Colloquia in the current semester
[colloque] Tue 3.10.2017, Phys 2.52 at 17h15 | more |
Prof. Manfred Trummer (Simon Fraser University Vancouver): Reconstruction of Dynamic SPECT Images
Single photon emission computed tomography (SPECT) is a diagnostic functional imaging modality wherein the distribution of a radioactive tracer inside the body is estimated based on data acquired from around the patient by a slowly rotating camera. Conventional SPECT image reconstruction assumes that this distribution remains constant during acquisition.
In this talk we investigate imaging of a time-varying distribution of radiotracer, which results in a highly underdetermined reconstruction problem. Recovery of an accurate dynamic image from this data requires the use of additional constraints, including temporal regularization. We use simple inequality constraints to restrict the temporal behaviour of the reconstructed image. Since dynamic tracer behavior in the human body arises as a result of continuous physiological processes, changes in tracer concentration should follow a smooth time activity curve (TAC). Our algorithm promotes smoothness by constraining the concavity of the TAC in every voxel of the reconstructed image. Digital phantom simulations show that the algorithm yields more accurate images, with smoother, more consistent TACs within dynamic regions of interest.
We will also describe some recent results obtained with a pinhole camera which allows for data acquisition from all directions simultaneously. In this case we again obtain much smoother TACs.
[colloque] Tue 17.10.2017, Phys 2.52 at 17h15 | more |
Prof. Jamal Najim (Université de Marne La Vallée): Large Random Matrices of Long Memory Stationary Processes: Asymptotics and fluctuations of the largest eigenvalue
Given $n$ i.i.d. samples $(\boldsymbol{\vec x}_1, \cdots, \boldsymbol{\vec x}_n)$ of a $N$-dimensional long memory stationary process, it has recently been proved that the limiting spectral distribution of the sample covariance matrix,
$$
\frac 1n \sum_{i=1}^n \boldsymbol{\vec x}_i \boldsymbol{\vec x}^*_i
$$
has an unbounded support as $N,n\to \infty$ and $\frac Nn\to c\in (0,\infty)$. As a consequence, its largest eigenvalue
$$
\lambda_{\max} \left( \frac 1n \sum_{i=1}^n \boldsymbol{\vec x}_i \boldsymbol{\vec x}^*_i
\right)
$$
goes to infinity. In this talk, we will describe its asymptotics and fluctuations, tightly related to the features of the underlying population covariance matrix, which is of a Toeplitz nature.
This is a joint work with Peng Tian and Florence Merlevčde.
[colloque] Tue 24.10.2017, Phys 2.52 at 17h15 | more |
Prof. Omid Amini (CNRS - Ecole Normale Supérieure Paris): Graph theory in arithmetic and algebraic geometry
Despite their simplicity of nature, graphs are among the richest mathematical objects whose links to other branches of mathematics constantly surprise.
In this talk, I will aim to explain how some concepts which have shaped the development of modern graph theory appear naturally in algebraic geometry, in the study of the space of solutions of systems of polynomial equations. I will discuss two
themes: geometric universality of graphs and control of arithmetic points of algebraic curves.
[colloque] Tue 7.11.2017, Phys 2.52 at 17h15 | more |
Prof. Christine Bachoc (Bordeaux): Sets avoiding the distance 1 in $\mathbb R^n$
What is the supremum density of a measurable set in $\mathbb R^n$ avoiding distance 1?
If the distance is the Euclidean distance, the answer is known only in
the trivial case n = 1. This problem is closely related to that of the
determination of the chromatic number of the Euclidean space, a
surprisingly difficult problem even in dimension 2, which is open
since it was posed by Nelson and Hadwiger in 1950. We will discuss
recent results obtained on this question and also on several variants
that rely on a combination of methods from convex optimization and
Fourier analysis.
In one of these variants, one replaces the Euclidean norm by a norm defined
by a convex symmetric polytope. When the polytope tiles the space by
translations, we conjecture that the answer is $\mathbb 1/2^n$ and that the
proof should be much easier that in the Euclidean setting.
We will present a proof of this conjecture in dimension 2 and discuss
a few other cases of Voronoi polytopes associated to lattices.
[colloque] Tue 14.11.2017, Phys 2.52 at 17h15 | more |
Prof. Yohei Komori (Waseda Univ, Tokyo): Growth functions of hyperbolic groups
Let $G$ be a finitely generated group and $S$ be a set of generators.
The growth rate of metric spheres of the Cayley graph of G with respect to S is one of the central topics in geometric group theory.
In this talk I will give an overview of recent progress on arithmetic aspects of growth rates of
hyperbolic groups, focussing mainly on hyperbolic Coxeter groups.
[colloque] Tue 28.11.2017, Phys 2.52 at 17h15 | more |
Prof. Urs Lang (ETHZ): Higher rank hyperbolicity in spaces of nonpositive curvature
The large scale geometry of Gromov hyperbolic metric spaces exhibits many distinctive
features, such as the stability of quasi-geodesics (the Morse Lemma), the linear isoperimetric filling
inequality for 1-cycles, the visibility property, and the homeomorphism between visual boundaries
induced by a quasi-isometry. After briefly reviewing these properties, I will describe a number of closely
analogous results for spaces of rank n > 1 in an asymptotic sense, under some weak assumptions
reminiscent of non-positive curvature. A central role is played by a suitable class of n-dimensional
surfaces of polynomial growth of order n, which serve as a substitute for quasi-geodesics.
[colloque] Tue 5.12.2017, Phys 2.52 at 17h15 | more |
Prof. Valentin Féray (Uni Zürich): What do random constraint permutations look like?
We will present a notion of limit of permutations,recently introduced by Hoppen et al.
The limiting objects, called permutons, describe the scaling limit of large random permutations.
We will consider various probability distributions on the set
of permutations of a given size
and present convergence results under these distributions in the sense of permutons (when the size tends to infinity).
In some family of examples, the limiting permutons are connected to the Brownian (or stable) excursions.
[colloque] Tue 12.12.2017, Phys 2.52 at 17h15 | more |
CANCELLED - Prof. Maryna Viazovska (EPFL): The sphere packing problem in dimensions 8 and 24
The sphere packing problem is to find an arrangement of non-overlapping unit spheres in the $d$-dimensional Euclidean space in which the spheres fill as large a proportion of the space as possible. In this talk we will present a solution of the sphere packing problem in dimensions 8 and 24. In 2003 N. Elkies and H. Cohn proved that the existence of a real function satisfying certain constrains leads to an upper bound for the sphere packing constant. Using this method they obtained almost sharp estimates in dimensions 8 and 24. We will show that functions providing exact bounds can be constructed explicitly as certain integral transforms of modular forms. Therefore, the sphere packing problem in dimensions 8 and 24 is solved by a linear programming method.
Other talks and events in the current semester
[oberseminar] Oberseminar Topologie, Mon 18.09.2017, Math II (Lonza) at 16h00 | more |
David González Álvaro (Fribourg): Non-negative sectional curvature on stable classes of vector bundles
We will discuss the following question, motivated by Cheeger-Gromoll's Soul Theorem: given a vector bundle E over a compact manifold, does the product E\times R^k admit a metric with non-negative sectional curvature for some k? We will give an affirmative answer for every vector bundle over (almost) any homogeneous space with positive curvature. We will extend this result to include further classes of homogeneous spaces, and we will show that the question above is stable under tangential homotopy equivalences. This is joint work with Marcus Zibrowius.
[oberseminar] Wed 27.09.2017, Math II (Lonza) at 10h20 | more |
Grigory Ivanov (EPFL): The cross-polytope and the cube, their sections and projections
We will speak about different bounds on the volume of a projection of the n-dimensional cross-polytope and a central section of the n-dimensional cube. There are several results about central sections of the cube. However, the dual problems are still open.
For instance, the celebrated Vaaler theorem asserts that the volume of a k-dimensional central section of the n-cube is greater or equal to the volume of the k-cube. But it is not known whether the volume of a k-dimensional projection of the n-dimensional cross-polytope is less or equal to the volume of the k-dimensional cross-polytope.
We will introduce a linear algebra approach to these problems which allows us to find some optimal bounds for the cross-polytope and the cube at the same time. Also we will discuss current view on the problems and formulate some open questions.
[oberseminar] Oberseminar Topologie, Mon 02.10.2017, Math II (Lonza) at 16h00 | more |
Anand Dessai (Fribourg): Topology of moduli spaces of nonnegative
curved metrics and eta-invariants
We will discuss results concerning the topology of moduli spaces of metrics of
nonnegative curvature on closed manifolds. All known results are confined
to manifolds of dimension (4k-1). We will explain an approach using eta-invariants
to study the moduli space for manifolds in other dimensions.
[oberseminar] Wed 04.10.2017, Math II (Lonza) at 10h20 | more |
Stephan Klaus (MFO): Dihedral angles and combinatorial Gauss-Bonnet Theorem
The classical Gauss-Bonnet Theorem in differential geometry connects the integrated Pfaffian curvature with the Euler characteristics of a manifold. We introduce a combinatorial version which is more general as it holds for general euclidean simplicial complexes. Special cases include tringulated manifolds, also with boundary, or combinatorial pseudo-manifolds. At the same time the proof is very simple and straightforward. Higher dihedral angles play a crucial role and we will recapitulate their properties.
[oberseminar] Wed 11.10.2017, Math II (Lonza) at 10:20 | more |
Yuri Suris (TU Berlin): On a discretization of confocal quadrics
We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system may be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. Various sequences correspond to various discrete parametrizations. The theory is illustrated with a variety of examples in two and three dimensions. A connection with incircular (IC) nets is established.
[mathematikon] Tue 24.10.2017, Phys 2.52 at 12h15 | more |
Ivan Izmestiev: Schliessungsätze
Please find the abstract here.
[oberseminar] Oberseminar Topologie, Mon 30.10.2017, Math II (Lonza) at 16h00 | more |
Michael Joachim (Münster): Twisted spin^c bordism and twisted K-homology
In our talk we present a twisted analogue of a result of Hopkins and Hovey who show that the functor which associates to a space $X$ the graded abelian group $\Omega^{spin}_{*}(X) \otimes_{\Omega^{spin}_{*}} KO_{*}(pt)$ yields a geometric description of $KO_{*}(X)$. Our analogue for twisted $K$-theory also gives further inside to a Brown-Douglas approach to twisted $K$-homology.
The results are joint work with Baum, Khorami and Schick.
[oberseminar] Oberseminar Topologie, Mon 27.11.2017, Math II (Lonza) at 16h15 | more |
Masoumeh Zarei (Peking Uni.): Equivariant classification of cohomogeneity one Alexandrov spaces in low dimension
In this talk, I will give an equivariant classification of cohomogeneity one Alexandrov spaces in dimensions 5; 6 and 7. As a result, we show that all orbifolds appeared in the classification are equivariantly homeomorphic to a smooth good orbifold of cohomogeneity one. Furthermore, I will discuss a characterization of cohomogeneity one Alexandrov spaces to be topological manifolds. In particular, we prove that such spaces are homeomorphic to smooth manifolds in low dimensions. This is a joint work with Fernando Galaz-Garcia.
[oberseminar] Oberseminar Geometrie, Wed 29.11.2017, Math II (Lonza) at 10h20 | more |
Plinio G. P. Murillo (Bern): Hyperbolic manifolds with large systole
See here
[oberseminar] Oberseminar Topologie, Mon 04.12.2017, Math II (Lonza) at 16h00 | more |
Michael Wiemeler (Münster): On the topology of moduli spaces of non-negative Ricci-curved metrics
By now there are a lot of examples of manifolds for which the space or moduli space of positive scalar curved metrics has non-trivial topology. There are much less examples where one has non-trivial topology for the space or moduli space of positive or non-negative Ricci-curved metrics. In this talk I will report on on-going joint work with Wilderich Tuschmann on the topology of moduli spaces of non-negative Ricci curved metrics on closed non-simply connected manifolds. We give examples of manifolds for which the fundamental group or the higher rational cohomology groups of these spaces are non-trivial.
[oberseminar] Oberseminar Geometrie, Wed 06.12.2017, Math II (Lonza) at 10h20 | more |
David González Álvaro (Fribourg): Manifolds with positive sectional curvature
See here
[oberseminar] Oberseminar Geometrie, Wed 13.12.2017, Math II (Lonza) at 10h20 | more |
Florian Besau (Frankfurt): Floating bodies in spherical and hyperbolic geometry
Imagine for a moment a solid body floating in water. From a physics point of view, Archimedes' principle states that the buoyant force acting on the body in an upward direction is equal to the weight of the water displaced by it. In other words, the body is floating if the weight of the
body is equal to the weight of the water displaced.
Now let the body roll around. If it is light enough, then there is a part inside the body that will always be above the water surface. This kernel is the (convex) floating body . To be more precise, for a compact
convex set with non-empty interior, we may define the floating body as the subset that is obtained by cutting away all caps of volume equal to a given positive constant δ .
This classic construction goes back to C. Dupin in the 19th century and one very remarkable fact is, that the floating body is an affine notion, that is, it behaves covariant with respect to affine transformations.
The limit, as δ goes to zero, has been of particular interest in differential geometry, because the derivative of the volume difference between the body and its floating body gives rise to the classical (equi-)affne invariant surface area. This affine surface area was introduced by Blascke in the 1920s for smooth convex bodies in dimension two and three and an extension to all convex bodies in all dimensions was established by Schütt and Werner in 1990 using the (convex) floating body.
We recently started investigating the floating body in spherical and hyperbolic space and in this talk I will give a brief overview on our results and open questions that still remain.
[oberseminar]
Oberseminar Topologie, Mon 18.12.2017, Math II (Lonza) at 16h00
| more |
Manuel Amann (Augburg): Orbifolds with all geodesics closed
The concept of a Riemannian orbifold generalises the one of a Riemannian
manifold by permitting certain singularities. In particular, one is able to
speak about several concepts known from classical Riemannian geometry
including geodesics. Whenever all geodesics can be extended for infinite time
and are all periodic, the orbifold is called a Besse orbifold—in analogy to
Besse manifolds. A classical result in the simply-connected manifold case
states that in odd dimensions only spheres may arise as examples of Besse
manifolds.
In this talk we shall illustrate that the same holds for Besse orbifolds,
namely that they are actually already manifolds whence they are spheres.
The talk is based on joint work in progress with Christian Lange and Marco
Radeschi.
[oberseminar] Oberseminar Geometrie, Wed 20.12.2017, Math II (Lonza) at 10h20 | more |
Roman Prosanov (Fribourg): Covering problems in discrete geometry
Let X and Y be subsets of some geometric space. What is the least possible k such that we can cover X using k copies of Y? (If X can not be covered with a finite numbers of copies of Y then we can introduce and investigate the {\it density} of covering.) This is the most basic scheme for covering problems in discrete geometry. These problems are important for different applications. The richest theory was developed in the case when the ``geometric'' space is R^n or Euclidean torus and copies are translates of Y.
The main milestone was the theorem of Rogers which can be reformulated as follows.
{\bf Theorem.} Let T^n be an n-dimensional Euclidean torus and K \subset T^n be a convex body. Define the relative volume \rho(K) = \frac{{\rm vol}(K)}{{\rm vol}(T^n)}. Then it is possible to cover T^n using no more than \rho(K)^{-1}(n \ln n + n \ln \ln n + 5n) translates of K.
This result had a big impact on discrete geometry and was reproved several times. We will discuss the recent approach proposed in 2015 by M. Nasz\'odi. It uses fractional coverings in hypergraphs. We will also speak about its possible perspectives to other problems.