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What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

I received an email today about the award of the 2020 Nobel Prize in Physics to Roger Penrose, Reinhard Genzel and Andrea Ghez. Roger Penrose receives one-half of the prize "for the discovery that black hole formation is a robust prediction of the…
zeraoulia rafik
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49
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3 answers

Is the Riemann zeta function surjective?

Is the Riemann zeta function surjective or does it miss one value?
Shimrod
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49
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5 answers

Fundamental group as topological group

Background Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the subset of maps mapping $1 \to x$ with the…
Martin Brandenburg
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49
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14 answers

Interactive model of the hyperbolic plane for a general public lecture

The following is not quite a research level question, but I still find this site appropriate for asking it. I hope I get it right here. I am preparing a talk for a general public and I want to discuss some hyperbolic geometry. I wish I had a good…
Uri Bader
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49
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8 answers

Published results: when to take them for granted?

Two kinds of papers. There are two kinds of papers: self-contained ones, and those relying on published results (which I believe are the vast majority). Checking the result. Of course, one should check carefully other's results before using them.…
Thomas Sauvaget
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1 answer

Why is the Frankl conjecture hard?

This is a naive question that could justifiably be quickly closed. Nevertheless: Q. Why is Péter Frankl's conjecture so difficult? If any two sets in some family of sets have a union that also belongs to the family, must some element belong to at…
Joseph O'Rourke
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49
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How undecidable is the spectral gap?

Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled Undecidability of the Spectral Gap, there is an extended version on arxiv which is 146 pages long. Here is from the abstract:"Many challenging open problems, such as the Haldane…
Conifold
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49
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2 answers

Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be reductive. Hence my question : why is reductive…
Desiderius Severus
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49
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4 answers

Why is there a duality between spaces and commutative algebras?

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated reduced algebras over $\mathbb{C}$. The equivalence associates to an affine variety its algebra of regular functions to…
Yonatan Harpaz
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49
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8 answers

Roadmap for studying arithmetic geometry

I have read Hartshorne's Algebraic Geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know how to use scheme theory and its cohomology to solve arithmetic…
kiseki
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49
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20 answers

Sources for BibTeX entries

Does anyone know of a good place to find already-done BibTeX entries for standard books in advanced math? Or is this impossible because the citation should include items specific to your copy? (I am seeing the latter as potentially problematic…
Charles Staats
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4 answers

Intuition behind the definition of quantum groups

Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to improve the situation. The definition of the…
asv
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49
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Can knot diagrams be monotonically simplified using under moves?

It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger move that 3-manifold topologists sometimes use,…
Dylan Thurston
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2 answers

Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?
David Callan
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49
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7 answers

Way to memorize relations between the Sobolev spaces?

Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel lost. Are there certain tricks to memorize the…
Orbicular
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