Riemann Roch: structure of genus 1 curves
This talk is about the Riemann Roch theorem in the spacial case of genus 1 curves or Riemann surface. We show that a compact Riemann surface satisfying the Riemann Roch theorem for g=1 is isomorphic to a nonsingular plane cubic. We show that this is topologically a torus, and use this to s
From playlist Algebraic geometry: extra topics
This talk is about some properties of plane curves used in the Riemann-Roch theorem. We first show that every nonsingular curve is isomorphic to a resolution of a plane curve with no singularities worse than ordinary double points (nodes). We then calculate the genus of plane curves with o
From playlist Algebraic geometry: extra topics
This talk is the first of two talks that give a proof of the Riemann Roch theorem, in the spacial case of nonsingular complex plane curves. We divide the Riemann-Roch theorem into 3 pieces: Riemann's theorem, a topological theorem identifying the three definitions of the genus, and Roch'
From playlist Algebraic geometry: extra topics
This talk is about the Riemann-Roch theorem for genus 3 curves. We show that any such curve is either hyperelliptic or a nonsingular plane quartic. We find the Weierstrass points and the holomorphic 1-forms and the canonical divisors of these curves. Finally we give a brief description of
From playlist Algebraic geometry: extra topics
This talk is about the Riemann-Roch theorem for genus 2 curves. We show that all genus 2 complex curves are hyperelliptic (meaning they are branched double covers of the projective line). We also describe the Weierstrass points and the holomorphic 1-forms explicitly. Finally we briefly su
From playlist Algebraic geometry: extra topics
I give a proof of the Cartan-Hadamard theorem on non-positively curved complete Riemannian manifolds. For more details see Chapter 7 of do Carmo's "Riemannian geomety". If you find any typos or mistakes, please point them out in the comments.
From playlist Differential geometry
We present a proof of the Hopf-Rinow theorem. For more details see do Carmo's "Riemannian geometry" Chapter 7.
From playlist Differential geometry
How to find the position function given the acceleration function
👉 Learn how to approximate the integral of a function using the Reimann sum approximation. Reimann sum is an approximation of the area under a curve or between two curves by dividing it into multiple simple shapes like rectangles and trapezoids. In using the Reimann sum to approximate the
From playlist Riemann Sum Approximation
Introduction to hyperbolic groups (Lecture – 01) by Mahan Mj
Geometry, Groups and Dynamics (GGD) - 2017 DATE: 06 November 2017 to 24 November 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has b
From playlist Geometry, Groups and Dynamics (GGD) - 2017
This lecture is part of an online course on algebraic geometry, following the book "Algebraic geometry" by Hartshorne. It is the first of a few elementary lectures on the Riemann-Roch theorem, mostly for compact complex curves. In this lecture we state the Riemann Roch theorem and explain
From playlist Algebraic geometry: extra topics
Geometry of the symmetric space SL(n,R)/SO(n,R)(Lecture – 01) by Pranab Sardar
Geometry, Groups and Dynamics (GGD) - 2017 DATE: 06 November 2017 to 24 November 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has b
From playlist Geometry, Groups and Dynamics (GGD) - 2017
Crash course in Riemanian geometry (Lecture - 03) by Harish Seshadri
Geometry, Groups and Dynamics (GGD) - 2017 DATE: 06 November 2017 to 24 November 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has b
From playlist Geometry, Groups and Dynamics (GGD) - 2017
Hyperbolic geometry, Fuchsian groups and moduli spaces (Lecture 1) by Subhojoy Gupta
ORGANIZERS : C. S. Aravinda and Rukmini Dey DATE & TIME: 16 June 2018 to 25 June 2018 VENUE : Madhava Lecture Hall, ICTS, Bangalore This workshop on geometry and topology for lecturers is aimed for participants who are lecturers in universities/institutes and colleges in India. This wi
From playlist Geometry and Topology for Lecturers
Riemannian Exponential Map on the Group of Volume-Preserving Diffeomorphisms - Gerard Misiolek
Gerard Misiolek University of Notre Dame; Institute for Advanced Study October 19, 2011 In 1966 V. Arnold showed how solutions of the Euler equations of hydrodynamics can be viewed as geodesics in the group of volume-preserving diffeomorphisms. This provided a motivation to study the geome
From playlist Mathematics
RT4.2. Schur's Lemma (Expanded)
Representation Theory: We introduce Schur's Lemma for irreducible representations and apply it to our previous constructions. In particular, we identify Hom(V,V) with invariant sesquilinear forms on V when (pi, V) is unitary. Course materials, including problem sets and solutions, availa
From playlist Representation Theory
D. Prandri - Weyl law for singular Riemannian manifolds
In this talk we present recent results on the asymptotic growth of eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable
From playlist Journées Sous-Riemanniennes 2018
Riemann Sum Defined w/ 2 Limit of Sums Examples Calculus 1
I show how the Definition of Area of a Plane is a special case of the Riemann Sum. When finding the area of a plane bound by a function and an axis on a closed interval, the width of the partitions (probably rectangles) does not have to be equal. I work through two examples that are rela
From playlist Calculus
An R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form which is a compact embedded totally geodesic Lagrangian submanifold.
From playlist Geometry
Z. Badreddine - Optimal transportation problem and MCP property on sub-Riemannian structures
This presentation is devoted to the study of mass transportation on sub-Riemannian geometry. In order to obtain existence and uniqueness of optimal transport maps, the first relevant method to consider is the one used by Figalli and Rifford which is based on the local semiconcavity of the
From playlist Journées Sous-Riemanniennes 2018
This talk is about the Riemann Roch theorem for genus 1 curves. We check the Riemann Roch theorem explicitly by using elliptic functions to find periodic functions with given divisors. We use this to show that the ring of functions on an affine elliptic curve is not a unique factorizati
From playlist Algebraic geometry: extra topics