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Top new questions this week:

A property of even continuous functions on the sphere

This question is inspired by On moments of inertia of planar and 3D convex bodies. Let $f:{\mathbb R}^3\setminus\{0\}\to{\mathbb R}$ be an even homogeneous ($f(kx)=f(x)$ for all real $k\neq 0$) ...

user avatar asked by Alexandre Eremenko Score of 22
user avatar answered by Saúl RM Score of 19

What is the origin/history of the following very short definition of the Lebesgue integral?

Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...

real-analysis ca.classical-analysis-and-odes ho.history-overview teaching lebesgue-measure  
user avatar asked by Gro-Tsen Score of 21
user avatar answered by Kostya_I Score of 18

Strategic vs. tactical closure

The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when ...

set-theory lo.logic forcing infinite-games  
user avatar asked by Monroe Eskew Score of 15
user avatar answered by Joel David Hamkins Score of 8

Is there a good theory of solid vector spaces?

Lately I have become interested in solid $F$-modules where $F$ is some discrete field. Ideally, one would want a category that is as nicely behaved as solid abelian groups or solid $\mathbb{F_p}$-...

ct.category-theory condensed-mathematics  
user avatar asked by Sofía Marlasca Aparicio Score of 13
user avatar answered by Peter Scholze Score of 10

What is the complexity of the winning condition in infinite Hex? In particular, is infinite Hex a Borel game?

Consider the game of infinite Hex, where two players Red and Blue alternately place their stones on the infinite hex grid, each aiming to create a winning configuration. Red wins after infinite play, ...

set-theory descriptive-set-theory combinatorial-game-theory determinacy infinite-games  
user avatar asked by Joel David Hamkins Score of 13

Finiteness for motivic local systems

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$. Say a complex local system $\mathbb{V}$ on $X$ is motivic if there exists a dense Zariski-open subset $U\subset X$, and a smooth proper ...

ag.algebraic-geometry arithmetic-geometry local-systems  
user avatar asked by Daniel Litt Score of 11

Why is integer factoring hard while determining whether an integer is prime easy?

In 2002, the discovery of the AKS algorithm proved that it is possible to determine whether an integer is prime in polynomial time deterministically. However, it is still not known whether there is an ...

prime-numbers computational-number-theory mathematical-philosophy  
user avatar asked by Craig Feinstein Score of 10
user avatar answered by Timothy Chow Score of 30

Greatest hits from previous weeks:

If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...

pr.probability euclidean-geometry  
user avatar asked by Michael Lugo Score of 82
user avatar answered by Kevin P. Costello Score of 108

What makes dependent type theory more suitable than set theory for proof assistants?

In his talk, The Future of Mathematics, Dr. Kevin Buzzard states that Lean is the only existing proof assistant suitable for formalizing all of math. In the Q&A part of the talk (at 1:00:00) he ...

set-theory soft-question foundations type-theory proof-assistants  
user avatar asked by MWB Score of 143
user avatar answered by Andrej Bauer Score of 221

Estimating the size of solutions of a diophantine equation

A. Is there natural numbers $a,b,c$ such that $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that $\frac{a}{b+c} + \...

nt.number-theory elliptic-curves diophantine-equations rational-points  
user avatar asked by alex alexeq Score of 163
user avatar answered by Michael Stoll Score of 124

What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...

open-problems definitions riemann-hypothesis  
user avatar asked by Keshav Srinivasan Score of 106
user avatar answered by Wilem2 Score of 24

The enigmatic complexity of number theory

One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ...

nt.number-theory mathematical-philosophy metamathematics  
user avatar asked by David G. Stork Score of 82
user avatar answered by Scott Aaronson Score of 109

Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...

dg.differential-geometry reading-list  
user avatar asked by GMRA Score of 81
user avatar answered by Alon Amit Score of 31

Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow. How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...

mathematical-writing soft-question big-list  
user avatar asked by Anton Geraschenko Score of 190
user avatar answered by Ilya Grigoriev Score of 139

Can you answer these questions?

Is there an explicit solution to the reaction diffusion system in the following special form?

Suppose $\Omega \subset R^N$ is a smooth bounded domain. Is there an explicit solution to the reaction diffusion equations (RDE) in the following special form? \begin{equation} \left\{\...

mp.mathematical-physics differential-equations  
user avatar asked by Young22 Score of 1

A function $f_r$ where $f_r (x)$ is defined as the ratio between $f(x)$ and the average value of $f$ over $B(x, r)$

Let $E := \mathbb R^d$. Let $f:E \to \mathbb R_{>0}$ be continuous and integrable. For $r>0$, we define $$ f_r (x) := \frac{f(x)}{ \frac{1}{|B(x, r)|} \int_{B(x, r)} f(y) \, \mathrm{d} y} \quad \...

reference-request real-analysis calculus-of-variations  
user avatar asked by Analyst Score of 1

Measure preserving maps of pseudo-Lebesgue measure in infinite-dimensional vector space

Let $I =([0,1),\mathcal{B},\lambda)$ stand for the unit interval with a Lebesgue measure constrained on it. This is just a uniform probability distribution, an infinite power $I^{\infty}$ is a well ...

fa.functional-analysis measure-theory  
user avatar asked by Nik Pronko Score of 1
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