Diophantine geometry | Theorems in number theory | Theorems in algebraic geometry
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field. (Wikipedia).
Lucia Mocz: A new Northcott property for Faltings height
Abstract: The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness stat
From playlist Algebraic and Complex Geometry
Jürg Kramer: Effective bounds for Faltings' delta function
In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces. For a given compact Riemann surface X of genus g, this invariant is roughly given as minus the logarithm of the distance of the point in the moduli space of genus g curve
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
A beautiful combinatorical proof of the Brouwer Fixed Point Theorem - Via Sperner's Lemma
Using a simple combinatorical argument, we can prove an important theorem in topology without any sophisticated machinery. Brouwer's Fixed Point Theorem: Every continuous mapping f(p) from between closed balls of the same dimension have a fixed point where f(p)=p. Sperner's Lemma: Ever
From playlist Cool Math Series
Introduction to additive combinatorics lecture 1.8 --- Plünnecke's theorem
In this video I present a proof of Plünnecke's theorem due to George Petridis, which also uses some arguments of Imre Ruzsa. Plünnecke's theorem is a very useful tool in additive combinatorics, which implies that if A is a set of integers such that |A+A| is at most C|A|, then for any pair
From playlist Introduction to Additive Combinatorics (Cambridge Part III course)
Introduction to additive combinatorics lecture 10.8 --- A weak form of Freiman's theorem
In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to
From playlist Introduction to Additive Combinatorics (Cambridge Part III course)
Faulhaber's polynomials via Excel | Algebraic Calculus One | Anna Tomskova
Dr Anna Tomskova shows how Excel is a powerful tool for generating important mathematical sequences and objects -- in this case the Faulhaber polynomials that capture the sums of powers of natural numbers, and that contain the Bernoulli numbers as coefficients. Excel is not just oriented
From playlist Algebraic Calculus One
David Corwin, Kim's conjecture and effective Faltings
VaNTAGe seminar, on Nov 24, 2020 License: CC-BY-NC-SA.
From playlist ICERM/AGNTC workshop updates
Proof & Explanation: Gauss's Lemma in Number Theory
Euler's criterion: https://youtu.be/2IBPOI43jek One common proof of quadratic reciprocity uses Gauss's lemma. To understand Gauss's lemma, here we prove how it works using Euler's criterion and the Legendre symbol. Quadratic Residues playlist: https://www.youtube.com/playlist?list=PLug5Z
From playlist Quadratic Residues
Jacob Tsimerman, Unlikely intersections and the André-Oort conjecture
VaNTAGe Seminar, December 7, 2021 License: CC-BY-NC-SA
From playlist Complex multiplication and reduction of curves and abelian varieties
VaNTAGe seminar, on Sep 15, 2020 License: CC-BY-NC-SA.
From playlist Rational points on elliptic curves
Karl Schwede: Ordinary reductions & F singularities
In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces. For a given compact Riemann surface X of genus g, this invariant is roughly given as minus the logarithm of the distance of the point in the moduli space of genus g curve
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
Smooth representations of reductive p-adic groups over arbitrary fields by Marie France Vigneras
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Ahmed Abbes - The p-adic Simpson correspondence: Functoriality by proper direct image and (...) 1/3
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors and whose properties have been developed according to several approaches. I will present in these lectures the approach I developed with Michel Gros,
From playlist Franco-Asian Summer School on Arithmetic Geometry (CIRM)
p r -Selmer companion modular forms by Somnath Jha
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
p-adic automorphic forms in the sense of Scholze (Lecture 1) by Aditya Karnataki
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Convolution Theorem: Fourier Transforms
Free ebook https://bookboon.com/en/partial-differential-equations-ebook Statement and proof of the convolution theorem for Fourier transforms. Such ideas are very important in the solution of partial differential equations.
From playlist Partial differential equations
Classicality of overconvergent modular symbols by Fabrizio Andreatta
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Theory of numbers: Congruences: Euler's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Euler's theorem, a generalization of Fermat's theorem to non-prime moduli, by using Lagrange's theorem and group theory. As an application of Fermat's theorem we show there are infinitely many prim
From playlist Theory of numbers