Colloquium program from August 2016 to January 2017
[colloque] Tue 20.9.2016, Phys 2.52 at 17h15 | more |
Prof. Daniel Grieser (Universität Oldenburg): A look at triangles from elementary geometry, spectral theory and singular analysis
Is there anything new to be said about Euclidean triangles in the 21st century? Yes!
I will present three recent results: The first one could be formulated as a problem in high school, but its solution requires higher analysis. The second result uses the first one to answer the question ‚Can you hear the shape of a triangle?‘ — this is about the spectral theory of the Laplace operator. The third result concerns the asymptotics of solutions of PDEs on ‚thin‘ triangles. It gives a new perspective on the second result, and its proof uses modern geometric methods of singular analysis.
[colloque] Tue 27.9.2016, Phys 2.52 at 17h15 | more |
Prof. Christoph Böhm (Universität Münster): Immortal homogeneous Ricci flows
We show that immortal homogeneous Ricci flow solutions subconverge (after suitable
rescalings) to a homogeneous expanding soliton. A key step in the proof of this result
is the construction of a new Lyapunov function, using methods from geometric invariant theory.
Then, several applications to homogeneous Ricci flow solutions on solvable Lie groups will be given.
[colloque] Tue 11.10.2016, Phys 2.52 at 17h15 | more |
Prof. Gerhard Opfer (Universität Hamburg): Zeros of unilateral quaternionic and coquaternionic
polynomials
Let ${\cal A}$ be a finite dimensional algebra over the reals.
For ${\cal A}$ we will consider
$\mathbb{H}$ (quaternions),
$\mathbb{H}_{\rm coq}$ (coquaternions),
$\mathbb{H}_{\rm nec}$ (nectarines),
and $\mathbb{H}_{\rm con}$ (conectarines),
and study the possibility of finding the zeros of unilateral polynomials
over these algebras, which are the four noncommutative algebras in~$\mathbb{R}^4$.
A polynomial $p$ will be defined by
$$p(z):=\sum_{j=0}^n a_jz^j,\quad a_j,z\in {\cal A},$$
and for finding the zeros we use of the so-called {\it companion polynomial}, which has real coefficients,
and is defined by
$$q(z):=\sum_{j,k=0}^n \overline{a_j}a_kz^{j+k}=\sum_{\ell=0}^{2n}b_\ell z^\ell \Rightarrow b_\ell\in\mathbb{R}.$$
See D. Janovsk\'a \& G. O.: SIAM J. Numer. Anal. {\bf48} (2010), 244-256,
for quaternionic polynomials and
ETNA {\bf 41} (2014), 133-158 for coquaternionic polynomials.
The real or complex roots of the companion polynomial $q$ will provide information on similarity classes which contain zeros of $p$.
It will be shown, that the companion polynomial $q$ has more capacity than formerly described in our papers, valid in all
noncommutative algebras of $\mathbb{R}^4$. There will be numerical examples.
[colloque] Tue, 25.10.2016, Phys 2.52 at 17h15 | more |
Dr. Wim Hordijk (Konrad Lorenz Institut, Klosterneuburg): Autocatalytic Sets:
The Origin and Organization of Life
Life is a chemical reaction. Or, more precisely, life is a functionally closed and self-sustaining
chemical reaction network. In other words, living systems produce their own components, in
such a way as to maintain and regulate the chemical reaction network that produced them.
During the 1970s, several researchers independently developed formal models of a minimal
living system based on the above definition. However, most of these models do not explain
how these systems could have emerged spontaneously from basic chemistry. They provide
insights into the organization of life, but not necessarily its origin.
Now, a new mathematical framework, based on the original notion of autocatalytic sets, is
able to shed more light on both of these aspects. Autocatalytic sets capture the functionally
closed and self-sustaining properties of life in a formal way, and detailed studies have shown
how such sets emerge spontaneously, and can then evolve further, in simple models of
chemical reaction networks. Furthermore, this new framework has been applied directly and
successfully to real chemical and biological networks. Thus, the autocatalytic sets framework
provides a useful and formal tool for studying and understanding both the origin and
organization of life.
In this talk, I will give a non-technical overview of the background, concepts, and main results
of the formal framework, and how it can perhaps be generalized beyond chemistry and the
origin of life to entire living systems, ecological networks, and possibly even social systems
like the economy.
[colloque] Tue 8.11.2016, Phys 2.52 at 17h15 | more |
Prof. Manfred Lehn (JG Universität Mainz): The theorem of Grothendieck-Brieskorn-Slodowy and symplectic singularities
Dynkin diagrams appear in various mathematical
contexts like the theory of simple Lie algebras, finite reflection
groups, representation theory, or surface singularities. One of
the connexions between these different fields is made explicit
by the theorem of Grothendieck-Brieskorn-Slodowy. It describes a
relation between the singularities associated to finite subgroups
of $\operatorname{SU}(2)$, that were studied by Felix Klein, and the geometry of
simple Lie algebras. In the talk I will illustrate this theorem in
detailed examples and discuss a far reaching generalisation
via the notion of Poisson deformations. This last part of the
talk reports on joint work with Christoph Sorger and Namikawa
Yoshinori.
[colloque] Tue 15.11.2016, Phys 2.52 at 17h15 | more |
Prof. Nicolas Curien (Université Paris-Sud Orsay): A panoramic introduction to random planar maps
In this talk we go through the history of methods for counting planar maps, namely Tutte's original approach, matrix integrals and the bijective methods recently developed by Schaeffer. We then dive into the fascinating geometry of planar maps and will outline certain links with 2D quantic gravity and the Gaussien Free Field.
[colloque] Tue 22.11.2016, Phys 2.52 at 17h15 | more |
Prof. Masahiko Yoshinaga (Hokkaido University): Characteristic polynomials of hyperplane arrangements
The characteristic polynomial of a hyperplane arrangement
is related to many counting problems. In the first half of this
talk, I will give an overview of several aspects of the
characteristic polynomial. Then I will focus on the relationship
with lattice points counting and Eulerian polynomials, in
connection with a conjecture by Postnikov and Stanley on
the location of roots of the characteristic polynomial of
certain arrangements.
[colloque] Tue 6.12.2016, Phys 2.52 at 17h15 | more |
Prof. Priyam Patel (UCSB): Quantitative methods in hyperbolic geometry
Peter Scott’s famous result states that the fundamental groups of hyperbolic surfaces are subgroup separable, which has many powerful consequences. For example, given any closed curve on such a surface, potentially with many self-intersections, there is always a finite cover to which the curve lifts to an embedding. It was shown recently that hyperbolic 3-manifold groups share this separability property, and this was a key tool in Ian Agol's resolution to the Virtual Haken and Virtual Fibering conjectures for hyperbolic 3-manifolds.
I will begin this talk by giving some background on separability properties of groups, hyperbolic manifolds, and these two conjectures. There are also a number of interesting quantitative questions that naturally arise in the context of these topics. These questions fit into a recent trend in low-dimensional topology aimed at providing concrete topological and geometric information about hyperbolic manifolds that often cannot be gathered from existence results alone. I will highlight a few of them before focusing on a quantitative question regarding the process of lifting curves on surfaces to embeddings in finite covers.
[colloque] Tue 13.12.2016, Phys 2.52 at 17h15 | more |
Prof. Christophe Soulé (IHES): La théorie d'Arakelov
On présentera une introduction à la théorie d'Arakelov des surfaces arithmétiques. On énoncera dans ce cadre une conjecture, due à Parshin et Moret-Bailly, qui implique la conjecture abc.
Other events from August 2016 to January 2017
[workshop] September 5-7 2016 - Pérolles campus | more |
Main speakers :
Vladimir Matveev (Jena) : Metric projective geometry
Valentin Ovsienko (Reims) : Projective geometry and combinatorics
Athanase Papadopoulos (Strasbourg) : Teichmüller spaces
[oberseminar] 10:20, Math II (Lonza) | more |
Zoé Philippe (Nantes): Maximal radius of quaternionic hyperbolic orbifold
It has been known since the end of the 1960’s with the work of Každan and Margulis, that any locally symmetric manifold of non-compact type contains an embedded ball of radius rG depending only on the group G of isometries of its universal cover. Given a symmetric space X, denoting by G = I(X) its isometry group, a lower bound on rG provides geometric information on any manifold obtained as a quotient of X: for instance, one can then deduce a lower bound on the maximal injectivity radius of any such manifold, and information about its sick-thin decomposition...
My main goal in this talk will be to explain how one can derive an explicit lower bound for rG in the case where the space X is the quaternionic hyperbolic space.
I will start of course with a short introduction to the quaternionic hyperbolic space, and then present the main ideas of the process.
We will then see that the bound obtained with the method i present decreases exponentially with the dimension. I will thus conclude this presentation with a discussion about the behaviour of the maximal radius with the dimension, and connect it with the behaviour of another important invariant of these manifolds, that is, their Margulis constant.
[oberseminar] Wed 28.09.2016, Math II (Lonza) at 10:20 | more |
Gaëtan Borot (MPI Bonn): Around a Riemann-Hilbert problem in P^{1} related to Seifert spaces
Let (A_{j})_{j = 1}^{n} be pairwise disjoint segments on the real line. We study the following problem: find a vector (W_{j}(z))_{j = 1}^{n} of analytic 1-forms with prescribed meromorphic singularities in P^{1} minus A_{j}, admitting boundary values W_{j}(z + i0) and W_{j}(z - i0) when z approaches A_{j} from Im(z) > 0 or Im(z) < 0, such that
where T
_{j,k} are real constants and V(z) a given meromorphic 1-form on P
^{1}.
There are many situations in mathematical physics where we desire to solve such problems, as explicitly as possible, for specific T_j,k and V, and I will first give some motivations from random matrices, random lozenge tilings in 2d, 3-manifold invariants, ...
I will explain an elementary approach to *, which in favorable (but not so common) cases lead to a full solution. * is indeed an instance of Riemann-Hilbert problem. It indeed allows us the construction of the monodromy group G of the solution (W_j), and it is then the matter of finding functions with prescribed monodromy and singularities. G here is a pseudo-reflection group defined by n generators, and has a natural action on an n-dimensional vector space. In case the group admits a non-trivial finite orbit (which can happen even if G is infinite), the solutions can be described by algebro-geometric methods.
Given T_j,k, we are therefore interested in identifying the group G, as well as its set of (finite or maybe not) orbits.
The third part of the talk will focus on a class of examples of T_j,k, attached to Seifert spaces. These are S
^{1}-bundles over an orbifold S
^{2}. Important geometric invariants of those spaces are the order of orbifold points on the base surface (a_1,...,a_r), and the orbifold Euler characteristic
c = 2 - r +
å_i 1/a_i. The geometrical type (spherical, euclidean, hyperbolic) of the 3-manifold is determined with a few exceptions by the sign of
c. It turns out that the large N limit (and actually the full large N expansion) of the SU(N) evaluation of LMO invariants of these Seifert spaces is determined by a solution to * for T_j,k determined by a_1,...,a_r. We find that the corresponding G, although not finite, admits finite orbits if and only if
c is nonnegative, and in these case solutions can (almost) effectively be computed, implying some algebraicity for LMO invariants. Many questions yet remain open about identification of G, both for
c > 0 and
c < 0, its relation to the geometry of the underlying Seifert space, and how to solve * in the cases for
c < 0.
This is based on a joint work with Bertrand Eynard.
[oberseminar] Wed 19.10.2016 ,Math II (Lonza) at 10h20 | more |
Masato Mimura (EPFL): Superintrinsic synthesis in fixed point properties
For a class X of metric spaces, we say a finitely generated group G has the fixed point property (F_{X}), relative to X, if all isometric G-actions on every member in X have global fixed points. Fix a class X of "non-positively curved spaces" (for instance, in the sense of Busemann) stable uder certain operations. We obtain new criteria to "synthesize" the "partial" (F_{X}) (more precisely, with respect to subgroups) into the "whole" (F_{X}). A basic example of such X is the class of all Hilbert spaces, and then (F_{X}) is equivalent to the celebrated property (T) of Kazhdan.
Our "synthesis" is intrinsic, in the sense of that our criteria do not depend on the choices of X. The point here is that, nevertheless, we exclude all of "Bounded Generation" axioms, which were the clue in previous inventive works by Y. Shalom (Publ. IHES, 1999 and ICM 2006) on intrinsic synthesis. As applications, we present a simpler proof of (T) for elementary groups over noncommutative rings (Ershov--Jaikin, Invent. Math., 2010). Moreover, our synthesis enables us to extend that to one in general L_p space settings for all finite p>1.
[oberseminar] Wed 26.10.2016, Math II (Lonza) at 10:20 | more |
Felix Schlenk (Neuchatel): Growth of the number of geodesics between two points in manifolds of non-finite type
Consider a closed Riemannian manifold M, and two non-conjugate points p,q in M.
The growth function CF(L;p,q) counts the number of geodesics from p to q
of length at most L. The study of this function is a traditional topic, with major results by Morse, Serre, Gromov, and Paternain-Petean. The answers are quite complete for manifolds of finite type, which means that the universal cover of M has finitely generated homology groups, or, equivalently, that the universal cover of M has the homotopy type of a finite CW complex. For manifolds of non-finite type, it was conjectured by Paternain and Petean that CF(L;p,q) grows exponentially. While geometric and topological approaches don't give much for this problem, one can use the Hopf algebra structure of the based loop space to prove "half of" this conjecture: If M is not of finite type, then
CF (L;p,q) ≥ e^{C √ T} for a constant C>0.
This is work joint with Urs Frauenfelder.
[mathematikon] Tue 8.11.2016, Phys 2.73 at 12h15 | more |
Joé Brendel: The Geometry of Complex Numbers
Please find the abstract here.
[oberseminar] Wed 09.11.2016, Math II (Lonza) at 10h20 | more |
Manfred Lehn (Universität Mainz): Irreducible holomorphic symplectic manifolds from cubic fourfolds
An irreducible holomorphic symplectic manifold (ihsm) by
definition
is a simply-connected compact Kähler manifold with holonomy group Sp(n).
Such
manifolds appear besides compact complex tori and Calabi-Yau manifolds as
building blocks in the Bogomolov-Beauville structure theorem for compact
Kähler manifolds with vanishing first Chern class. The basic example of
an ihsm is a K3-surface. Though irreducible holomorphic symplectic
manifolds
have been studied intensively in the last decades and despite the fact that
only a few topological types are known, even a topological
classification seems
out of reach at present. Higher dimensional examples most often arise as
moduli spaces of sheaves on K3-surfaces. However, by a classical theorem of
Beauville and Donagi, the Fano scheme of lines on a smooth hypersurface Y of
degree 3 in 5-dimensional projective space also is an ihsm. Recently, more
constructions related to such cubic fourfolds have surfaced. In the talk,
I want to describe one such example and possible explanations for its
existence.
This part is a report on joint work with C. Lehn, C. Sorger, D. van Straten
and with N. Addington.
[oberseminar] Wed 23.11.2016, Math II (Lonza) at 10h20 | more |
Stefano Riolo (University of Pisa)
By gluing copies of a deforming polytope, we describe some deformations of
complete, finite-volume hyperbolic cone four-manifolds.
Despite the fact that hyperbolic lattices are locally rigid in dimension
greater than three (Garland-Raghunathan), we see a four-dimensional
analogue of Thurston's hyperbolic Dehn filling: a path of cone-manifolds
M_{t} interpolating between two cusped hyperbolic four-manifolds M_{0} and M_{1}.
This is a joint work with Bruno Martelli.
[oberseminar] Wed 30.11.2016, Math II (Lonza) at 10h20 | more |
Edoardo Dotti and Simon Drewitz: Right-Angled Polygons in Hyperbolic Space
Fenchel studied right-angled hexagons in the hyperbolic plane.
Recently Delgove and Retailleau considered right-angled hexagons in hyperbolic 5-space.
We generalize their work in order to describe oriented right-angled polygons in hyperbolic space of arbitrary dimension. In this talk we present an algorithm to construct such an n-gon given n-3 parameters living in a Clifford algebra by exploiting the geometric interpretation of the cross-ratio.
[oberseminar] Wed 7.12.2016, Math II (Lonza) at 10h20 | more |
Andrew Yarmola (University of Luxembourg): Hyperbolic 3-manifolds with low cusp volume
For cusped hyperbolic 3-manifolds, one can consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present preliminary results for classifying the infinite families of hyperbolic 3-manifolds of cusp volume < 2.62 and the implications of this classification. These families are of particular interest as they exhibit the largest number of exceptional Dehn fillings. As in some other results on hyperbolic 3-manifolds of low volume, our technique utilizes a rigorous computer assisted search. This talk will focus on providing sufficient background to explain our approach and describe our conclusions. This work is joint with David Gabai, Robert Meyerhoff, Nathaniel Thurston, and Robert Haraway.
[oberseminar] Wed 14.12.2016, Math II (Lonza) at 10h20 | more |
Rafael Guglielmetti (Fribourg): Clifford algebras and isometries of (infinite dimensional) hyperbolic spaces
The aim of this talk is to explain how we can use Clifford algebras to induce isometries of (infinite dimensional) hyperbolic spaces. First, I will present basic material about Hilbert spaces which are used to construct models of infinite dimensional hyperbolic spaces. Then, we will see basic properties of isometries in the upper half-space model. Finally, I will explain how to construct Clifford matrices in the infinite dimensional setting and how we can use them to get isometries of infinite dimensional hyperbolic spaces.
[oberseminar] Thur, 15.12.2016, Earth Sciences 1.309 at 11h00 | more |
Jamal Najim (LIGM - Université de Marne La Vallée): Non-hermitian random matrices with a variance profile
For each $n$, let $A_n = (\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n = (X_{ij})$ be an $n\times n$ random matrix with i.i.d.~centered entries of unit variance. We are interested in the asymptotic behavior of the empirical spectral distribution $\mu_{Y_n}$ of the rescaled entry-wise product
$$
Y_n = \frac 1{\sqrt{n}}\left(\sigma_{ij}X_{ij}\right) .
$$
For our main result, we provide a deterministic sequence of probability measures $\mu_n$, each described by a family of Master Equations, such that the difference $\mu_{Y_n}- \mu_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries $\sigma_{ij}$ to vanish, provided that the standard deviation profiles An satisfy a certain quantitative irreducibility property.
This is a joint work with Nick Cook, Walid Hachem and David Renfrew.
[oberseminar] Wed 21.12.2016, Math II (Lonza) at 10h20 | more |
Marton Naszodi (EPFL, Lausanne & ELTE, Budapest)
A light source p in the d-dimensional real space illuminates a boundary point b of a convex body K, if the ray emanating from b in the direction pb intersects the interior of K. As is easy to see, the minimum number of light sources needed so that each boundary point is illuminated is the same as the minimum number of translates of the interior of K that cover K.
A famous open problem in discrete geometry is the Boltjanskii-Hadwiger conjecture, according to which, for every convex body, this number is at most 2^{d}. In this introductory talk, we will discuss different approaches to the problem, and its relationship with other questions.