## 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 $k\neq 0$) ...

at.algebraic-topology
 asked by Alexandre Eremenko Score of 22
 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
 asked by Gro-Tsen Score of 21
 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
 asked by Monroe Eskew Score of 15
 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
 asked by Sofía Marlasca Aparicio Score of 13
 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
 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
 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
 asked by Craig Feinstein Score of 10
 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
 asked by Michael Lugo Score of 82
 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
 asked by MWB Score of 143
 answered by Andrej Bauer Score of 221

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 $\frac{a}{b+c} + \... nt.number-theory elliptic-curves diophantine-equations rational-points  asked by alex alexeq Score of 163  answered by Michael Stoll Score of 124 ### What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis? In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function$T(s)$, known as the Todd function. My question is, what is the ... open-problems definitions riemann-hypothesis  asked by Keshav Srinivasan Score of 106  answered by Wilem2 Score of 24 ### The enigmatic complexity of number theory One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ... nt.number-theory mathematical-philosophy metamathematics  asked by David G. Stork Score of 82  answered by Scott Aaronson Score of 109 ### Reading list for basic differential geometry? I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ... dg.differential-geometry reading-list  asked by GMRA Score of 81  answered by Alon Amit Score of 31 ### Examples of great mathematical writing This question is basically from Ravi Vakil's web page, but modified for Math Overflow. How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ... mathematical-writing soft-question big-list  asked by Anton Geraschenko Score of 190  answered by Ilya Grigoriev Score of 139 ## Can you answer these questions? ### Is there an explicit solution to the reaction diffusion system in the following special form? Suppose$\Omega \subset R^N$is a smooth bounded domain. Is there an explicit solution to the reaction diffusion equations (RDE) in the following special form? \left\{\... mp.mathematical-physics differential-equations  asked by Young22 Score of 1 ### A function$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  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
 asked by Nik Pronko Score of 1
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