11 research outputs found
Ovoids and spreads of finite classical generalized hexagons and applications
One intuitively describes a generalized hexagon as a point-line geometry full of ordinary hexagons, but containing no ordinary n-gons for n<6. A generalized hexagon has order (s,t) if every point is on t+1 lines and every line contains s+1 points. The main result of my PhD Thesis is the construction of three new examples of distance-2 ovoids (a set of non-collinear points that is uniquely intersected by any chosen line) in H(3) and H(4), where H(q) belongs to a special class of order (q,q) generalized hexagons. One of these examples has lead to the construction of a new infinite class of two-character sets. These in turn give rise to new strongly regular graphs and new two-weight codes, which is why I dedicate a whole chapter on codes arising from small generalized hexagons. By considering the (0,1)-vector space of characteristic functions within H(q), one obtains a one-to-one correspondence between such a code and some substructure of the hexagon. A regular substructure can be viewed as the eigenvector of a certain (0,1)-matrix and the fact that eigenvectors of distinct eigenvalues have to be orthogonal often yields exact values for the intersection number of the according substructures. In my thesis I reveal some unexpected results to this particular technique. Furthermore I classify all distance-2 and -3 ovoids (a maximal set of points mutually at maximal distance) within H(3). As such we obtain a geometrical interpretation of all maximal subgroups of G2(3), a geometric construction of a GAB, the first sporadic examples of ovoid-spread pairings and a transitive 1-system of Q(6,3). Research on derivations of this 1-system was followed by an investigation of common point reguli of different hexagons on the same Q(6,q), with nice applications as a result. Of these, the most important is the alternative construction of the Hölz design and a subdesign. Furthermore we theoretically prove that the Hölz design on 28 points only contains Hermitian and Ree unitals (previously shown by Tonchev by computer). As these Hölz designs are one-point extensions of generalized quadrangles, we dedicate a final chapter to the characterization of the affine extension of H(2) using a combinatorial property
Doubly transitive parabolic ovals in affine planes of even order
In the present paper all parabolic ovals in affine planes of even order which are preserved by a collineation group isomorphic to are determined. It turns out that hey are either parabolas or translation ovals
Ahlfors circle maps and total reality: from Riemann to Rohlin
This is a prejudiced survey on the Ahlfors (extremal) function and the weaker
{\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e.
those (branched) maps effecting the conformal representation upon the disc of a
{\it compact bordered Riemann surface}. The theory in question has some
well-known intersection with real algebraic geometry, especially Klein's
ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a
gallery of pictures quite pleasant to visit of which we have attempted to trace
the simplest representatives. This drifted us toward some electrodynamic
motions along real circuits of dividing curves perhaps reminiscent of Kepler's
planetary motions along ellipses. The ultimate origin of circle maps is of
course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass.
Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found
in Klein (what we failed to assess on printed evidence), the pivotal
contribution belongs to Ahlfors 1950 supplying an existence-proof of circle
maps, as well as an analysis of an allied function-theoretic extremal problem.
Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree
controls than available in Ahlfors' era. Accordingly, our partisan belief is
that much remains to be clarified regarding the foundation and optimal control
of Ahlfors circle maps. The game of sharp estimation may look narrow-minded
"Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to
contemplate how conformal and algebraic geometry are fighting together for the
soul of Riemann surfaces. A second part explores the connection with Hilbert's
16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by
including now Rohlin's theory (v.2
Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
Our monograph presents the foundations of the theory of groups and semigroups
acting isometrically on Gromov hyperbolic metric spaces. Our work unifies and
extends a long list of results by many authors. We make it a point to avoid any
assumption of properness/compactness, keeping in mind the motivating example of
, the infinite-dimensional rank-one symmetric space of
noncompact type over the reals. The monograph provides a number of examples of
groups acting on which exhibit a wide range of phenomena not
to be found in the finite-dimensional theory. Such examples often demonstrate
the optimality of our theorems. We introduce a modification of the Poincar\'e
exponent, an invariant of a group which gives more information than the usual
Poincar\'e exponent, which we then use to vastly generalize the Bishop--Jones
theorem relating the Hausdorff dimension of the radial limit set to the
Poincar\'e exponent of the underlying semigroup. We give some examples based on
our results which illustrate the connection between Hausdorff dimension and
various notions of discreteness which show up in non-proper settings. We
construct Patterson--Sullivan measures for groups of divergence type without
any compactness assumption. This is carried out by first constructing such
measures on the Samuel--Smirnov compactification of the bordification of the
underlying hyperbolic space, and then showing that the measures are supported
on the bordification. We study quasiconformal measures of geometrically finite
groups in terms of (a) doubling and (b) exact dimensionality. Our analysis
characterizes exact dimensionality in terms of Diophantine approximation on the
boundary. We demonstrate that some Patterson--Sullivan measures are neither
doubling nor exact dimensional, and some are exact dimensional but not
doubling, but all doubling measures are exact dimensional.Comment: A previous version of this document included Section 12.5 (Tukia's
isomorphism theorem). The results of that subsection have been split off into
a new document which is available at arXiv:1508.0696
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Gauge Theories, Toric Varieties and Machine Learning
We discuss some recent study on quiver gauge theories in the setting of toric geometry. After mentioning some basic geometric and topological properties, we consider some mathematical concepts, namely the Mahler measure and the dessins d’enfants, in this context. We then focus on the quiver BPS algebras and their connections to diferent aspects in physics. We also have a discussion on the stability of chiral rings for more general geometry. Besides, we make some comments on the applications of machine learning to relevant topics. This thesis is based on the works [1–16]
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum