14,021 research outputs found
Serre's Modularity Conjecture
These are the lecture notes from a five-hour mini-course given at the Winter
School on Galois Theory held at the University of Luxembourg in February 2012.
Their aim is to give an overview of Serre's modularity conjecture and of its
proof by Khare, Wintenberger, and Kisin, as well as of the results of other
mathematicians that played an important role in the proof. Along the way we
remark on some recent (as of 2012) work concerning generalizations of the
conjecture
Normal zeta functions of the Heisenberg groups over number rings II -- the non-split case
We compute explicitly the normal zeta functions of the Heisenberg groups
, where is a compact discrete valuation ring of characteristic zero.
These zeta functions occur as Euler factors of normal zeta functions of
Heisenberg groups of the form , where is the
ring of integers of an arbitrary number field~, at the rational primes which
are non-split in~. We show that these local zeta functions satisfy
functional equations upon the inversion of the prime.Comment: 19 pages; to appear in Israel J. Mat
Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations
Cubic polyhedra with icosahedral symmetry where all faces are pentagons or
hexagons have been studied in chemistry and biology as well as mathematics. In
chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal
symmetry, whereas in biology they describe the structure of spherical viruses.
Parameterized operations to construct all such polyhedra were first described
by Goldberg in 1937 in a mathematical context and later by Caspar and Klug --
not knowing about Goldberg's work -- in 1962 in a biological context. In the
meantime Buckminster Fuller also used subdivided icosahedral structures for the
construction of his geodesic domes. In 1971 Coxeter published a survey article
that refers to these constructions. Subsequently, the literature often refers
to the Goldberg-Coxeter construction. This construction is actually that of
Caspar and Klug. Moreover, there are essential differences between this
(Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We
will sketch the different approaches and generalize Goldberg's approach to a
systematic one encompassing all local symmetry-preserving operations on
polyhedra
Bolza quaternion order and asymptotics of systoles along congruence subgroups
We give a detailed description of the arithmetic Fuchsian group of the Bolza
surface and the associated quaternion order. This description enables us to
show that the corresponding principal congruence covers satisfy the bound
sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the
Bolza group as a congruence subgroup, and calculate out a few examples of
"Bolza twins" (using magma). Like the Hurwitz triplets, these correspond to the
factoring of certain rational primes in the ring of integers of the invariant
trace field of the surface. We exploit random sampling combined with the
Reidemeister-Schreier algorithm as implemented in magma to generate these
surfaces.Comment: 35 pages, to appear in Experimental Mathematic
Finite-Bandwidth Calculations for Charge Carrier Mobility in Organic Crystals
Finite-bandwidth effects on the temperature dependence of the mobility of
injected carriers in pure organic crystals are explored for a simplifed case of
impurity scattering. Temperature-dependent bandwidth effects are discussed
briefly through a simplified combination of band and polaronic concepts
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