2,103 research outputs found
Elementary abelian subgroups: from algebraic groups to finite groups
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian p-subgroups for torsion primes p in finite groups of exceptional Lie type. Such classification results are important for determining the maximal p-local subgroups and p-radical subgroups, both of which play a crucial role in modular representation theory
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bringâs curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bringâs curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
On the essential torsion finiteness of abelian varieties over torsion fields
The classical Mordell-Weil theorem implies that an abelian variety over a
number field has only finitely many -rational torsion points. This
finitude of torsion still holds even over the cyclotomic extension by a result of Ribet. In this article, we
consider the finiteness of torsion points of an abelian variety over the
infinite algebraic extension obtained by adjoining the coordinates of all
torsion points of an abelian variety . Assuming the Mumford-Tate conjecture,
and up to a finite extension of the base field , we give a necessary and
sufficient condition for the finiteness of in terms of
Mumford--Tate groups. We give a complete answer when both abelian varieties
have dimension both three, or when both have complex multiplication.Comment: 35 page
Realizing a Fake Projective Plane as a Degree 25 Surface in
Fake projective planes are smooth complex surfaces of general type with Betti
numbers equal to that of the usual projective plane. Recent explicit
constructions of fake projective planes embed them via their bicanonical
embedding in . In this paper, we study Keum's fake projective
plane and use the equations of \cite{Borisov} to
construct an embedding of fake projective plane in . We also
simplify the 84 cubic equations defining the fake projective plane in .Comment: 11 pages, 1 table. Mathematica, Magma, and Macaulay2 code and key
equations from the paper are included in separate files for convenienc
Minimal PD-sets for codes associated with the graphs Qm2, m even
Please read abstract in the article.The National Research Foundation of South Africahttp://link.springer.com/journal/2002021-12-08hj2021Mathematics and Applied Mathematic
-theoretic counterexamples to Ravenel's telescope conjecture
At each prime and height , we prove that the telescopic and
chromatic localizations of spectra differ. Specifically, for
acting by Adams operations on , we prove that the
-localized algebraic -theory of is not -local. We also show that Galois
hyperdescent, -invariance, and nil-invariance fail for the
-localized algebraic -theory of -local
-rings. In the case and we make complete
computations of , for certain finite Galois extensions
of the -local sphere. We show for that the algebraic -theory
of the -local sphere is asymptotically -local.Comment: 100 pages. Comments very welcom
Hodge-Tate stacks and non-abelian -adic Hodge theory of v-perfect complexes on rigid spaces
Let be a quasi-compact quasi-separated -adic formal scheme that is
smooth either over a perfectoid -algebra or over some ring of
integers of a complete discretely valued extension of with
-finite residue field. We construct a fully faithful functor from perfect
complexes on the Hodge-Tate stack of up to isogeny to perfect complexes on
the v-site of the generic fibre of . Moreover, we describe perfect complexes
on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp.
Higgs-Sen modules. This leads to a derived -adic Simpson functor. We deduce
new results about the -adic Simpson correspondence in both cases
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Forward Limit Sets of Semigroups of Substitutions and Arithmetic Progressions in Automatic Sequences
This thesis deals with symbolic sequences generated by semigroups of substitutions acting on finite alphabets.
First, we investigate the underlying structure of certain automatic sequences by studying the maximum length A(d) of the monochromatic arithmetic progressions of difference d appearing in these sequences. For example, for the Thue-Morse sequence and a class of generalised Thue-Morse sequences, we give exact values of A(d) or upper bounds on it, for certain differences d. For aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively, we study the asymptotic growth rate of A(d). In particular, we prove that there exists a subsequence (d_n) of differences along which A(d_n) grows at least polynomially in d_n. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution considered.
Next, we introduce the forward limit set Î of a semigroup S generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that Î is the unique maximal closed and strongly S-invariant subset of the space of all infinite words, and we prove that it is the closure of the image under S of the set of all fixed points of S. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, Î is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension
Generating Polynomials of Exponential Random Graphs
The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, ErdoÌs and ReÌnyi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM).
In the past, determining when a probability distribution has strong negative dependence has proven to be difficult ([Pem00, BBL09]). The negative dependence of a probability distribution is characterized by properties of its corresponding generating polynomial ([BBL09]). This thesis bridges the theory of exponential random graphs with the geometry of their generating polynomials, namely, when and how they satisfy the stable or Lorentzian properties ([Wag09, BBL09, BH20, AGV21]). We provide necessary and sufficient conditions as well as full characterizations of the parameter space for when this model has a stable or Lorentzian generating polynomial. This is done using a well-developed dictionary between probability distributions and their corresponding multiaffine generating polynomials.
In particular, we characterize when the generating polynomial of a random graph model with a large symmetry group is irreducible. We assert that the edge parameter of the exponential random graph model does not affect stability and that the triangle and k-star parameters are necessarily related if the model is stable or Lorentzian. We also provide full Lorentzian and stable characterizations for the model on K3 and a Lorentzian characterization for specializations of the model on K4
Left braces of size 8 p
We describe all left braces aof size 8 p for an odd p diferent from 3 and 7.Postprint (author's final draft
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