MathOverflow Community Digest

Top new questions this week:

Does there exist a complete implementation of the Risch algorithm?

Is there a generally available (commercial or not) complete implementation of the Risch algorithm for determining whether an elementary function has an elementary antiderivative? The Wikipedia article ...

algorithms integration computer-algebra differential-algebra  
asked by Timothy Chow 32 votes
answered by Sam Blake 16 votes

Function of $(x_1,x_2,x_3,x_4)$ that factors in two ways as $\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$

Suppose we have a function $f(x_1 ,x_2 ,x_3 ,x_4).$ We know that we can factor it in two ways as $f(x_1 ,x_2 ,x_3 ,x_4)=\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$ Show that ...

co.combinatorics graph-theory ra.rings-and-algebras  
asked by Daniel Li 22 votes
answered by Tony Huynh 22 votes

Need for support and guidance for my near future as a PhD student (or: has stable homotopy theory become an overly algebraic theory ?)

The question in brackets in the title is my main mathematical question, but does not reflect my initial motivation for writing this post. It is in fact above all for personal reasons that I'm ...

at.algebraic-topology gt.geometric-topology soft-question stable-homotopy  
asked by buck 21 votes

Is being simply connected very rare?

Essentially, my question is how strong a restriction it is to be simply connected. Here is a way of making this precise: Let's say we want to count simplicial complexes (of dimension 2, though that ...

co.combinatorics at.algebraic-topology gt.geometric-topology computational-topology  
asked by Karim Adiprasito 19 votes
answered by Tim Campion 7 votes

Are the Morse inequalities sharp for 5-manifolds

Given a compact smooth manifold $M$ denote by $b_i(M)$ the $i$-th Betti number and denote by $q_i(M)$ the minimal number of generators for $H_i(M)$. Let $f$ be a Morse function on $M$. The Morse ...

gt.geometric-topology  
asked by Stefan Friedl 18 votes
answered by Danny Ruberman 8 votes

Is the Alexander horned sphere a cofibration?

The Alexander horned sphere is a closed embedding of $S^2$ into $S^3$ which is not flat because otherwise the Schoenflies Theorem would be true for it. However, not being flat is not the same as not ...

at.algebraic-topology gt.geometric-topology  
asked by daniel 15 votes
answered by Tyrone 17 votes

Pontrjagin-Thom model for units of the sphere spectrum?

Is there a framed bordism model for the units of the sphere spectrum, $gl_1(S)$? At the level of individual homotopy groups, the Pontrjagin-Thom construction identifies the group of bordism classes of ...

at.algebraic-topology  
asked by pupshaw 14 votes

Greatest hits from previous weeks:

Proofs where higher dimension or cardinality actually enabled much simpler proof?

I am very interested in proofs that become shorter and simpler by going to higher dimension in $\mathbb R^n$, or higher cardinality. By "higher" I mean that the proof is using higher dimension or ...

ag.algebraic-geometry dg.differential-geometry graph-theory gt.geometric-topology big-list  
asked by Claus Dollinger 67 votes
answered by Gabe K 55 votes

Have any long-suspected irrational numbers turned out to be rational?

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...

nt.number-theory real-analysis soft-question big-list  
asked by I. J. Kennedy 199 votes
answered by Gjergji Zaimi 242 votes

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 77 votes
answered by Scott Aaronson 105 votes

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...

reference-request big-list examples intuition slick-proof  
asked by Mariano Suárez-Álvarez 350 votes
answered by Mariano Suárez-Álvarez 506 votes

Coin pusher game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins). Essentially, one has a distribution of ...

pr.probability reference-request st.statistics stochastic-processes  
asked by Alex R. 11 votes
answered by Diamond Jim 5 votes

John Nash's Mathematical Legacy

It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 (reference). My condolences to his family and friends. Maybe this is an appropriate time to ask a ...

riemannian-geometry big-list game-theory  
asked by Paul Siegel 221 votes
answered by Deane Yang 61 votes

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...

soft-question big-list gm.general-mathematics  
asked by Ilya Nikokoshev 193 votes
answered by Daniel Moskovich 202 votes

Can you answer these questions?

Fourier Transforms exhibiting symmetries about their critical points

Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...

reference-request fourier-analysis fourier-transform symmetry critical-point-theory  
asked by John Clever 1 vote

Fiber of the Hitchin map

Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...

ag.algebraic-geometry vector-bundles moduli-spaces  
asked by Aoki 1 vote

A bilinear estimates involving critical Sobolev norms

Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering ...

fa.functional-analysis real-analysis sobolev-spaces  
asked by Capublanca 1 vote
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