248 research outputs found
Niceness theorems
Many things in mathematics seem lamost unreasonably nice. This includes
objects, counterexamples, proofs. In this preprint I discuss many examples of
this phenomenon with emphasis on the ring of polynomials in a countably
infinite number of variables in its many incarnations such as the representing
object of the Witt vectors, the direct sum of the rings of representations of
the symmetric groups, the free lambda ring on one generator, the homology and
cohomology of the classifying space BU, ... . In addition attention is paid to
the phenomenon that solutions to universal problems (adjoint functors) tend to
pick up extra structure.Comment: 52 page
Delzant's T-invariant, Kolmogorov complexity and one-relator groups
We prove that ``almost generically'' for a one-relator group Delzant's
-invariant (which measures the smallest size of a finite presentation for a
group) is comparable in magnitude with the length of the defining relator. The
proof relies on our previous results regarding isomorphism rigidity of generic
one-relator groups and on the methods of the theory of Kolmogorov-Chaitin
complexity. We also give a precise asymptotic estimate (when is fixed and
goes to infinity) for the number of isomorphism classes of
-generator one-relator groups with a cyclically reduced defining relator of
length : Here
means that .Comment: A revised version, to appear in Comment. Math. Hel
Thermalisation for Wigner matrices
We compute the deterministic approximation of products of Sobolev functions
of large Wigner matrices and provide an optimal error bound on their
fluctuation with very high probability. This generalizes Voiculescu's seminal
theorem [Voiculescu 1991] from polynomials to general Sobolev functions, as
well as from tracial quantities to individual matrix elements. Applying the
result to for large , we obtain a precise decay rate
for the overlaps of several deterministic matrices with temporally well
separated Heisenberg time evolutions; thus we demonstrate the thermalisation
effect of the unitary group generated by Wigner matrices.Comment: 24 pages, 4 figure
Numerical invariants and moduli spaces for line arrangements
Using several numerical invariants, we study a partition of the space of line
arrangements in the complex projective plane, given by the intersection lattice
types. We offer also a new characterization of the free plane curves using the
Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.Comment: v3: A new proof of a result due to Tohaneanu, giving the
classification of line arrangements with a Jacobian syzygy of minimal degree
2 is given in Theorem 4.11. Some other minor change
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C*-Algebren
The theory of C*-algebras plays a major role in many areas of modern mathematics, like Non-commutative Geometry, Dynamical Systems, Harmonic Analysis, and Topology, to name a few. The aim of the conference “C*-algebras” is to bring together experts from all those areas to provide a present day picture and to initiate new cooperations in this fast growing mathematical field
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