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- Bjoern Muetzel

# Bjoern Muetzel

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16

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Introduction

My work is in the area of low dimensional geometry and topology. I'm interested in systolic geometry and harmonic forms on surfaces. Most of my work is on hyperbolic surfaces.

Research Experience

Jul 2015

Apr 2014 - Sep 2014

Sep 2011 - Oct 2013

- Institut de Mathématiques et de Modélisation de Montpellier (I3M)
- Montpellier, France

Position

- PostDoc Position

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Research Items (16)

In this paper we construct quasiconformal embeddings from Y-pieces that
contain a short boundary geodesic into degenerate ones. These results are used
in a companion paper to study the Jacobian tori of Riemann surfaces that
contain small simple closed geodesics.

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call
a systole a shortest simple closed geodesic in $S$ and denote by
$\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that
$\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus
$g$. We call a (globally) maximal surface $S_{max}$ a compact Riemann surface
of genus $g$ whose systole has length $\mathop{msys}(g)$. In Section 2 we use
cutting and pasting techniques to construct compact hyperbolic Riemann surfaces
with large systoles from maximal surfaces. This enables us to prove several
inequalities relating $\mathop{msys}(\cdot)$ of different genera. In Section 3
we derive similar intersystolic inequalities for non-compact hyperbolic Riemann
surfaces with cusps.

To any compact Riemann surface of genus g one may assign a principally
polarized abelian variety of dimension g, the Jacobian of the Riemann surface.
The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian
is called a period Gram matrix. This paper provides upper and lower bounds for
all the entries of the period Gram matrix with respect to a suitable homology
basis. These bounds depend on the geometry of the cut locus of non-separating
simple closed geodesics. Assuming that the cut loci can be calculated, a
theoretical approach is presented followed by an example where the upper bound
is sharp. Finally we give practical estimates based on the Fenchel-Nielsen
coordinates of surfaces of signature (1,1), or Q-pieces. The methods developed
here have been applied to surfaces that contain small non-separating simple
closed geodesics in [BMMS].

In section 1 we give an improved lower bound on Hermite's constant
$\delta_{2g}$ for symplectic lattices in even dimensions ($g=2n$) by applying a
mean-value argument from the geometry of numbers to a subset of symmetric
lattices. Here we obtain only a slight improvement. However, we believe that
the method applied has further potential. In section 2 we present new families
of highly symmetric (symplectic) lattices, which occur in dimensions of powers
of two. Here the lattices in dimension $2^n$ are constructed with the help of a
multiplicative matrix group isomorphic to $({\Z_2}^n,+)$. We furthermore show
the connection of these lattices with the circulant matrices and the
Barnes-Wall lattices.

Using a new method we give elementary estimates for the capacity of
non-contractible annuli on cylinders and provide examples, where these
inequalities are sharp. Here the lower bound depends only on the area of the
annulus. In the case of constant curvature this lower bound is obtained with
the help of a symmetrization process that results in an annulus of minimal
capacity. In the case of variable negative curvature we obtain the lower bound
by constructing a comparison annulus with the same area but lower capacity on a
cylinder of constant curvature. The methods developed here have been applied to
estimated the energy of harmonic forms on Riemann surfaces in \cite{mu}.

- Oct 2018

We study the energy distribution of harmonic 1-forms on a compact hyperbolic Riemann surface $S$ that has a small separating closed geodesic. The result is applied to the question how close the Jacobian torus of $S$ comes to a torus that splits. The aim is to answer this and related questions in terms of geometric data of $S$ such as its injectivity radius and the lengths of geodesics that form a homology basis. This is version 1 of an extended paper in which also non separating small geodesics are considered.

- Sep 2018

Let $S$ be a translation surface of genus $g > 1$ with $n$ cone points $(p_i)_{i=1,\ldots,n}$ with cone angle $2\pi \cdot (k_i+1)$ at $p_i$, where $k_i \in \mathbb{N}$. In this paper we investigate the systolic landscape of these translation surfaces for fixed genus.

We give a general method for constructing compact K\"ahler manifolds $X_1$ and $X_2$ whose intermediate Jacobians $J^k(X_1)$ and $J^k(X_2)$ are isogenous for each $k$, and we exhibit some examples. The method is based upon the algebraic transplantation formalism arising from Sunada's technique for constructing pairs of compact Riemannian manifolds whose Laplace spectra are the same. We also show that the method produces compact Riemannian manifolds whose Lazzeri Jacobians are isogenous.

Let $(M,g)$ be a closed, oriented, Riemannian manifold of dimension $m$. We call a \textit{systole} a shortest non-contractible loop in $\mn$ and denote by $\sy(M,g)$ its length. Let $\sr(M,g)=\frac{{\sy\mn}^m}{\vol \mn}$ be the \textit{systolic ratio} of $\mn$. Denote by $\sr(k)$ the supremum of $\sr(S,g)$ among the surfaces of fixed genus $k \neq 0$. In Section 2 we construct surfaces with large systolic ratio from surfaces with systolic ratio close to the optimal value $\sr(k)$ using cutting and pasting techniques. For all $k_i \geq 1$, this enables us to prove:
\[
\frac{1}{\sr( k_1 + k_2)} \leq \frac{1}{\sr(k_1)} + \frac{1}{\sr(k_2)}.
\]
We furthermore derive the equivalent intersystolic inequality for $\srh(k)$, the supremum of the homological systolic ratio. As a consequence we greatly enlarge the number of genera $k$ for which the bound $\srh(k) \geq \sr(k) \gtrsim \frac{4}{9\pi} \frac{\log(k)^2}{k}$ is valid and show that $\srh(k) \leq \frac{(\log(195k)+8)^2}{\pi(k-1)}$ for all $k \geq 76$. In Section 3 we expand on this idea. There we construct product manifolds with large systolic ratio from lower dimensional manifolds.

Given a closed, oriented surface M, the algebraic intersection of closed
curves induces a symplectic form Int(.,.) on the first homology group of M. If
M is equipped with a Riemannian metric g, the first homology group of M
inherits a norm, called the stable norm. We study the norm of the bilinear form
Int(.,.), with respect to the stable norm.

In section 1 we reformulate a theorem of Blichfeldt in the framework of
manifolds of nonpositive curvature. As a result we obtain a lower bound on the
number of homotopically distinct geodesic loops emanating from a common point q
whose length is smaller than a fixed constant. This bound depends only on the
volume growth of balls in the universal covering and the volume of the manifold
itself. We compare the result with known results about the asymptotic growth
rate of closed geodesics and loops in section 2.

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus and we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. The aim of this thesis is to contribute to the Schottky problem, which is to discern the Jacobians among the PPAVs. Buser and Sarnak approached this problem by means of a geometric invariant, the first successive minimum. They showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector, in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. The main goal of this work was to improve this result and to get insight into the connection between the geometry of a compact Riemann surface that is given in hyperbolic geometric terms, and the geometry of its Jacobian. We show the following general findings: For a hyperelliptic surface the first successive minimum is bounded from above by a universal constant. The square of the second successive minimum of the Jacobian of a Riemann surface of genus g is equally of order log(g). We provide refined upper bounds on the consecutive successive minima if the surface contains several disjoint small simple closed geodesics and a lower bound for the norm of certain lattice vectors of the Jacobian, if the surface contains small non-separating simple closed geodesics. If the concrete geometry of the Riemann surface is known, more precise statements can be made. In this case we obtain theoretical and practical estimates on all entries of the period Gram matrix. Here we establish upper and lower bounds based on the geometry of the cut locus of simple closed geodesics and also on the geometry of Q-pieces. In addition the following two results have been obtained: First, an improved lower bound for the maximum value of the norm of the shortest non-zero lattice vector among all PPAVs in even dimensions. This follows from an averaging method from the geometry of numbers applied to a family of symmetric PPAVs. Second, a new proof for a lower bound on the number of homotopically distinct geodesic loops, whose length is smaller than a fixed constant. This lower bound applies not only to geodesic loops on Riemann surfaces, but on arbitrary manifolds of non-positive curvature.

To a compact Riemann surface of genus g can be assigned a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Schottky problem is to discern the Jacobians among the PPAVs. Buser and Sarnak showed, that the square of the first successive minimum, the squared norm of the shortest non-zero vector in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. We show, that in the case of a hyperelliptic surface this geometric invariant is bounded from above by a constant and that for any surface of genus g the square of the second successive minimum is equally of order log(g). We obtain improved bounds for the k-th successive minimum of the Jacobian, if the surface contains small simple closed geodesics.

Genome-wide expression, sequence and association studies typically yield large sets of gene candidates, which must then be further analysed and interpreted. Information about these genes is increasingly being captured and organized in ontologies, such as the Gene Ontology. Relationships between the gene sets identified by experimental methods and biological knowledge can be made explicit and used in the interpretation of results. However, it is often difficult to assess the statistical significance of such analyses since many inter-dependent categories are tested simultaneously.
We developed the program package FUNC that includes and expands on currently available methods to identify significant associations between gene sets and ontological annotations. Implemented are several tests in particular well suited for genome wide sequence comparisons, estimates of the family-wise error rate, the false discovery rate, a sensitive estimator of the global significance of the results and an algorithm to reduce the complexity of the results.
FUNC is a versatile and useful tool for the analysis of genome-wide data. It is freely available under the GPL license and also accessible via a web service.

We have analyzed gene expression in various brain regions of humans and chimpanzees. Within both human and chimpanzee individuals, the transcriptomes of the cerebral cortex are very similar to each other and differ more between individuals than among regions within an individual. In contrast, the transcriptomes of the cerebral cortex, the caudate nucleus, and the cerebellum differ substantially from each other. Between humans and chimpanzees, 10% of genes differ in their expression in at least one region of the brain. The majority of these expression differences are shared among all brain regions. Whereas genes encoding proteins involved in signal transduction and cell differentiation differ significantly between brain regions within individuals, no such pattern is seen between the species. However, a subset of genes that show expression differences between humans and chimpanzees are distributed nonrandomly across the genome. Furthermore, genes that show an elevated expression level in humans are statistically significantly enriched in regions that are recently duplicated in humans.

Microarray technologies allow the identification of large numbers of expression differences within and between species. Although environmental and physiological stimuli are clearly responsible for changes in the expression levels of many genes, it is not known whether the majority of changes of gene expression fixed during evolution between species and between various tissues within a species are caused by Darwinian selection or by stochastic processes. We find the following: (1) expression differences between species accumulate approximately linearly with time; (2) gene expression variation among individuals within a species correlates positively with expression divergence between species; (3) rates of expression divergence between species do not differ significantly between intact genes and expressed pseudogenes; (4) expression differences between brain regions within a species have accumulated approximately linearly with time since these regions emerged during evolution. These results suggest that the majority of expression differences observed between species are selectively neutral or nearly neutral and likely to be of little or no functional significance. Therefore, the identification of gene expression differences between species fixed by selection should be based on null hypotheses assuming functional neutrality. Furthermore, it may be possible to apply a molecular clock based on expression differences to infer the evolutionary history of tissues.

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Scholarship · Sep 2011

Feodor Lynen Fellowship for Postdoctoral Researchers / Alexander von Humboldt Foundation