Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points
Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I: Generic Covers and Covers with Many Branch Points

Author: Robert M. Guralnick

Publisher: American Mathematical Soc.

Published: 2007

Total Pages: 142

ISBN-13: 0821839926

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Considers indecomposable degree $n$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $d$. The authors show that if the cover has five or more branch points then the genus grows rapidly with $n$ unless either $d = n$ or the curves have genus zero, there are precisely five branch points and $n =d(d-1)/2$.

Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I
Symmetric and Alternating Groups as Monodromy Groups of Riemann Surfaces I

Author: R. Guralnick

Publisher: American Mathematical Society(RI)

Published: 2014-09-11

Total Pages: 142

ISBN-13:

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Considers indecomposable degree $n$ covers of Riemann surfaces with monodromy group an alternating or symmetric group of degree $d$. The authors show that if the cover has five or more branch points then the genus grows rapidly with $n$ unless either $d = n$ or the curves have genus zero, there are precisely five branch points and $n =d(d-1)/2$.

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces
Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

Author: William Mark Goldman

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 86

ISBN-13: 082184136X

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This expository article details the theory of rank one Higgs bundles over a closed Riemann surface $X$ and their relation to representations of the fundamental group of $X$. The authors construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of $X$. The authors describe the moduli spaces and their geometry in terms of the Riemann period matrix of $X$.

Computational Algebraic and Analytic Geometry
Computational Algebraic and Analytic Geometry

Author: Mika Seppälä

Publisher: American Mathematical Soc.

Published: 2012

Total Pages: 242

ISBN-13: 0821868691

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This volume contains the proceedings of three AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties held January 8, 2007, in New Orleans, LA; January 6, 2009, in Washington, DC; and January 6, 2011, in New Orleans, LA. Algebraic, analytic, and geometric methods are used to study algebraic curves and Riemann surfaces from a variety of points of view. The object of the study is the same. The methods are different. The fact that a multitude of methods, stemming from very different mathematical cultures, can be used to study the same objects makes this area both fascinating and challenging.

Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings
Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings

Author: Wolfgang Bertram

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 218

ISBN-13: 0821840916

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The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

Invariant Differential Operators for Quantum Symmetric Spaces
Invariant Differential Operators for Quantum Symmetric Spaces

Author: Gail Letzter

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 104

ISBN-13: 0821841319

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This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.

Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories
Complicial Sets Characterising the Simplicial Nerves of Strict $\omega $-Categories

Author: Dominic Verity

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 208

ISBN-13: 0821841424

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The primary purpose of this work is to characterise strict $\omega$-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the ``complicial sets'' defined and named by John Roberts in his handwritten notes of that title (circa 1978). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's $\omega$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.

Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets
Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets

Author: Jun Kigami

Publisher: American Mathematical Soc.

Published: 2009-04-10

Total Pages: 110

ISBN-13: 0821842927

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This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? These three problems turn out to be closely related. The author introduces a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and gives complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel.

Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

Author: Volkmar Liebscher

Publisher: American Mathematical Soc.

Published: 2009-04-10

Total Pages: 124

ISBN-13: 0821843184

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In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in $[0,1]$ or $\mathbb R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types $\mathrm{I}_n$, $\mathrm{II}_n$ and $\mathrm{III}$ is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.

The Topological Dynamics of Ellis Actions
The Topological Dynamics of Ellis Actions

Author: Ethan Akin

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 166

ISBN-13: 0821841882

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An Ellis semigroup is a compact space with a semigroup multiplication which is continuous in only one variable. An Ellis action is an action of an Ellis semigroup on a compact space such that for each point in the space the evaluation map from the semigroup to the space is continuous. At first the weak linkage between the topology and the algebra discourages expectations that such structures will have much utility. However, Ellis has demonstrated that these actions arise naturallyfrom classical topological actions of locally compact groups on compact spaces and provide a useful tool for the study of such actions. In fact, via the apparatus of the enveloping semigroup the classical theory of topological dynamics is subsumed by the theory of Ellis actions. The authors'exposition describes and extends Ellis' theory and demonstrates its usefulness by unifying many recently introduced concepts related to proximality and distality. Moreover, this approach leads to several results which are new even in the classical setup.