All Questions
152,871
questions
3
votes
0
answers
241
views
Failure of periodic cyclic homology to be a localizing invariant
A localizing invariant $E: dgcat_{sm} \rightarrow Ch_k$ from small dg categories to chain complexes over a field $k$ (say $k = \mathbb{C}$) is a dg functor which sends localization sequences to exact ...
2
votes
1
answer
89
views
Is the "composition" of two dense subsets of functions dense?
Given $F \subseteq C_C(\mathbb{R}^d, \mathbb{R}^p)$, $F$ is dense in $C_C(\mathbb{R}^d, \mathbb{R}^p)$ in the supremum norm $\|\cdot\|_\infty$. Also given $G \subseteq C_C(\mathbb{R}^p, \mathbb{R}^s)$,...
3
votes
1
answer
150
views
Product of types in stable theories
Let $T$ be stable and $M \models T$. Given two types $p(x) \in S_x(M)$ and $q(y) \in S_y(M)$ there is a canonical way to get a type in $S_{xy}(M)$. Define
$$p(x) \otimes q(y) = tp_{xy}(ab/M)$$
where $...
4
votes
1
answer
696
views
Cohomology of $BE_8$ and $BSU(2)$
What are the cohomology of the classifying space of $E_8$ group and $SU(2)$ group, $H^*(BE_8;\mathbb{Z})$ and $H^*(BSU(2);\mathbb{Z})$?
In the paper
http://homepages.math.uic.edu/~bshipley/...
1
vote
0
answers
109
views
Toral subgroup acting regularly on the homogeneous space
Let $G$ be a connected second countable compact Hausdorff group, and let $H\subset G$ be a closed subgroup such that the homogeneous space $G/H$ has continuum cardinality. For every $x\in G/H$ let $...
6
votes
0
answers
195
views
Units in $B\otimes_{A}B$, where $B/A$ is a finite Galois extension of number rings
I need information or directions to the literature regarding the structure of the group of units of $B\otimes_{A}B$, where $B/A$ is a finite Galois extension of rings of integers, say associated to a ...
2
votes
0
answers
59
views
Maximum number of edges on $2^{k-1}+s$ vertices of a $k$-dimensional cube?
Let $k$ be an even number. For a $k$-dimensional cube (http://mathworld.wolfram.com/HypercubeGraph.html) $Q_k$, let $G$ be a subgraph of $Q_k$ with $2^{k-1}+s$ vertices, for $1\le s\le 2^{k-1}-1$. I ...
13
votes
2
answers
647
views
The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
1
vote
0
answers
77
views
Compute action of the gauge group in deformation theory of an algebra
I am working on Balazs Szendroi's introduction to deformation theory, but I got stuck on the exercise on the bottom of page 6.
Consider a vector space $A$ with a multiplication $m$ that makes it into ...
1
vote
0
answers
195
views
How can we construct a derived scheme as a gluing of derived schemes?
More precisely, consider a Segal groupoid $X_*$ in an infinity category of derived schemes : dSch
In Toen's note, 'Derived Algebraic Geometry', he defines a 1-Artin stacks as a homotopy colimits of ...
16
votes
3
answers
2k
views
Taking a proper class as a model for Set Theory
When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the ...
1
vote
1
answer
176
views
Is there an elementary reason for why $SL_2(\mathbb{F}_p)$ for $p>5$ does not embed into $SL_2(\mathbb{Z}_p[w])?$
This is an exercise from Serre's book on Galois cohomology.
Let $p>5$ and consider the groups $SL_2(\mathbb{F}_p)$ and $SL_2(\mathbb{Z}_p[w])$ where $w$ is a primitive $p$th root of unity.
Is ...
9
votes
1
answer
412
views
Countable support iteration of proper forcings and the tree property
I'm mainly concerned with countable support iterations of proper forcings that add reals of some large cardinal length. It is known that countable support iteration of Sacks forcing/Cohen forcing of ...
4
votes
0
answers
927
views
Base change of integral scheme of finite type over a field
Let $X$ be an integral scheme of finite type over a field $k$. If $k'\supseteq k$ is a field extension, then $X' = X\otimes_k k'$ is not necessarily integral.
Why does each irreducible component of ...
5
votes
0
answers
200
views
Resultant of a binomial and a trinomial
Does anyone know of any papers dealing with the resultant of a binomial $x^n+a$ and a trinomial $x^r+bx^s+c$ ?
Even special cases would be of interest.
(The resultant of two binomials is well known.)...
7
votes
0
answers
290
views
Generalizing Gromov Hausdorff distance using Vietoris topology
There are two notions of convergence of a sequence of metric space. One is by the Gromov Hausdorff distance for compact metric spaces, another one is the pointed Gromov Hausdorff convergence for ...
13
votes
1
answer
577
views
Gabriel-Ulmer duality for $\infty$-categories
Gabriel-Ulmer duality states that 2-categories $\mathrm{Lex}$ (of small finitely complete categories and functors preserving finite limits) and $\mathrm{LFP}$ (of locally finitely presentable ...
17
votes
3
answers
1k
views
Axioms for constructive Euclidean geometry
In the summer I will be teaching a course in (plane) Euclidean geometry to future high school teachers and I am looking for a suitable axiom system (unlike College (Euclidean) geometry textbook ...
8
votes
1
answer
513
views
Are the following subsets of a Hilbert space always homeomorphic?
Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...
8
votes
1
answer
4k
views
Properties of Zero Line-Sum Matrices
By a Zero Line-Sum (ZLS) matrix I mean matrices with the property, that each row sum and each column sum equals zero:
$$A\in\mathbb{R}^{m\times n}:\ \sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{m}a_{ij}=0$$
...
2
votes
1
answer
153
views
On sections into Banach bundles over a compact manifold
Let $M$ be a smooth, compact manifold and $\xi: \mathcal B \to M$ a smooth complex Banach bundle over $M$. Here, smooth is understood to be in the Fréchet-sense. Further, let $p: V \to M$ be an ...
11
votes
1
answer
855
views
Simulate coin tossing by die tossing
On the one hand we toss $n$ times a fair coin, and we sum the outcomes (+1 for heads, -1 for tails). Let $f:\mathbb{N}\to\mathbb{R}$ describe the probability distribution of the outcome.
On the other ...
6
votes
1
answer
238
views
Regulator of abelian extensions of Q
Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$:
$$\zeta_K(s) \sim s^{r_1+r_2-1}h(K)R(K)/w(K) $$
where $h,R,w$ stand for ...
1
vote
1
answer
138
views
does $Aut^0$ act trivially on the Neron-Severi group?
Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does $\mathrm{Aut}^0_{X/k}(k)$ act trivially on $NS(X)$?
5
votes
1
answer
728
views
On faces of convex sets of positive semidefinite matrices
A face $F$ of a convex set $C$ is a set such that, if $x \in F$ is a convex combination of other elements in $C$, then they must also be in $F$. I will denote by $F(C,x)$ the smallest face of $C$ ...
7
votes
2
answers
403
views
Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give
Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing ...
6
votes
0
answers
206
views
Dimension of space of K-fixed vectors
If $G$ is an unramified group over an $p$-adic field $F$, the Satake isomorphism identifies the spherical Hecke algebra with respect to a special maximal compact subgroup $K$. In particular,
(1) $H(G(...
5
votes
0
answers
209
views
Non existence of commutative singular cochains vs quasi iso between cochains on BT and its cohomology
I believe that there isn't a commutative model for the DGA of cochains on a space, because of cohomology operations. This question has some nice explanations of why this is so. For example, there is a ...
17
votes
1
answer
2k
views
What is known about the relationship between Fermat's last theorem and Peano Arithmetic?
As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem?
In particular, what is known about the arithmetic systems $PA + \...
3
votes
1
answer
538
views
Absolute Hodge cycles
Let $X$ is a smooth projective variety defined over a finite extension $K/\mathbf{Q}$, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{...
1
vote
1
answer
476
views
The combinations of a finite multiset [closed]
Suppose there is a basket $S$ containing $3\ \color{blue}{blue}$, $2\ \color{green}{green}$ and $1\ \color{red}{red}$ balls. A subject can extract any $k$ number of balls (including $0$) at random ...
0
votes
0
answers
149
views
Connected unipotent groups acting on an affine variety (re: stabilizers)
Let $U$ be a connected unipotent algebraic group over a field of characteristic $p>0$. Assume $U$ acts on an affine variety $X$ by regular maps.
Is it true that the stabilizers of rational points ...
2
votes
1
answer
218
views
Does cohomology with compact support contain "ample" elements
Let $X$ be a quasi-projective integral variety over $\mathbb{C}$. If $X$ is projective, then $\mathrm{H}^2(X,\mathbb{Z})$ contains "ample" classes. These "ample" classes are defined as being the image ...
4
votes
1
answer
222
views
No lifts in an exact sequence of profinite groups?
In pg. 24 of his book on Galois cohomology, Serre gives the following exercise:
"Give an example of an extension $1 \to P \to E \to G \to 1$ of profinite groups with the following properties:
(i) $...
1
vote
1
answer
195
views
What are some relatively unknown solution concepts in cooperative game theory that are useful in a specific context?
In cooperative game theory, the payoffs for the grand coalition can be distributed in a number of ways. Each of those ways is a solution concept. Well-known examples of solution concepts include the ...
2
votes
0
answers
69
views
A possible obstruction for existence of limit cycle for analytic vector field on $S^2$
Is there an analytic vector field $X$,on $S^2$ which possess a limit cycle but $X $, satisfy $\nabla_X X =0$ or satisfy $\nabla_X JX= 0$ where $J$ is the standard almost ...
-2
votes
1
answer
73
views
What is an algorithm for generating a set of null (spacetime) vectors that add to zero? [closed]
I am interested in generating a list of n 4-vectors (t,x,y,z) such that -t^2+x^2+y^2+z^2=0 for each vector and the sum of the n 4-vectors equals zero. All of the t,x,y,z are real. I, particular, I am ...
3
votes
2
answers
209
views
$Ind_H^G 1_H-1_G$ as direct sum of monomials
If $G$ is a finite solvable group, it is known (for example, Murty & Raghuram 2000, lemma 2.4) that $Ind_H^G 1_H-1_G$ can be expressed directly (with all coefficients $=1$) as a sum of monomial ...
5
votes
2
answers
241
views
How to prove that a certain curve does not lie in a proper abelian subvariety of abelian variety
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ by the equation
$$E: y^2=x^3-Ax+B=:f(x).$$
Consider the abelian variety $E^3:=E \times E \times E \subset \mathbb{P}^2 \times \mathbb{P}^2 \...
0
votes
2
answers
283
views
quasi-affine-ness [closed]
Let $G$ be a group. Let $H$ be a subgroup of $R_u(G)$. Then $G/H\rightarrow G/R_u(G)$ is a $R_u(G)/H$ fibration. It is well known that
$R_u(G)/H=\mathbb{A}^n$. Is $G/H$ a quasi-affine variety?
1
vote
0
answers
62
views
Bounds on the tensor and border rank ratios of tensor unfoldings?
Consider $T_1 \in \mathbb{C}^{d_1 \times \cdots \times d_k}$ as an order-$k$ tensor of the format $(d_1, \cdots , d_k)$. Now, let $T_2$ be an unfolding of $T_1$ that is obtained by merging $\mathbb{C}^...
6
votes
1
answer
290
views
Explicit local normal form symplectomorphism at torus fixed point of a coadjoint orbit
Let $K$ be a compact, connected (probably also simple) Lie group and with a maximal torus $T$. Regular coadjoint orbits $\mathcal{O}_{\lambda} \cong K/T$, parameterized by a regular element $\lambda \...
10
votes
0
answers
209
views
2-generator subgroups of an Artin group of small type
Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...
0
votes
1
answer
232
views
The Abstraction of Equality [closed]
In finitely presented groups, we can define equivalence classes simply by writing equations in the generators : $abc=d$. In this equivalence class we find elements like this $a(aa^{-1})bc$. We can ...
3
votes
1
answer
154
views
A matrix monotonicity question
Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane....
2
votes
1
answer
234
views
Can degenerations have derived equivalent fibres
Let $\pi: X \rightarrow B$ be a proper flat morphism of varieties and $0 \in B$ be a closed point such that on $B \setminus 0$ the morphism $\pi$ is trivial, isomorphic to $(B \setminus 0) \times F$. ...
1
vote
1
answer
135
views
Proving the hyponormality of $A\otimes B$
Let $E$ be a complex Hilbert space. We recall that an operator $T\in\mathcal{L}(E)$ is said to be hyponormal if $[T^*, T]\geq 0$ (i.e. $\langle (T^*T-TT^*)x,x \rangle\geq 0$ for all $x\in E$). Let $E\...
1
vote
1
answer
499
views
Number of zeros of quadratic equation over finite fields
Let $\mathbb{F}_q$ denote the finite field with $q$ elements and Ch$\mathbb{F}_q\neq 2$. What is the number of solutions of the quadratic equation
$X_1^2+\cdots + X_r^2=0$ in $\mathbb{F}_q^m$ for $1\...
12
votes
1
answer
331
views
History of the classification of mathematical subjects
I would like to know if there are sources on the history of the classification of mathematical subjects.
Gérard Lang
3
votes
1
answer
118
views
Meaningful generalization of viscosity solutions to higher order equations
Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?