Systolic geometry | Riemann surfaces
In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms. They are also referred to as Hurwitz curves, interpreting them as complex algebraic curves (complex dimension 1 = real dimension 2). The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the (ordinary) (2,3,7) triangle group. The finite quotient group is precisely the automorphism group. Automorphisms of complex algebraic curves are orientation-preserving automorphisms of the underlying real surface; if one allows orientation-reversing isometries, this yields a group twice as large, of order 168(g − 1), which is sometimes of interest. A note on terminology – in this and other contexts, the "(2,3,7) triangle group" most often refers, not to the full triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which is index 2. The group of complex automorphisms is a quotient of the ordinary (orientation-preserving) triangle group, while the group of (possibly orientation-reversing) isometries is a quotient of the full triangle group. (Wikipedia).
MATH331: Riemann Surfaces - part 1
We define what a Riemann Surface is. We show that PP^1 is a Riemann surface an then interpret our crazy looking conditions from a previous video about "holomorphicity at infinity" as coming from the definition of a Riemann Surface.
From playlist The Riemann Sphere
The Routh-Hurwitz Stability Criterion
In this video we explore the Routh Hurwitz Stability Criterion and investigate how it can be applied to control systems engineering. The Routh Hurwitz Stability Criterion can be used to determine how many roots of a polynomial are in the right half plane. Topics and time stamps: 0:00 –
From playlist Control Theory
Fabio Tanturri: On the unirationality of Hurwitz spaces
Abstract: In this talk I will discuss about the unirationality of the Hurwitz spaces H_g,d parametrizing d-sheeted branched simple covers of the projective line by smooth curves of genus g. I will summarize what is already known and formulate some questions and speculations on the general
From playlist Algebraic and Complex Geometry
Maxim Kazarian - 3/3 Mathematical Physics of Hurwitz numbers
Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge num
From playlist Physique mathématique des nombres de Hurwitz pour débutants
Maxim Kazarian - 2/3 Mathematical Physics of Hurwitz numbers
Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge num
From playlist Physique mathématique des nombres de Hurwitz pour débutants
Maxim Kazarian - 1/3 Mathematical Physics of Hurwitz numbers
Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge num
From playlist Physique mathématique des nombres de Hurwitz pour débutants
Dimitri Zvonkine - Hurwitz numbers, the ELSV formula, and the topological recursion
We will use the example of Hurwitz numbers to make an introduction into the intersection theory of moduli spaces of curves and into the subject of topological recursion.
From playlist Physique mathématique des nombres de Hurwitz pour débutants
Algebraic geometry 45: Hurwitz curves
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It discusses Hurwitz curves and sketches a proof of Hurwitz's bound for the symmetry group of a complex curve.
From playlist Algebraic geometry I: Varieties
Rahul Pandharipande - Enumerative Geometry of Curves, Maps, and Sheaves 2/5
The main topics will be the intersection theory of tautological classes on moduli space of curves, the enumeration of stable maps via Gromov-Witten theory, and the enumeration of sheaves via Donaldson-Thomas theory. I will cover a mix of classical and modern results. My goal will be, by th
From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory
Billiards in quadrilaterals, Hurwitz spaces, and real multiplication of Hecke type - Alex Wright
Members' Seminar Topic: Billiards in quadrilaterals, Hurwitz spaces, and real multiplication of Hecke type Speaker: Alexander Wright Affiliation: Stanford University; Member, School of Mathematics Monday, November 30 Video Link: https://video.ias.edu/membsem/2015/1130-Wright After a brief
From playlist Mathematics
John Voight, Belyi maps in number theory: a survey
VaNTAGe Seminar, August 17, 2021 License CC-BY-NC-SA
From playlist Belyi maps and Hurwitz spaces
Edray Goins, Critical points of toroidal Belyi maps
VaNTAGe seminar, August 31, 2021 License CC-BY-NC-SA
From playlist Belyi maps and Hurwitz spaces
Algebraic geometry 46: Examples of Hurwitz curves
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It gives examples of complex curves of genus 2 and 3 with the largest possible symmetry groups .
From playlist Algebraic geometry I: Varieties
Integral points on Markoff-type cubic surfaces - Amit Ghosh
Special Seminar Topic: Integral points on Markoff-type cubic surfaces Speaker: Amit Ghosh Affiliation: Oklahoma State University Date: December 8, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics
David Roberts, Hurwitz Belyi maps
VaNTAGe seminar, October 12, 2021 License: CC-BY-NC-SA
From playlist Belyi maps and Hurwitz spaces
Michael Magee: Thermodynamical formalism and Markoff-Hurwitz equations
The lecture was held within the framework of the Hausdorff Trimester Program "Dynamics: Topology and Numbers": Conference on “Transfer operators in number theory and quantum chaos” Abstract: Beginning with the simple question ’when is the sum of the squares of a tuple of integersequal to
From playlist Conference: Transfer operators in number theory and quantum chaos
Computational Aspects in the Braid Group and Applications to Cryptography - Mina Teicher
Mina Teicher Bar-Ilan University; Member, School of Mathematics March 12, 2012 The braid group on n strands may be viewed as an infinite analog of the symmetric group on n elements with additional topological phenomena. It appears in several areas of mathematics, physics and computer scien
From playlist Mathematics
An invitation to higher Teichmüller theory – Anna Wienhard – ICM2018
Geometry Invited Lecture 5.11 An invitation to higher Teichmüller theory Anna Wienhard Abstract: Riemann surfaces are of fundamental importance in many areas of mathematics and theoretical physics. The study of the moduli space of Riemann surfaces of a fixed topological type is intimatel
From playlist Geometry