1,990 research outputs found
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 Bring’s 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 Bring’s 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
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Quantum ergodicity on the Bruhat-Tits building for in the Benjamini-Schramm limit
We study eigenfunctions of the spherical Hecke algebra acting on
where with a
non-archimedean local field of characteristic zero, with the ring of integers of , and
is a sequence of cocompact torsionfree lattices. We prove a form of
equidistribution on average for eigenfunctions whose spectral parameters lie in
the tempered spectrum when the associated sequence of quotients of the
Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table
Quantum traces for -skein algebras
We establish the existence of several quantum trace maps. The simplest one is
an algebra map between two quantizations of the algebra of regular functions on
the -character variety of a surface equipped with an ideal
triangulation . The first is the (stated) -skein algebra
. The second
is the Fock and Goncharov's
quantization of their -moduli space. The quantum trace is an algebra
homomorphism
where the reduced skein algebra is a
quotient of . When the quantum parameter is 1, the
quantum trace coincides with the classical Fock-Goncharov
homomorphism. This is a generalization of the Bonahon-Wong quantum trace map
for the case . We then define the extended Fock-Goncharov algebra
and show that can be lifted to
. We show
that both and are natural with respect to the change of
triangulations. When each connected component of has non-empty
boundary and no interior ideal point, we define a quantization of the
Fock-Goncharov -moduli space
and its extension . We then show that there
exist quantum traces
and
,
where the second map is injective, while the first is injective at least when
is a polygon. They are equivalent to the -versions but have
better algebraic properties.Comment: 111 pages, 35 figure
On the use of senders for minimal Ramsey theory
This thesis investigates problems related to extremal and probabilistic graph theory. Our focus lies on the highly dynamic field of Ramsey theory. The foundational result of this field was proved in 1930 by Franck P. Ramsey. It implies that for every integer t and every sufficiently large complete graph Kn, every colouring of the edges of Kn with colours red and blue contains a red copy or a blue copy of Kt.
Let q ⩾ 2 represent a number of colours, and let H1,..., Hq be graphs. A graph G is said to be q-Ramsey for the tuple (H1,...,Hq) if, for every colouring of the edges of G with colours {1, . . . , q}, there exists a colour i and a monochromatic copy of Hi in colour i. As we often want to understand the structural properties of the collection of graphs that are q-Ramsey for (H1,..., Hq), we restrict our attention to the graphs that are minimal for this property, with respect to subgraph inclusion. Such graphs are said to be q-Ramsey-minimal for (H1,..., Hq).
In 1976, Burr, Erdős, and Lovász determined, for every s, t ⩾ 3, the smallest minimum degree of a graph G that is 2-Ramsey-minimal for (Ks, Kt). Significant efforts have been dedicated to generalising this result to a higher number of colours, q⩾3, starting with the ‘symmetric’ q-tuple (Kt,..., Kt). In this thesis, we improve on the best known bounds for this parameter, providing state-of-the-art bounds in different (q, t)-regimes. These improvements rely on constructions based on finite geometry, which are then used to prove the existence of extremal graphs with certain key properties. Another crucial ingredient in these proofs is the existence of gadget graphs, called signal senders, that were initially developed by Burr, Erdős, and Lovász in 1976 for pairs of complete graphs. Until now, these senders have been shown to
exist only in the two-colour setting, when q = 2, or in the symmetric multicolour setting, when H1,..., Hq are pairwise isomorphic. In this thesis, we then construct similar gadgets for all tuples of complete graphs, providing the first known constructions of these tools in the multicolour asymmetric setting. We use these new senders to prove far-reaching generalisations of several classical results in the area
The galaxy of Coxeter groups
In this paper we introduce the galaxy of Coxeter groups -- an infinite
dimensional, locally finite, ranked simplicial complex which captures
isomorphisms between Coxeter systems. In doing so, we would like to suggest a
new framework to study the isomorphism problem for Coxeter groups. We prove
some structural results about this space, provide a full characterization in
small ranks and propose many questions. In addition we survey known tools,
results and conjectures. Along the way we show profinite rigidity of triangle
Coxeter groups -- a result which is possibly of independent interest.Comment: 30 pages, 6 figures. v2: Incorporated referee's suggestions;
Corrected a mistake in the proof of Theorem 4.25 (formerly 4.24), improved
other proofs and text; Final version, to appear in the Journal of Algebr
Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders
Orientations of supersingular elliptic curves encode the information of an
endomorphism of the curve. Computing the full endomorphism ring is a known hard
problem, so one might consider how hard it is to find one such orientation. We
prove that access to an oracle which tells if an elliptic curve is
-orientable for a fixed imaginary quadratic order
provides non-trivial information towards computing an endomorphism
corresponding to the -orientation. We provide explicit algorithms
and in-depth complexity analysis.
We also consider the question in terms of quaternion algebras. We provide
algorithms which compute an embedding of a fixed imaginary quadratic order into
a maximal order of the quaternion algebra ramified at and . We
provide code implementations in Sagemath which is efficient for finding
embeddings of imaginary quadratic orders of discriminants up to , even
for cryptographically sized
Measured foliations at infinity and constant mean curvature surface in quasi-Fuchsian manifolds close to the Fuchsian locus
Given an orientied, closed hyperbolic surface , we study quasi-Fuchsian
hyperbolic manifolds homeomorphic to . We study two
questions regarding them: one is on \textit{measured foliations at infinity}
and the other is on \textit{foliation by constant mean curvature surfaces}.
Measured foliations at infinity of quasi-Fuchsian manifolds are a natural
analog at infinity to the measured bending laminations on the boundary of its
convex core. Given a pair of measured foliations
which fill a closed hyperbolic surface
and are {arational}, we prove that for sufficiently small
and can be uniquely realised as the
measured foliations at infinity of a quasi-Fuchsian manifold homeomorphic to
, which is sufficiently close to the Fuchsian locus. The
proof is based on that of Bonahon in \cite{bonahon05} which shows that a
quasi-Fuchsian manifold close to the Fuchsian locus can be uniquely determined
by the data of filling measured bending laminations on the boundary of its
convex core. Finally, we interpret the result in half-pipe geometry. For the
second part of the thesis we deal with a conjecture due to Thurston asks if
almost-Fuchsian manifolds admit a foliation by CMC surfaces. Here,
almost-Fuchsian manifolds are defined as quasi-Fuchsian manifolds which contain
a unique minimal surface with principal curvatures in and it is known
that in general, quasi-Fuchsian manifolds are not foliated by surfaces of
constant mean curvature (CMC) although their ends are. However, we prove that
almost-Fuchsian manifolds which are sufficiently close to being Fuchsian are
indeed monotonically foliated by surfaces of constant mean curvature. This work
is in collaboration with Filippo Mazzoli and Andrea Seppi.Comment: Ph.D. thesis, combination of the papers arXiv:2111.01614 and
arXiv:2204.0573
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