Colloquia in the current semester
[colloque] Tue 1.3.2016, Phys 2.52 at 17h15 - Festvortrag zum 80. Geburtstag von Ernst Ruh | more |
Prof. Dr. Ernst Heintze (Uni Augsburg): Affine root systems and submanifolds of Hilbert spaces
We review the relationship between compact Lie groups, symmetric spaces, root systems, and isoparametric submanifolds and indicate its extension to infinite dimensions in the context of affine Kac-Moody algebras.
[colloque] Tue 8.3.2016, Phys 2.52 at 17h15 | more |
Prof. Joan Porti (Barcelona): Reidemeister torsion and hyperbolic 3-manifolds
Reidemeister torsion was introduced in 1935, as an invariant that classifies lens spaces. In this talk I plan to discuss first some aspects of its history and then I will focus on its applications to hyperbolic 3-manifolds.
[colloque] Tue 22.3.2016, Phys 2.52 at 17h15 | more |
Prof. Uwe Semmelmann (Uni. Stuttgart): Invariant 4-forms and exceptional Lie algebras
In my talk I will consider Lie algebras which are defined
using finite dimensional representations. In particular
I want to explain how the Jacobi identity can be reformulated
into a geometric or representation theoretic condition.
As an application I will present a simple construction
for exceptional Lie algebras. My talk is based on a
joint work with Andrei Moroianu.
[colloque] Tue 12.4.2016, Phys 2.52 at 17h15 | more |
Prof. Dr. Benjamin Sudakov (ETH Zürich): Induced matchings, arithmetic progressions and communication
Extremal combinatorics is one of the central branches of discrete mathematics that deals with the problem of estimating the maximum possible size of a combinatorial structure which satisfies certain restrictions. Often, such problems also have applications to other areas including theoretical computer science, additive number theory and information theory. In his talk, we will illustrate this fact using several closely related examples that focus on a recent work with Alon and Moitra.
[colloque] Tue 19.4.2016, Phys 2.52 at 17h15 | more |
Prof. Christophe Bavard (Bordeaux): Points conjugués en géométrie riemannienne et lorentzienne
Les points conjugués jouent un rôle important en géométrie riemannienne et
lorentzienne, en particulier pour l'étude du rayon d'injectivité. Dans le cadre riemannien,
l'absence de points conjugués impose des contraintes assez fortes sur la topologie de la
variété, et parfois même sur sa géométrie. Ainsi, un théorème de E. Hopf (1948), généralisé
par Burago et Ivanov (1994), affirme qu'un tore riemannien sans points conjugués
est nécessairement plat, c'est-à-dire à courbure nulle. La situation s'avère moins rigide
dans le cas lorentzien : dans cet exposé, je montrerai l'existence d'un tore lorentzien
sans points conjugués et non plat. Il s'agit d'un travail conjoint avec Pierre Mounoud.
[colloque] Tue 26.4.2016, Phys 2.52 at 17h15 | more |
Prof. Roman Sauer (KIT): Invariant random subgroups
When one studies families of topological spaces, groups or other mathematical objects, it is often helpful to assemble the objects in an abstract (topological) space in which the objects become points.
We introduce the space of invariant random subgroups, which is a probabilistic version of the space of subgroups of a group. Lattices and locally symmetric spaces can be regarded as points of this space. Are topological invariants, as Betti numbers, continuous functionals on the space of invariant random subgroups? How is this related to more classical results about Betti numbers?
[colloque] Tue 3.5.2016, Phys 2.52 at 17h15 | more |
Prof. Hans Josef Pesch (Uni Bayreuth): Optimal Control of Dynamical Systems Governed by Partial Differential Equations
When analyzing mathematical models for complex dynamical systems, their analysis and numerical simulation
is often only a first step. Thereafter, one often wishes to complete the analysis by an optimization step
to exploit inherent degrees of freedom for optimizing a desired performance index with the dynamical system
as side condition. This generally leads to optimization problems of extremely high complexity
if the underlying system is described by (time dependent) partial differential equations (PDEs)
or, more generally, by a system of partial differential algebraic equations (PDAEs).
In the talk we will report on some of the lastest achievements on the field of optimization with PDEs
and exhibit the challenges we are facing and have to cope with to solve such tasks.
In the introduction three problems from engineering sciences are presented:
Hot cracking in welding of aluminium alloys
Intercontinental flights at hypersonic speeds
The optimal control of certain fuell cell systems for an environmentally friendly production of electricity
After this motivation an outline of the mathematical theory of optimal control problems for one elliptic equation
is given to depict the purpose of solving optimal control problems by first order necessary conditions. Thereafter
two numerical concepts, namely First optimize then discretize and First discretize then optimize are discussed with
respect to their pros and cons as well as an overlook on the mathematical toolbox from the literature is given.
The main part of the talk then deals with the results for optimal load changes when applying the two aforementioned
methodologies including a method for the practical realisation of the computed optimal solutions based on model
reduction techniques.
[colloque] Tue 10.5.2016, Phys 2.52 at 17h15 | more |
Prof. Sergiu Moroianu (Bucharest): Renormalized volume in hyperbolic geometry
Hyperbolic $3$-manifolds of finite geometry
and infinite volume have an intrinsic
compactification by Riemann surfaces.
I will survey the properties of the renormalized
volume of such $3$-manifolds, viewed as a function
on the Teichmüller space of the boundary at infinity.
[colloque] Tue 17.5.2016, Phys 2.52 at 17h15 | more |
Prof. Vincent Beffara (CNRS Grenoble): Introduction to the Schramm-Loewner Evolution
The introduction by Schramm in 1999 of the Stochastic Lowner Evolution (now rather known as Schramm-Loewner Evolution or simply SLE) led to some of the most spectacular recent progress in the study of critical two-dimensional statistical physics models. The construction of SLE involves both probabilistic techniques and tools from complex analysis; the goal of this talk is to give a mostly self-contained introduction to this process, as well as to some of the results that were later proved using it as a tool. I will also mention a few related question that remain open.
[colloque] Tue 24.5.2016, Phys 2.52 at 17h15 | more |
Dr. Roger Zuest (Fribourg): Regularity theory for almost minimal surfaces
The general statement in regularity theory for geometric variational problems is of the form: Outside a small singular set, a solution of such a problem is a submanifold. As a specific example the notion of almost area minimizing sets was introduced by Almgren and Taylor subsequently showed that such sets have the structure of soap films as predicted by Plateau in the 19th century.
In this talk we first review part of the history and framework of regularity theory related to almost area minimizing surfaces and later discuss some general strategies used in proofs of such regularity results.
[colloque] Tue 31.5.2016, Phys 2.52 at 17h15 | more |
Prof. Wolfgang Lück (Universität Bonn / HIM Bonn): Introduction to $L^2$-Betti numbers
We give an introduction to $L^2$-homology and $L^2$-Betti numbers which generalizes the well-known classical notions of homology and Betti numbers. They have suprising applications to problems in topology, geometry, and group theory which a priori seem not be related, but whose proofs require $L^2$-techniques. We also discuss some open conjectures.
[colloque] Tue 7.6.2016, Phys 2.52 at 15h30 | more |
Prof. Alexandru Suciu (Northeastern University, Boston): Fundamental groups in algebraic geometry and three-dimensional topology
In this talk I will present some of the rich interplay between complex algebraic geometry and low-dimensional topology, as it occurs when studying the fundamental groups of algebraic varieties (such as complements of hyperplane arrangements) and 3-dimensional manifolds. The bridge between the two settings is provided by the Alexander polynomial and the cohomology jump loci of those space.
[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.
[colloque] Tue 7.3.2017, Phys 2.52 at 17h15 | more |
Prof. Sergei Tabachnikov (Pennsylvania State University): Skewers
The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space: points and lines in the plane are replaced by lines is space, the incidence between a line and a point translates as the intersection of two lines at right angle, and the operations of connecting two points by a line or by intersecting two lines at a point translate as taking the skewer of two lines. These configuration theorems hold in elliptic, Euclidean, and hyperbolic geometries.
This correspondence principle extends to plane configuration theorems involving polarity. For example, the theorem that the three altitudes of a triangle are concurrent corresponds to the Petersen-Morley theorem that the common normals of the opposite sides of a space right-angled hexagon have a common normal.
I shall also discuss the skewer versions of the Sylvester problem: given a finite collection of pairwise skew lines such that the skewer of any pair intersects at least one other line at right angle, do all the line have to share a skewer? The answer is positive in the elliptic and Euclidean geometries, but negative in the hyperbolic one.
[colloque] Tue 14.3.2017, Phys 2.52 at 17h15 | more |
Prof. Philipp Habegger (Uni Basel): Complex Multiplication or is $e^{\pi \sqrt{163}}$ an Integer?
Roots of unity are algebraic values of the exponential
function at rational arguments shifted by $\pi i$. Kronecker's Jugendtraum was
to find analytic functions that mimic this behavior for algebraic numbers of higher degree.
The theory of complex multiplication of elliptic curves provides a rich
trove of examples of such functions with many surprising symmetries. It
originated in the 19th century in work of Kronecker and Weber and
underwent a remarkable development in the 20th century by Hilbert,
Shimura, Deligne and many others.
In this talk I will provide a glimpse into some classical aspects of complex multiplication
from a diophantine point of view. Then I will discuss recent questions
connected to problems in diophantine geometry, some of them are joint
work with Jonathan Pila.
[colloque] Tue 21.3.2017, Phys 2.52 at 17h15 | more |
Prof. Jan Draisma (Uni Bern): tba
[colloque] Tue 25.4.2017, Phys 2.52 at 17h15 | more |
Prof. Antti Knowles (Uni Genève): tba
[colloque] Tue 16.5.2017, Phys 2.52 at 17h15 | more |
Prof. Pierre Nolin (ETHZ): tba
Other talks and events in the current semester
[oberseminar] Oberseminar Geometrie, Wed 4.3.2015, Math II (Lonza) at 10:20 | more |
Martin Herrmann (Fribourg): Almost nonnegative curvature operator, homogeneous spaces and a question of Karsten Grove
I will talk about the construction of almost nonnegative curvature operator on
principal torus bundles (joint work with Dennis Sebastian and Wilderich Tuschmann).
In the second part of the talk I will give an overview about a question of Karsten
Grove whether the bounds of Gromov’s Betti number theorem imply finiteness of
rational homotopy types and show how one can used the construction of the first
part to get improved counterexamples to this question among homogeneous spaces
with almost nonnegative curvature operator.
[oberseminar] Oberseminar Geometrie, Wed 11.3.2015, Math II (Lonza) at 10:20 | more |
Mike Roberts (Durham): A methodical approach to the construction of Coxeter polytopes in hyperbolic n-space with n+3 facets
Coxeter polytopes which tessellate hyperbolic n-space have been fully classified in the cases with
n+1 and n+2 facets but not yet for those with n+3 facets a classification only exists for compact polytopes. This seminar will focus on the construction of new non-compact Coxeter polytopes and present many new results that have been discovered of such polytopes which are non-simple.
[oberseminar] Combinatorics seminar, Fri 13.3.2015, Math II (Lonza) at 13:15 | more |
Ornella greco (KTH Stockholm): The Betti table of Veronese Modules
In this talk, I will concentrate on the minimal free resolution of the Veronese modules, and I will give a formula for their Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. As applications, I will characterize their Cohen-Macaulayness and the linearity of their resolution. Moreover, I will give a complete description of the Betti table of all the Veronese modules in 2 variables.
[oberseminar] Oberseminar Geometrie, Wed 18.3.2015, Math II (Lonza) at 10:20 | more |
Kate Juschenko (Northwestern): Techniques and concepts of amenability of discrete groups
The subject of amenability essentially begins in 1900's with Lebesgue. He asked
whether the properties of his integral are really fundamental and follow from more
familiar integral axioms. This led to the study of positive, finitely additive and
translation invariant measure on different spaces. In particular the study of isometry-
invariant measure led to the Banach-Tarski decomposition theorem in 1924. The
class of amenable groups was introduced and studied by von Neumann in 1929 and
he explained why the paradox appeared only in dimensions greater or equal to three.
In 1940's and 1950's a major contribution was made by M. Day in his paper on
amenable semigroups. We will give an introductory to amenability talk, and explain
more recent developments in this field.
[oberseminar] Oberseminar Geometrie, Wed 26.3.2015, Math II (Lonza) at 10:20 | more |
Stefan Rosemann (Jena): Kähler metrics with Hamiltonian 2-forms, their classification via projectively equivalent metrics and applications
In this talk I show how a local description of a Kähler metric with a Hamiltonian 2-form can be obtained via a local description of projectively equivalent metrics (i.e., metrics having the same unparametrized
geodesics). I will explain how to single out those metrics admitting more than one linearly independent Hamiltonian 2-form and I will show how to apply the local description to obtain global results.
[oberseminar] Oberseminar Geometrie, Wed 15.4.2015, Math II (Lonza) at 10:20 | more |
Bernhard Mühlherr (Giessen): Tits-indices, affine Coxeter groups and buildings
Tits-indices are decorated Coxeter diagrams which have been introduced by Tits in the context of the classification of semi-simple algebraic groups over arbitrary fields. They have a pure combinatorial interpretation on the level of Coxeter groups which provides a tool to embed a (relative) Coxeter system into an (absolute) Coxeter system such that their length functions are compatible. It is an easy consequence of the definitions that the relative Coxeter system is spherical if and only if the absolute Coxeter system is spherical. Whether this statement remains true if one replaces 'spherical'
by 'affine' is less obvious.
In the first part of my talk I present the combinatorial theory of Tits-indices and a characterization of affine Coxeter systems; the latter provides a tool to show that the answer to the question above is affimative. Both subjects are 'side products' of some recent results on buildings which I will explain in the second part:
In analogy to the theory of algebraic groups one can use Tits-indices to define Galois-descent in buildings. The resulting theory is part of joint work with H. Petersson and R. Weiss on the local structure
of exceptional Bruhat-Tits buildings.
The characterization of affine Coxeter groups relies on joint with K. Struyve on the non-exististence of hyperbolic triple buildings.
[oberseminar] Oberseminar Geometrie, Wed 22.4.2015, Math II (Lonza) at 10:20 | more |
Viktoriya Ozornova (Bremen / MPIM Bonn): Factorability structures
A factorability structure on a group or a monoid is a choice of geodesic normal forms with respect to a fixed generating system with certain compatibility properties. Factorability structure yields a smaller chain complex than the bar complex for computing the homology of the given monoid or group. Moreover, under some additional assumptions, factorability yields a complete string rewriting system on the monoid. Examples of factorability structures can be found on symmetric groups, braid groups (and more general, Garside groups), and on arbitrary Artin monoids.
(Joint with A. Hess)
[oberseminar] Combinatorics seminar, Fri 24.04.2015, Time and place TBA | more |
Matteo Varbaro (Genova): On dual graphs of complete intersections
Let X be a projective subvariety of the projective space over an algebraically closed field. The dual graph of X, denoted by G(X), has the irreducible components of X as nodes, and two nodes are connected by an edge if and only if the corresponding irreducible components intersect in codimension 1. A classical result of Hartshorne states that, if the (quotient by) an ideal defining X is Cohen-Macaulay, then G(X) is connected. In particular G(X) is connected whenever X is the zero locus of c polynomials, where c is the codimension of X. If the (only) radical ideal defining X is defined by c polynomials (i.e. if X is a complete intersection) and X is the union of linear subspaces (i.e. X is a subspace arrangement) we recently proved with Bruno Benedetti that G(X) is r-connected, where r = d_1+…+d_c-c and the d_i’s are the degrees of the polynomials. In the talk I will discuss this result, do some related examples and, time permitting, explain a more general version of the theorem holding true for any arithmetically Gorenstein projective scheme.
[oberseminar] Oberseminar Geometrie, Wed 29.4.2015, Math II (Lonza) at 10:20 | more |
Jordane Granier (Fribourg): Sous-groupe discret de PU(2,1) avec l'éponge de Menger pour ensemble limite
See here
[oberseminar] Oberseminar Geometrie, Wed 6.5.2015, Math II (Lonza) at 10:20 | more |
Vincent Emery (EPF Lausanne): Volumes des variétés hyperboliques et actions de Galois
See here
[oberseminar] Oberseminar Geometrie, Wed 20.5.2015, Math II (Lonza) at 10:20 | more |
Misha Wasem (ETHZ): Isometric extensions at low regularity and codimension
here
[oberseminar] Oberseminar Geometrie, Wed 27.5.2015, Math II (Lonza) at 10:20 | more |
Martin Deraux (Grenoble): Groupes de triangles en géométrie hyperbolique complexe