possibly in dimension three.
- Catherine Searle: The maximal symmetry rank conjecture for
non-negatively curved manifolds
Abstract. I'll talk about joint work with Christine Escher related to
the conjecture, which states:
Maximal Symmetry Rank Conjecture. Let $T^k$ act isometrically and
effectively on
$M^n$, a closed, simply-connected, non-negatively curved Riemannian manifold.
Then
(1) $k\leq \lfloor 2n/3\rfloor$;
(2) When $k= \lfloor 2n/3\rfloor$,
$M^n$ is equivariantly diffeomorphic to $$Z= \prod_{i\leq r} S^{2n_i+1}
\times\prod_{i>r} S^{2n_i}, \, \, \, \, {\textrm{ with }} r= 2\lfloor
2n/3\rfloor-n,$$ or the quotient of $Z$ by a free linear action of a torus of
rank less than or equal to $2n$ mod $3$.
- Craig Sutton: Detecting the Moments of Inertia of a
Molecule via its Rotational Spectrum
Abstract. Spectral geometry has connections with the field of
spectroscopy where one is interested in recovering the structure and
composition of a molecule or compound from various spectral data. We
demonstrate that the moments of inertia of a molecule can be recovered from its
rotational spectrum. Geometrically speaking this means that the isometry
classes of left-invariant metrics on $\operatorname{SO}(3)$ can be mutually
distinguished via their spectra. In fact, they can be distinguished by their
first four heat invariants. More generally, we demonstrate that among compact
homogeneous three-manifolds a non-trivial isospectral pair must consist of
spherical three-manifolds possessing non-isomorphic cyclic fundamental groups
and each is equipped with a so-called Type I metric: at present, no such
isospectral pairs exist in the literature. This is joint work with Ben Schmidt.
- Hiroshi Tamaru: Left-invariant metrics and submanifold
geometry
Abstract. For a left-invariant metric on a given Lie group,
we can construct a submanifold,
where the ambient space is the space of all left-invariant metrics on that Lie
group.
We expect that nice left-invariant metrics (such as Einstein or Ricci soliton)
are corresponding to nice submanifolds.
In this talk, we introduce our framework,
and mention some results related to such correspondence.
- Cynthia Will: Lie groups with negative Ricci curvature
Abstract. In this talk we will be interested in Lie groups admitting a
metric with negative Ricci curvature. Although a great progress has been made
lately, specially in the solvable case, the general case seems to be far from
being completely understood. We will begin by introducing the known results and
then we will show many new families of examples. Some of them have compact Levi
factor, which is unexpected in some sense, and provide new topologies. On the
opposite side, we will exhibit a family of examples where the Levy factor is
almost any semisimple Lie algebra with no compact factors.
- Dmytro Yeroshkin: Group Actions on Manifolds with
Density
Abstract. Riemannian manifolds with measure, and more general metric
spaces with measure have been studied under different guises for decades. In
this talk I'll present a new geometric approach to these spaces that introduces
a torsion free affine connection, as opposed to a measure, as the fundamental
object of study. This approach allows us to provide several results relating to
group actions on manifolds with density, such as maximal and almost maximal
symmetry ranks classifications as well as results on fixed point homogeneity
for manifolds with positive weighted sectional curvature. This is joint work
with Lee Kennard and Will Wylie.
- Wolfgang Ziller: SURVEY LECTURE. Manifolds with
nonnegative or positive curvature under the presence of symmetry
Abstract. We give a survey about examples and obstructions to the
existence of metrics with nonnegative or positive curvature, under the
assumption that the isometry group of the metric is “sufficiently large”.
- Wolfgang Ziller: Finsler metrics with constant
curvature
Abstract. In the realm of Finsler metrics there is a surprisingly rich
class of metrics with constant (flag) curvature. We study the geodesic flow of
such metrics and show that the conjugacy class of the flow only depends on the
length of the shortest closed geodesic. We will also discuss Frankel's theorem
for Finsler metrics.