Low-dimensional topology | Conjectures that have been proved | Smooth manifolds | Theorems in topology

The Smale conjecture, named after Stephen Smale, is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in 1983 by Allen Hatcher. (Wikipedia).

What is the Riemann Hypothesis?

This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation

From playlist Mathematics

The Most Difficult Math Problem You've Never Heard Of - Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture is a millennium prize problem, one of the famed seven placed by the Clay Mathematical Institute in the year 2000. As the only number-theoretic problem in the list apart from the Riemann Hypothesis, the BSD Conjecture has been haunting mathematicians

From playlist Math

Mertens Conjecture Disproof and the Riemann Hypothesis | MegaFavNumbers

#MegaFavNumbers The Mertens conjecture is a conjecture is a conjecture about the distribution of the prime numbers. It can be seen as a stronger version of the Riemann hypothesis. It says that the Mertens function is bounded by sqrt(n). The Riemann hypothesis on the other hand only require

From playlist MegaFavNumbers

Bruce KLEINER - Ricci flow, diffeomorphism groups, and the Generalized Smale Conjecture

The Smale Conjecture (1961) may be stated in any of the following equivalent forms: • The space of embedded 2-spheres in R3 is contractible. • The inclusion of the orthogonal group O(4) into the group of diffeomorphisms of the 3-sphere is a homotopy equivalence. • The s

From playlist Riemannian Geometry Past, Present and Future: an homage to Marcel Berger

Albert Einstein, Holograms and Quantum Gravity

In the latest campaign to reconcile Einstein’s theory of gravity with quantum mechanics, many physicists are studying how a higher dimensional space that includes gravity arises like a hologram from a lower dimensional particle theory. Read about the second episode of the new season here:

From playlist In Theory

Maciej Zworski - From redshift effect to classical dynamics : microlocal proof of Smale's conjecture

Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted "that a positive ans

From playlist Ecole d'été 2014 - Analyse asymptotique en relativité générale

What is the Birch and Swinnerton-Dyer Conjecture? - Manjul Bhargava [2016]

notes for this talk: https://drive.google.com/file/d/14K3JS0qDBWhsyyJXWHlw-jietZ6K2nXH/view?usp=sharing 2016 Clay Research Conference 28 September 2016 15:30 Manjul Bhargava What is the Birch and Swinnerton-Dyer Conjecture, and what is known about it? Introduced by Andrew Wiles Manjul B

From playlist Number Theory

In this video, we explore the "pattern" to prime numbers. I go over the Euler product formula, the prime number theorem and the connection between the Riemann zeta function and primes. Here's a video on a similar topic by Numberphile if you're interested: https://youtu.be/uvMGZb0Suyc The

From playlist Other Math Videos

ABC Intro - part 1 - What is the ABC conjecture?

This videos gives the basic statement of the ABC conjecture. It also gives some of the consequences.

From playlist ABC Conjecture Introduction

R. Haslhofer - The moduli space of 2-convex embedded spheres

We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}, i.e. the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). By Hatcher’s proof of the Smale conjecture, M_2 is contractible. This is a highly nontrivial theorem generalizing in particular the Schoenflies theor

From playlist Ecole d'été 2016 - Analyse géométrique, géométrie des espaces métriques et topologie

R. Bamler - Uniqueness of Weak Solutions to the Ricci Flow and Topological Applications 1

I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, wh

From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics

A (compelling?) reason for the Riemann Hypothesis to be true #SOME2

A visual walkthrough of the Riemann Zeta function and a claim of a good reason for the truth of the Riemann Hypothesis. This is not a formal proof but I believe the line of argument could lead to a formal proof.

From playlist Summer of Math Exposition 2 videos

R. Bamler - Uniqueness of Weak Solutions to the Ricci Flow and Topological Applications 1 (vt)

I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, wh

From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics

Caustics of fronts and the arborealization conjecture - Daniel Alvarez-Gavela

Short talks by postdoctoral members Topic: Caustics of fronts and the arborealization conjecture Speaker: Daniel Alvarez-Gavela Affiliation: Member, School of Mathematics Date: September 25, 2018 For more video please visit http://video.ias.edu

From playlist Mathematics

R. Bamler - Uniqueness of Weak Solutions to the Ricci Flow and Topological Applications 4 (vt)

I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, wh

From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics

R. Bamler - Uniqueness of Weak Solutions to the Ricci Flow and Topological Applications 4

From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics

The Million Dollar Problem that Went Unsolved for a Century - The Poincaré Conjecture

Topology was barely born in the late 19th century, but that didn't stop Henri Poincaré from making what is essentially the first conjecture ever in the subject. And it wasn't any ordinary conjecture - it took a hundred years of mathematical development to solve it using ideas so novel that

From playlist Math

Gabriel Rivière: Correlation spectrum of Morse-Smale flows

Abstract: I will explain how one can get a complete description of the correlation spectrum of a Morse-Smale flow in terms of the Lyapunov exponents and of the periods of the flow. I will also discuss the relation of these results with differential topology. This a joint work with Nguyen V

From playlist Topology