On modular homology in projective space

AbstractFor a vector space V over GF(q) let Lk be the collection of subspaces of dimension k. When R is a field let Mk be the vector space over it with basis Lk. The inclusion map ∂:Mk→Mk−1 then is the linear map defined on this basis via ∂(X)≔∑Y where the sum runs over all subspaces of co-dimension 1 in X. This gives rise to a sequenceM:0←M0←M1←⋯←Mk−1←Mk←⋯which has interesting homological properties if R has characteristic p>0 not dividing q. Following on from earlier papers we introduce the notion of π-homological, π-exact and almost π-exact sequences where π=π(p,q) is some elementary function of the two characteristics. We show that M and many other sequences derived from it are almost π-exact. From this one also obtains an explicit formula for the Brauer character on the homology modules derived from M. For infinite-dimensional spaces we give a general construction which yields π-exact sequences for finitary ideals in the group ring RPΓL(V)

Strongly constrained maximal subgroups and Sims order bounds for finite almost simple linear and unitary groups

The main topics of this thesis are the strongly constrained maximal subgroups of the finite almost simple linear and unitary groups, i.e. maximal subgroups whose generalized Fitting subgroup coincides with the largest normal p-subgroup for a prime p. The motivation to consider this particular class of local maximal subgroups arises by the Sims conjecture, a theorem of Wielandt and the O'Nan-Scott theorem: In the middle of the 1960s, Sims conjectured that for a finite primitive permutation group G the order of a point stabilizer is bounded by a function f in terms of an arbitrary non-trivial subdegree d of G. A function f which satisfies the conditions of the conjecture of Sims (for a collection H of finite primitive permutation groups) is called a Sims order bound (for H). By a theorem of Wielandt, one can establish an explicit Sims order bound for the collection consisting of the finite primitive permutation groups whose point stabilizers are not strongly constrained. Furthermore, using the O'Nan-Scott theorem, one may show that an explicit Sims order bound f can be determined if an explicit Sims order bound for the collection consisting of the finite almost simple primitive permutation groups is known. So, to obtain an explicit Sims order bound f it is sufficient to investigate the case of finite almost simple primitive permutation groups which have a strongly constrained point stabilizer. In this thesis we consider finite almost simple groups with linear or unitary socle. In Chapter 2 we determine the pairs (G,M) where G is a finite almost simple linear or unitary group and M a strongly constrained maximal subgroup of G. In particular, we classify all strongly constrained maximal subgroups of the finite almost simple linear and unitary groups. Here, we use the classification of the maximal subgroups of the finite almost simple classical groups, obtained by Aschbacher, Kleidman, Liebeck, Bray, Holt and Roney-Dougal. Often, we determine also additional facts, such as the determination of the largest normal p-subgroup O_{p}(M) of the strongly constrained maximal subgroup M of G (p the appropriate prime), or the determination of the centralizer of O_{p}(M) in G. Using the classification obtained in Chapter 2, in Chapter 3 we determine an explicit Sims order bound for the collection consisting of the finite almost simple primitive permutation groups whose socle is isomorphic to a projective special linear or unitary group and which have a strongly constrained point stabilizer. By this result, we deduce that the Wielandt order bound wdt(d)=d!((d-1)!)^{d} is a Sims order bound for the collection consisting of the finite primitive permutation groups which are almost simple linear or unitary groups

Actions of compact groups on spheres and on generalized quadrangles

Alle Wirkungen kompakter zusammenhängender Gruppen von genügend großer Dimension auf Sphären und auf zwei Arten von verallgemeinerten Vierecken werden im einzelnen beschrieben. Für Sphären läßt sich das Ergebnis wie folgt zusammenfassen: Jede treue stetige Wirkung einer kompakten zusammenhängenden Gruppe, deren Dimension 1 + dim SO(n-2) übersteigt, auf einer n-Sphäre ist linear, also äquivalent zur natürlichen Wirkung einer Untergruppe von SO(n+1). Unter ähnlichen Voraussetzungen untersuchen wir Wirkungen auf endlichdimensionalen kompakten verallgemeinerten Vierecken, deren Punktreihen Dimension 1 oder 4 haben. Hier zeigen wir, daß jede treue Wirkung einer kompakten Gruppe von genügend großer Dimension äquivalent ist zu einer Wirkung auf einem Moufang-Viereck, also auf einer Nebenklassengeometrie einer einfachen Lie-Gruppe, die durch ein BN-Paar beschrieben wird. Die vorliegende Arbeit steht in der Tradition der Untersuchung kompakter projektiver Ebenen und neuerdings anderer kompakter verallgemeinerter Polygone durch Salzmann und seine Schule. Der dabei entstandene Leitgedanke, nur die Wirkung einer Gruppe von genügend großer Dimension vorauszusetzen, wird in dieser Arbeit erstmals für verallgemeinerte Vierecke durchgeführt. Wir setzen zusätzlich voraus, daß die Gruppe kompakt ist, um die hochentwickelte Theorie der Wirkungen kompakter Gruppen auf (Kohomologie-) Mannigfaltigkeiten für die topologische Inzidenzgeometrie weiter zu erschließen. Umgekehrt ermöglicht erst die spezifische Salzmannsche Fragestellung die Ergebnisse über Sphären, die ja dem Bereich der klassischen Theorie angehören. Indem die Klassifikation der kompakten Lie-Gruppen konsequent ausgenutzt wird, läßt sich das Problem auf die Behandlung weniger Serien von Gruppen zurückführen. Bei verallgemeinerten Vierecken zeigt man dagegen zuerst die Transitivität der Wirkung und benutzt dann die bestehende (teilweise hier neu bewiesene) Klassifikation.The actions of sufficiently high-dimensional compact connected groups on spheres and on two types of compact Tits buildings are classified explicitly. The result for spheres may be summarized as follows: every effective continuous action of a compact connected group whose dimension exceeds 1 + dim SO(n-2) on an n-sphere is linear, i.e. it is equivalent to the natural action of a subgroup of SO(n+1). Under similar hypotheses, we study actions on finite-dimensional compact generalized quadrangles whose point rows have dimension either 1 or 4. We find that every effective action of a sufficiently high-dimensional compact group is equivalent to an action on a Moufang quadrangle, i.e. on a coset geometry associated to a BN-pair in a simple Lie group. Both for spheres and for generalized quadrangles, the classification arises from an explicit description of the actions. One main source for this thesis is the investigation of compact projective planes and, recently, other compact generalized polygons by Salzmann and his school. They developed the specific hypothesis of a sufficiently large group dimension, which here is applied to generalized quadrangles for the first time. Compactness of the group is a strong additional assumption which allows us to introduce the sophisticated theory of actions of compact groups on (cohomology) manifolds further into topological incidence geometry. Conversely, the results about spheres, which lie completely within the scope of the classical theory, are rendered possible by Salzmann's specific question. When combined with a thorough exploitation of the classification of compact Lie groups, it essentially reduces the problem to the consideration of a small number of series of groups. To obtain the results about generalized quadrangles, we first show transitivity of the action and then use, and partly re-prove, recent classification results

Computational methods in string theory and applications to the swampland conjectures

The goal of the swampland program is the classification of low energy effective theories which can be consistently coupled to quantum gravity. Due to the vastness of the string landscape most results of the swampland program are still conjectures, yet the web of conjectures is ever growing and many interdependencies between different conjectures are known. A better understanding or even proof of these conjectures would result in restrictions on the allowed effective theories. The aim of this thesis is to develop the necessary mathematical tools to explicitly test the conjectures in a string theory setup. To this end the periods of Calabi-Yau manifolds are computed numerically as well as analytically. Furthermore, tools applicable to general string phenomenological models are discussed, including the computation of target space Calabi-Yau metrics, line bundle cohomologies and Strebel differentials. These periods are used to test two conjectures, the refined swampland distance conjecture as well as the dS conjecture. The first states that an effective field theory is only valid up to a certain value of field excursions. If larger field values are included, the effective description breaks down due to an infinite tower of states becoming exponentially light. The conjecture is tested explicitly by computing the distances in the moduli space of CY manifolds. Challenging this conjecture requires the computation of the periods of different Calabi-Yau spaces. The dS conjecture on the other hand forbids vacua with positive cosmological constant. To test this conjecture, the KKLT construction is examined in detail and some steps of the construction are carried out explicitly. Moreover, the validity of the assumed effective theory in a warped throat is investigated. Besides these traditional approaches more exotic ones are followed, including the construction of dS theories using tachyons as well as modifying the signature of space time.Das Ziel des Swampland Programms ist die Klassifizierung effektiver, zu Quantengravitationstheorien vervollständigbarer Theorien. Aufgrund der enormen Anzahl an möglichen Stringvacua, zusammengefasst in der sogenannten Stringlandschaft, sind die meisten der bisherigen Resultate des Programms Vermutungen. Jedoch existiert ein beständig wachsendes dichtes Netz aus Abhängigkeiten zwischen diesen Vermutungen. Ein besseres Verständnis oder ein Beweis dieser Vermutungen würde die erlaubten Niederenergietheorien einschränken. Das Ziel dieser Arbeit ist deshalb die Entwicklung mathematischer Methoden, die explizite Tests der Swampland Vermutungen in stringtheoretischen Modellen ermöglichen. Insbesondere werden Perioden von Calabi-Yau Mannigfaltigkeiten auf numerischem und analytischem Weg berechnet. Darüber hinaus werden Methoden zur Berechnung von Calabi-Yau Metriken, Linienbündelkohomologien und Strebeldifferentialen behandelt. Diese werden zur Überprüfung zweier Vermutungen eingesetzt, zum Test der Swampland Distanzvermutung sowie zum Test der dS Vermutung. Erstere besagt, dass eine effektive Theorie nur bis zu bestimmten Feldwerten gültig sein kann. Werden diese überschritten werden unendlich viele nicht berücksichtigte Zustände exponentiell leicht und die verwendete effektive Beschreibung bricht zusammen. Diese Vermutung wird durch eine explizite Berechnung von Distanzen zwischen effektiven Theorien in Calabi-Yau Moduliräumen getestet. Die dS Vermutung verbietet hingegen stabile Vacua mit positiver kosmologischer Konstante. Um diese Vermutung zu überprüfen, wird ein Teil der KKLT-Konstruktion explizit durchgeführt. Darüber hinaus wird die Validität der zugrundeliegenden effektiven Theorie in einem warped throat analysiert. Neben diesen traditionellen Herangehensweisen werden exotischere Ansätze für die Konstruktion von dS Räumen untersucht. Dies umfasst Tachyonenkondensation sowie andere Raumzeitsignaturen

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute