Im Schulunterricht wird seit einiger Zeit mehr Wert auf darauf gelegt, Mathematik auf die Wirklichkeit anzuwenden.  Eine mathematische Modellierung ist jedoch ein komplexer Prozess mit mehreren Schritten, die ineinander greifen.  Einzelne Teile dieses Prozesses für sich liefern kaum einen Erkenntnisgewinn. Insbesondere bringen die mathematischen Handlungen, losgelöst von der mathematischen Modellierung, kaum Einblick in das Anwendungsproblem.  Viel kritisiert werden daher Pseudoanwendungen, die weder zum Verständnis der Wirklichkeit noch zum mathematischen Verständnis etwas beitragen.  In diesem Seminar soll der Prozess der mathematischen Modellierung daher als Ganzes betrachtet und an einer Reihe von Beispielen eingeübt werden.  Daneben bietet das Seminar auch die Gelegenheit, wichtige Konzepte aus der Differenzial- und Integralrechnung zu vertiefen.  Oft möchte man qualitative Aussagen über das Verhalten der mathematischen Modelle herleiten und verwendet dabei oft mehr oder weniger tiefliegende mathematische Sätze.

Es gibt interessante mathematische Modellierungsprobleme in den verschiedensten Bereichen, ob Physik, Biologie, Verkehrsplanung oder die Preisfindung in der Wirtschaft.  In diesem Seminar werde ich mich auf biologische Probleme konzentrieren.  Eine Rechtfertigung hierfür ist der Wunsch, mit dem Seminar auch einen kleinen Beitrag zum Nachhaltigkeitsziel der Biodiversität zu leisten.  Einige überraschende Vorgänge in komplexen Ökosystemen lassen sich auch schon in einfachen mathematischen Modellen beobachten.  Mit Hilfe dieser Modelle kann man die Vorgänge in Ökosystemen daher besser verstehen und erklären.  Ein typisches Beispiel hierfür ist das Umkippen von Gewässern.  Wird zu viel Phosphat in ein Gewässer eingeleitet, so kippt es um in einen lebensfeindlichen Zustand. Danach erholt es sich erst wieder, wenn die Phosphateinleitung unter eine andere Schwelle gesenkt wird, die deutlich niedriger ist als die Schwelle, ab der das Gewässer umkippt.

Die Wirklichkeit ist so komplex, dass „exakte“ mathematische Modelle weder möglich noch nützlich sind.  Denn je genauer ein Modell, desto mehr Parameter braucht es für die Beschreibung.  All diese Parameter müssen aus den vorhandenen Daten geschätzt werden, aber die Auswirkungen dieser Schätzungen auf die Vorhersagen des Modells werden mit der Komplexität des Modells immer undurchsichtiger.  Für bestimmte Probleme wie zum Beispiel die Wettervorhersage braucht es natürlich sehr komplexe Modelle.  Wir werden uns im Seminar allerdings auf einfachere Modelle beschränken, deren Ziel das Verständnis naturwissenschaftlicher Phänomene ist.  Solche Modelle enthalten eigentlich immer vereinfachende Annahmen, die offensichtlich falsch sind.  Zum Beispiel werden Populationsgrößen in vielen Modellen als reelle Zahlen angesehen, die bestimmten Differenzialgleichungen genügen sollen -- obwohl Populationsgrößen offensichtlich ganzzahlig sind.  Darum verdient auch die Übersetzung von der Wirklichkeit zum Mathematischen Modell und zurück besondere Beachtung.  Diese Schritte sind auch konzeptionell schwieriger als die Lösung der mathematischen Probleme, die ein Modell stellt, weil sie sowohl mathematisches als auch fachwissenschaftliches Verständnis benötigen.

Die mathematische Modellierung bietet auch die Gelegenheit, unter Computereinsatz mit Daten zu experimentieren.  Man kann das mathematische Modell programmieren und danach ausprobieren, wie es sich verhält, wenn man Anfangswerte und Parameter verändert.  Solche Simulationen sind eine verbreitete Methode in den Naturwissenschaften, so dass es sich lohnt, wenn dies auch im Mathematikunterricht in der Schule vorkommt.  Deutlich schwieriger ist es, Modelle auch quantitativ auf reale Daten anzuwenden.  Dabei müssen Parameter geschätzt, dann Vorhersagen gemacht und mit der Wirklichkeit verglichen werden.  Hierbei sind jedoch verschiedene Aspekte schwierig, so dass ich in diesem Seminar darauf verzichten werde.  Auf die computergestützte Modellierung möchte ich jedoch nicht verzichten.  In den Vorträgen des Seminars sollten daher die folgenden diversen Aspekte jeweils vorkommen, jedoch in unterschiedlicher Gewichtung:

•  Übersetzen von wirklichen Problemstellungen in mathematische, und Motivation der dabei auftretenden systematischen Modellierungsfehler;
• Mathematische Sätze beweisen;
• Qualitative Aussagen über spezifische Modelle herleiten und ihre Anwendung auf die Wirklichkeit diskutieren;
• Computerexperimente mit den mathematischen Modellen, Parameter variieren und qualitative Aussagen durch Computerexperimente überprüfen.

Der Computereinsatz ist auch deshalb für den Einsatz der Modellierung in Schulen wesentlich, weil er es erlaubt, die in der mathematischen Biologie verbreiteten Modelle mit Differenzialgleichungen durch Modelle mit Differenzengleichungen zu ersetzen.  Letztere setzen keine Differenzialrechnung voraus, so dass ihr Einsatz im Unterricht nicht auf die Oberstufe beschränkt bleibt.  Sie sind auch gerade in der Biologie sogar dichter an der Wirklichkeit, weil viele modellierte Größen von Natur aus diskret und nicht kontinuierlich sind.

Geplante Vorträge:

1. Grundlegendes zum mathematischen Modellieren, am Beispiel eines hängenden Kabels (Bauer, §1.1, §2.2 Extrema unter Nebenbedingungen, explikative statt deskriptive Modellierung)
2. Was ist die „optimale“ Geschwindigkeit für den Verkehrsfluss (Bauer §2.1), dazu entweder weitere Literatur zu anderen Verkehrsmodellierungsfragen oder §1.2 Trassierungsaufgaben und der Begriff der Krümmung
3. Lineare Differenzengleichungen und ihre Lösungen, logistische Modelle, Linearisierung, Stabilität von Gleichgewichten und Zyklen, Modellierung von bestimmten Insektenpopulationen, Bekämpfung von Insekten, Modellparameter anpassen, (Bauer §4. Brauer-Castillo-Chavez §2.8)
4. Populationsmodelle mit Differenzialgleichungen, Qualitatives Verhalten der Lösungen, Trennung der Variablen und damit Existenz von Lösungen (Bauer §5.1–2, §5.5)
5. Fischfang, Jagdmodelle, (§5.3, Aufgaben)
6. Blutalkoholkonzentration: Beispiel einer Abituraufgabe, explikative versus deskriptive Modelle (§5.7)
7. Um- und zurückkippen von Seen, Hysterese, andere Beispiele für Hysterese, grafische Erklärung (Bauer Aufgabe 5.4. Brauer-Castillo-Chavez §1.6)?
8. Räuber-Beute-Modell 1: Lotka-Volterra, Existenz von Lösungen, Betrachtung empirischer Daten (Luchs und Schneehase) (§6.1–3)
9. Räuber-Beute-Modell 2: Ein komplexeres Modell, inklusive Modellbildung, Theorie: Linearisierungssatz Grobman und Hartman) (§6.4)
10. Räuber-Beute-Modell 3: Ein weiteres Modell, inklusive Modellbildung, Theorie: Satz von Poincaré-Bendixson) (§6.5)
11. Population von Tannenwickler und Balsamtanne, inklusive numerische Simulation und asymptotische Betrachtung 1 (§5.4 und §7.3)
12. Population von Tannenwickler und Balsamtanne, inklusive numerische Simulation und asymptotische Betrachtung 2 (§5.4 und §7.3)
13. Das SIR-Modell für die Ausbreitung von Epidemien, Steuerung der Epidemieausbreitung, wie wirken Impfungen oder Kontaktbeschränkungen (Teile von Brauer-Castillo-Chavez §9)
14. Varianten des SIR-Modells: Endemische Krankheiten, Herdenimmunität  (Teile von Brauer-Castillo-Chavez §10)
    Weitere mögliche Themen Verteilung bei mehreren Virusvarianten, Verteilung der Infektionen auf Altersgruppen oder räumliche Aspekte (kürzere Forschungsarbeiten)

Literatur:

Bauer 2021: Mathematisches Modellieren als fachlicher Hintergrund für die Sekundarstufe I+II
Brauer-Castillo-Chavez 2012: Mathematical models in population biology and epidemiology
 

Which topics do I offer?

My research interests are quite diverse, so that I may accommodate many different directions for bachelor's and master's theses. This includes, among others:

•  C*-Algebras
•  K-Theory
• some parts of Algebraic Topology
•  some parts of Functional Analysis, especially involving bornologies
• Homological Algebra
• bicategories
• groupoids.

The list of courses below and the previous bachelor's and master's theses listed below give you more concrete ideas of possible topics.

Zwei-Fächer-Bachelor und Master of Education?

Ich habe bisher eine Bachelorarbeit im 2-FBA betreut und einige ko-betreut, aber noch keine Masterarbeit im Master of Education.  Ich bin mir bewusst, dass Studierende in diesen Studiengängen nur wenige Mathematikvorlesungen belegen können, so dass ich andere Themen wähle, die weniger voraussetzen.  Ein dankbares Gebiet hierfür ist die mathematische Biologie, zum Beispiel Modellierung von Epidemien, ich kann mir aber auch andere Themen vorstellen.

Should I take some particular courses before?

As a result, there is not one course that you must have taken to write a thesis with me, but you should have taken courses that allow you to get quickly into some research question in the areas listed above.  At the moment, there is a cycle of lectures in noncommutative geometry, with courses about

• Abstract Harmonic Analysis
• C*-algebras, especially Toeplitz and Cuntz-Pimsner algebras
• Mathematical modelling of topological materials through K-theory
• C*-algebras and groupoids.

More details are listed below.  These courses would, of course, be an ideal preparation to write a thesis with me.  At the same time, I already supervised a number of master's theses that did not involve any of these topics.  In particular, category theory would be a possible entry point for theses in the direction of bicategories.  I have recorded a number of courses in recent years, which are available for self-study.  They may also be very useful preparation for writing a thesis with me. 

When and how best to contact me?

Please send me an email to ask for an appointment.  It is useful for me if you write also about the kind of mathematics that you particularly like and/or some of the subjects that you have already learnt or are learning right now, especially those that you enjoy doing and would like to figure in your thesis project.  If you approach me well in advance, there would be some time for you to take courses suitable to prepare you for a thesis topic in a particular direction.  That is why I recommend to approach me early if you are interested in a thesis with me.   If you come to a week before you want to start working on your thesis, it may work out nicely or it might not.  This depends on whether you have already taken enough relevant courses for me to choose an interesting and doable topic for you.

Previous Bachelor's Theses

• Examples of covariance rings (2022).  This thesis studies a construction of rings from generalised dynamical systems that is inspired by crossed products for group actions and Cuntz–Pimsner algebras for C*-correspondences. It shows that purely algebraic analogues of them such as twisted group rings and Leavitt path algebras fit into this language, and examines to what extent partial and twisted partial actions of groups on rings may be covered as well.
• Convolution of measures on locally compact groupoids (2021).  This thesis explores a possible generalisation of the Haar system on a locally compact groupoid where invariant measures are replaced by quasi-invariant measures. The somewhat surprising outcome is that if such a generalised Haar system exists, then there must be a Haar system in the usual sense.
• Epidemienmodellierung mit Differenzengleichungen (2021, 2-Fächer-Bachelor).  This thesis grew out of a seminar for students in the Master of Education about mathematical modelling of the Covid pandemic. It studies difference equations that are analogous to the differential equations that are used in the literature. These difference equations are much more accessible for school children.
• Continuity of joint spectra (2021).  This thesis studies the continuity of spectral for families of operators on Hilbert spaces. The starting point is the dissertation of S.~Beckus, who showed that for bounded normal operators, the spectra vary continuously if and only if the operators form the sections of a continuous field of C*-algebras. This is extended to several commuting normal operators. The unbounded case is also explored, but the results are more messy in that case.
• Classifying spaces over topological groupoids (2021).  This thesis studies the classifying space of a topological groupoid, with an emphasis on imposing no normality or Hausdorffness assumptions on the groupoids and spaces. Instead of restricting to paracompact spaces, it restricts to numerable covers in the definition of a topological groupoid and a principal bundle.
• On the K-theory and KK-theory of Cuntz-Pimsner algebras (2020).  This thesis studied Pimsner's proof that the Toeplitz C*-algebra of a C*-correspondence is KK-equivalent to the underlying C*-algebra of the C*-correspondence. This proof is rewritten using quasi-homomorphisms instead of KK-theory.
• Exakte Moduln über dem von Manuel Köhler beschriebenen Ring (2018).  This thesis studies modules over a certain rather complicated ring that occurs in the classification of group actions on certain C*-algebras. The Bachelor's thesis helped me to write the article On the classification of group actions on C*-algebras up to equivariant KK-equivalence.

Previous Master's Theses

• Towards Hochschild and cyclic homology of dagger algebras (2024).  This master's thesis brings together noncommutative differential geometry in the form of cyclic homology and algebraic geometry. The main results compute the Hochschild, cyclic and periodic cyclic homology for crossed product algebras of finite groups acting on dagger algebras over nonarchimedean fields. A dagger algebra is a completion of a commutative algebra, where the completion is defined using bornologies. The first step is to prove that some results about crossed products remain true for algebras over all fields of characteristic 0. In particular, it is irrelevant whether or not the ground field is algebraically closed. While this is already claimed in the literature, the proofs are not always carried out in the more general situation. The second step treats the relevant bornological completions and argues why certain results for dagger algebras apply also to crossed product situations.
• A bicategorical perspective on Steinberg algebras (2024).  This master's thesis lifts several results known for groupoid C*-algebras to the level of purely algebraic analogues called Steinberg algebras. First, it is shown that the Steinberg algebra construction is part of a homomorphism of bicategories from a bicategory of ample groupoids to the bicategory of rings. Secondly, for those diagrams where the shape category is an Ore monoid and the bimodules in the diagram are finitely generated and projective as right modules, the covariance ring is described concretely using inductive limits and endomorphism rings of right modules. Third, for a diagram of tight groupoid correspondences, it is shown that the covariance ring is the Steinberg algebra of the groupoid model of the diagram.
• Operator algebras for Hamiltonians with mobility gaps (2024).  This thesis explored a possibility to classify topological phases in the presence of Anderson localization. Here the Fermi energy does not lie in a spectral gap, so that the projections defined by the functional calculus do not belong to a nice observable C*-algebra. Even worse, to make certain homotopies continuous, one must use a topological algebra that contains also certain unbounded operators. This thesis identified that algebra and studied some of its K-theoretic properties, both in Bellissard-like models and in models that use the Roe C*-algebra to model disorder.
• Ideal structure of Nica–Toeplitz algebras (2023).  This thesis describes the gauge-invariant ideals in the Nica–Toeplitz algebra of a proper product system over a free commutative monoid. It led to a journal publication.
• Stammeier's C*-algebras for several injective group endomorphisms as C*-algebras of diagrams of étale groupoid correspondences (2023).  Nicolai Stammeier has studied a class of C*-algebras, which are associated to irreversible algebraic dynamical sytems. This thesis shows that these C*-algebras are special cases of C*-algebras defined by diagrams of groupoid correspondences, where the groupoid correspondences are associated to injective endomorphisms of a group. In particular, it follows that these C*-algebras are groupoid C*-algebras in case the endomorphisms have finite index. In this case, the properties of this groupoid that are related to simplicity and pure infiniteness of the groupoid C*-algebra are studied.
• A classification of 2-groups (2022).  This thesis reproves the classical Whitehead classification of crossed modules by reducing it to the classification of skeletal bigroups, which turns out to be rather easy. To reduce to this case, the Whitehead Theorem that characterises when a homomorphism of bicategories is an equivalences of bicategories is shown directly, without using the MacLane Coherence Theorem.
• Invariants for topological insulators coming from decompositions of coarse spaces (2021). This thesis continues the study of topological insulators using the K-theory of Roe C*-algebras. It simplifies an argument in an article by Ludewig and Thiang using the coarse Mayer–Vietoris sequence and then studies the universality of the boundary map in van Daele's K-theory, following previous work by George Elliott and myself.
• On groupoid models for diagrams of groupoid correspondences (2021).  This thesis proves that any diagram of proper correspondences between locally compact étale groupoids has a groupoid model. The issue is mainly to prove that a certain category of actions contains a final object, and this is shown by verifying that any action extends to a certain relative Stone–Čech compactification.  Some results of this thesis are published in the joint articles Existence of groupoid models for diagrams of groupoid correspondences and The bicategory of groupoid correspondences
• Leavitt path algebras as Cohn localisations and their Hochschild homology (2021).  This thesis proves that Leavitt path algebras of directed graphs are quasi-free and computes their Hochschild homology in complete generality. This extends previous results for row-finite graphs. The main method is to realise the Leavitt path algebras as a Cohn localisation of a variant of the path algebra. Issues that require some care are to define this variant carefully for irregular graphs and to handle nonunital rings in case the graph has infinitely many vertices.
• Geometric construction of Hamiltonians (2022).  This thesis starts from some computations by Prodan and Schulz-Baldes for a certain Hamiltonian for topological materials. These Hamiltonians are defined in a uniform manner in all dimensions and depend on a mass parameter. The question is which topological phase they have depending on the mass. This is computed using explicit generators of van Daele's K-theory for spheres and geometric bivariant K-theory to relate K-theory groups for spheres and tori. The computations are also done in the case of “real” K-theory. This is a cohomology theory, which was not treated in the earlier work on bivariant K-theory. It is checked in the thesis that geometric bivariant K-theory also works for this cohomology theory.  Some results of this thesis are published in the joint article Geometric construction of classes in van Daele's K-theory.
• Partial actions on C*-algebras (2022).  This thesis explores partial actions of groups and inverse semigroups on C*-algebras from the point of view of morphisms to the bicategory of C*-correspondences. It is already known that homomorphisms correspond to ordinary actions of groups and inverse semigroups. Both morphisms and partial actions are more general than that, but not the same. It turns out that extra conditions are needed on a morphism to correspond to a partial action and that extra conditions for cones over a morphism are needed to get the correct notion of covariant representation, which is used to define the crossed product for a partial action. As a class of examples, the article studies Toeplitz algebras of automorphisms and correspondences.

My main area of research is noncommutative geometry, a generalisation of our idea of what space and geometry are. This theory is inspired by quantum mechanics. The Heisenberg commutation relation predicts that the position and momentum observables in quantum mechanics form a noncommutative algebra. This noncommutative algebra describes the phase space of a quantum mechanical system. Thus some noncommutative algebras seem to describe geometric objects. Which geometric ideas carry over to such spaces, and how?

Algebraic topology provides many tools to distinguish different spaces by some homological invariants. Most of these invariants do not carry over to noncommutative spaces. The main invariant in that setting is K-theory, which for ordinary spaces classifies the vector bundles over a space. In noncommutative geometry, K-theory is part of a bivariant theory, which provides powerful tools to compute K-theory and a rich categorical structure. One important aspect of my work is extra structure in bivariant K-theory and how to use it, among others, for the classification of C*-algebras.

Another important homological invariant for noncommutative algebras are cyclic cohomology theories. Periodic cyclic cohomology generalises the de Rham homology of smooth manifolds, but only makes sense for algebras that behave like smooth functions. Variants of it like analytic and local cyclic cohomology also give good results for larger algebras like C*-algebras, which behave more like continuous functions. The original periodic cyclic cohomology is computable through its relationship to Hochschild cohomology. One important aspect of my work has been to extend various tools of homological algebra to algebras that carry some functional analytic structure, such as algebras of smooth or holomorphic functions and their noncommutative analogues. I found that bornologies are better behaved than topologies when functional analysis meets algebra.  In recent research of mine about analytic cyclic homology for algebras over nonarchimedean fields, this turned out to be so in a nice way, giving a good analytic interpretation for the weak completions that are used in the context of Monsky-Washnitzer and rigid cohomology for varieties over finite fields.  We may also want to compare the homological algebra for two different algebras that describe the same noncommutative space with more or less extra structure. One may be an algebra of polynomial functions, the other an algebra of smooth functions on that noncommutative space. We would expect the homological algebra for these to be related, and my notion of an isocohomological embedding gives one way to make precise how.

Ordinary symmetries of spaces are usually described through group actions. Local symmetries may lead to groupoid actions. In noncommutative geometry, however, the more general notion of space should also lead to drastically different types of geometry. One Ansatz to describe noncommutative symmetries are quantum groups, which are noncommutative C*-algebras with extra structure that resembles that in a group.  Another Ansatz which I found even more interesting is higher category theory.  Several constructions in the study of C*-algebras become very clear if we treat C*-algebras as a bicategory with C*-correspondences as arrows and isomorphisms of correspondences as 2-arrows.

For instance, the noncommutative torus may be thought of as the algebra of functions on the quotient of the circle group by a dense subgroup. This quotient is a rather badly behaved non-Hausdorff space, which is replaced in noncommutative geometry first by a groupoid and then by its groupoid C*-algebra. The group structure on this quotient does not correspond to a quantum group structure on noncommutative tori. The philosophy of noncommutative geometry suggests to replace a non-Hausdorff quotient group by a bigroup or crossed module. There is indeed an action of this crossed module on the noncommutative torus that describes the translation action of the underlying non-Hausdorff quotient group on itself.

Besides providing more general notions of symmetry, bicategories also allow clarify the role of Fell bundles: a saturated Fell bundle over a group or groupoid is equivalent to an action by equivalences. For non-invertible dynamics, we also get a common treatment for crossed products and Cuntz-Pimsner algebras. An action of a monoid by C*-correspondences on a C*-algebra is equivalent to a product system over that monoid. If the product system is proper, then the Cuntz-Pimsner algebra of this product system is characterised by a universal property that is analogous to that of crossed products. This universal property is a bicategorical analogue of the definition of a limit or colimit.

Such limit constructions make sense also in suitable bicategories of rings and groupoids, as opposed to C*-algebras. Ongoing work by my students and me interprets various constructions of rings and groupoids as such bicategorical limits.

Ich habe in den Semestern Winter 2020/1 bis Winter 2021/2 und Winter 2022/3 bis Winter 2023/4 jeweils die Vorlesungen Mathematik für Studierende der Physik 1-3 (MaPhy) angeboten.  Bei der MaPhy3-Vorlesung im Winter 2023/4 habe ich ein Skript erstellt, siehe https://www.studip.uni-goettingen.de/sendfile.php?force_download=1&type=0&file_id=212157ea4fd453638fb075134159be8b&file_name=main.pdf

Die Vorlesungen wurden jeweils aufgezeichnet, und diese Vorlesungen sind in Stud.IP weiterhin verfügbar und grundsätzlich zum Selbststudium geeignet.

I recorded several more or less advanced courses in the past, and some of them are suitable for self-study:

• Topological insulators and topological K-theory for Z/2-graded algebras (Winter Term 2024/5)
• C*-Algebras, especially Toeplitz and Cuntz-Pimsner algebras (Summer Term 2024)
• Bicategories in noncommutative geometry (Summer Term 2022)
• Noncommutative differential geometry (Summer Term 2020)
• Modelling topological phases of matter (Winter Term 2018–19)
• Category theory in context (Winter Term 2021–22)

If you are a student at Göttingen University and intend to take one of these courses for credit, please write me an email about the possibility of offering them as a reading course.

Mathematics is already difficult enough.  Write concisely.  Use simple grammar to make your writing easier to read.

When you construct overly complicated, such as this one, which is an intentionally bad example designed to show what can go wrong, the reader will almost certainly have already forgotten what it was that you were discussing at the very beginning of the sentence, when he reaches the end of the sentence, forcing her to go back to the beginning and read it again, maybe several times, until eventually after many false attempts understanding what you want to say, or giving up and turning to another author, who is capable of writing sentences that are easier to follow, being more concise.

Grammar also subtly changes the meaning of a sentence.  In a sentence, there is more emphasis on the subject and the verb, and also on the very first words in it, and less on subordinate clauses such as relative clauses.  For instance:

Continuous functions on compact spaces are bounded.
Good sentence, everything is said straight, without subtle shades of emphasis or room for misunderstanding.

It is clear that continuous functions on compact spaces are bounded.
This is a sentence about “it”, which “is clear”.  Whatever is clear is less important, hardly worth mentioning.  I see no use for this construction.

Clearly, continuous functions  on compact spaces are bounded.
This is a sentence about continuous functions being bounded, but it is made very clear that this is a trivial statement.  Use this if you really want to emphasise the triviality of the statement, but nevertheless want to mention the statement.  So this is rarely useful.

It is proved in Lemma 1.5 that continuous functions on compact spaces are bounded.
Once again, this is a statement about “it”, which “is proved”.  The second-most important thing in the sentence is Lemma 1.5, which is where the action takes place.  The continuity of bounded functions is the least important thing in this sentence.  Once again, not recommended.

Lemma 1.5 proves that continuous functions on compact spaces are bounded.
Now Lemma 1.5 is the main player, and it proves something.  Once again, what is being proved is marked as less important.  Actually, the sentence does not directly assert that continuous functions are bounded.  To deduce this, the reader has to combine it with the common knowledge that statements that are proved are also true.   This sentence may be useful if you are discussing the structure of an article, to lead the reader to interesting results.

We prove in Lemma 1.5 that continuous functions on compact spaces are bounded.
Now “We proving” something is the main action.  Use this to emphasise your own contribution to this proof.

Continuous functions on compact spaces are bounded by Lemma 1.5.
Now the continuity of bounded functions is the main statement; that it is proved in Lemma 1.5 is only a sideremark, for the reader wanting to check this fact.  This is probably how you want to refer to this fact during a proof.

By Lemma 1.5, continuous functions on compact spaces are bounded.
Now Lemma 1.5 is emphasised through the unusual word order.  The continuity of bounded functions is asserted neutrally.  I can imagine this construction being useful sometimes, but rarely.

From the Stone-Weierstraß Theorem, we have the following result.
This is a result about us having something.  I see no situation where this formulation is good style.  The comma is needed here, by the way, to excuse the unusual word order.  Better write “The Stone-Weierstraß Theorem implies/gives/yields the following result.” or “The following result follows from the Stone-Weierstraß Theorem.”, depending on whether you consider the Stone-Weierstraß Theorem or the following result to be slightly more important.  I often see the phrase “we have that”, but I rarely see a use for it because its does not add to the meaning and complicates the grammar by turning a main clause into a subordinate clause. 

Well, a general style guideline says that a sentence should not begin with a mathematical formula.  So the sentence “We have that c²=a²+b².” fulfils this rule, unlike the simpler “c²=a²+b².”  But the meaning of the rule is that such a formula should be put into context, by saying, say, “Hence c²=a²+b².” or “Now we use the Pythagorean Theoremc²=a²+b².”

Note that continuous functions on compact spaces are bounded.
Here “Note that” turns the main statement into a subordinate clause, hinting that this statement is not so important.  If you really want to mark statements as being less important, rather say clearly why they are not so important, but still important enough that you need to write them.

“Such that” or “satisfying” versus “with”
What is the difference between “We seek x with f(x)=0.” and “We seek x such that f(x)=0.” or “We seek x satisfying f(x)=0.”?  The second and third formulations create a subordinate clause, namely, “f(x)=0”, which is treated here as an assertion, not as an equation.  An equation may have two roles: it is a statement, f(x) is equal to 0, and a mathematical formula and thus an object in its own right.  The assertion “f(x)=0” is so simple that I prefer the first formulation with its simpler grammar, treating the equation as an object in its own right.  I would use “such that” if what is coming afterwards is so complicated that a new subordinate clause is needed or helpful to understand what we require of x.  For instance: “We seek x such that the sign of f changes at x.”


Punctuation marks have a function.
The punctuation marks “.!?;:” all end gramatically complete sentences.  For instance, “Given a C*-algebra A.” is not correct because it is not a sentence.  “Let A be a C*-algebra.” is correct.  An experienced reader expects a full stop as a sign to pause and digest the sentence, before going on.  Over-long sentences make this hard, so avoid them.

It is perfectly OK if every sentence expresses only one idea.  Sometimes, relationships between ideas are easier to express by combining them in one sentence together with appropriate links, such as “We will show that the function is continuous in order to prove that it is bounded.”; this points out that you only care about the function being bounded, but prove continuity instead.  The alternative formulation “We will show that the function is continuous; hence it is bounded.” says almost the same, but without commenting on whether continuity or boundedness is what you really care about.

The colon “:” differs from the the full stop “.” in asking the reader to read further, indicating that this is needed to understand the last sentence, as in “The following are equivalent:”.  I use the semicolon “;” to highlight a close relationship between the assertions before and after it.

For commas, there are many rules where they must or must not appear.  Some words are always separated from the rest of the text by commas.  This includes furthermore, moreover, however, therefore, of course, namely, that is, for example, say, and many more.  A comma must not be put around a necessary relative clause, that is, if the meaning of the sentence is changed or lost by leaving out the relative clause; example: “The function which we construct is continuous.”.  In contrast, a comma must be put around unnecessary relative clauses; example: “This function, which is very important for the following construction, is continuous.”

When no rule requires or forbids a comma, I write a comma if it helps the reader to structure a complicated sentence.

Ich bin Sprecher des Chapters Mathematik und Informatik bei Alumni Göttingen.

• Ich bin Vorsitzender des Allgemeinen Fakultätentags.
• Ich bin Mitglied im Beirat der Konferenz Mathematischer Fachbereiche (KMathF).
• Ich bin Mitglied bei den Grünen.
• Ich singe im Göttinger Vokalensemble.
• Ich singe auch im Vokalensemble Lenglern.

Die Möglichkeit, das Studium durch ein Stipendium zu finanzieren, ist vielen entweder unbekannt, oder sie lassen sich von dem Wort „Begabtenförderungswerke“ abschrecken. Dabei werden die Stipendien nicht nur nach Schul- und Studiennoten vergeben, sondern es kommt besonders auf Persönlichkeit und Engagement an. Sind Sie im Sportverein, in einer Kirchengemeinde oder einem Chor aktiv? Diese und ähnliche Aktivitäten gelten als „ehrenamtliches Engagement“. Wenn Ihre Noten nun noch mindestens im Zweier-Bereich liegen, lohnt eine Bewerbung auf jeden Fall. Sogar, wenn beim Abischnitt eine drei vorm Komma steht, Sie aber beispielsweise im Studium viel besser sind, sich besonders sozial engagieren, kann eine Bewerbung chancenreich sein.

An staatsfinanzierten Stiftungen gibt es neben der Studienstiftung (bei der man sich nur eingeschränkt selbst bewerben kann) Stiftungen verschiedener Parteien, Kirchen und Gewerkschaften. Ein gemeinsames Internetportal informiert über diese Stiftungen. Außerdem gibt es eine sehr große Zahl meist kleinerer privater Stiftungen. Häufig sind deren Stipendien auf bestimmte Zielgruppen, Studienfächer oder Orte ausgerichtet. Auch hier lohnt sich aber die Suche, denn wenn Sie das Glück haben, zur Zielgruppe einer dieser Stiftungen zu gehören, ist eine Bewerbung dort besonders chancenreich, weil weniger Bewerber(innen) um diese Stipendien konkurrieren. Eine Datenbank über Stipendien privater Stiftungen

Die Initiative Arbeiterkind bietet neben Links auf Stipendiengeber und -datenbanken auch Interviews mit erfolgreichen Bewerber(inne)n und Bewerbungstipps.

Einige Bewerbungsfristen, die ich vor langer Zeit im Internet recherchiert habe, die aber eventuell nicht mehr aktuell sind: 

Konrad-Adenauer-Stiftung: 15. Januar und 1. Juli
Friedrich-Ebert-Stiftung: keine Fristen
Friedrich-Naumann-Stiftung: 31. Mai und 30. November
Hans-Böckler-Stiftung: einige Zeit vor dem 1. Februar und 1. September (mehrstufiges Verfahren: Stipendiat(inn)en vor Ort müssen Bewerbungen prüfen und bis zu diesem Datum weiterleiten)
Heinrich-Böll-Stiftung: 1. März 2009
Evangelisches Studienwerk e.V. Villigst: 1. März und 1. September