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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 ...
Harrison Chen's user avatar
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 $G \subseteq C_C(\mathbb{R}^p, \mathbb{R}^s)$,...
msd15213's user avatar
  • 123
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 $...
Levon Haykazyan's user avatar
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/...
Xiao-Gang Wen's user avatar
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 $...
Bedovlat's user avatar
  • 1,939
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 ...
Cristian D. Gonzalez-Aviles's user avatar
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 ...
Connor's user avatar
  • 251
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 ...
Sascha's user avatar
  • 506
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 ...
Drew's user avatar
  • 1,469
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 ...
keaton's user avatar
  • 421
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 ...
Elie Ben-Shlomo's user avatar
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 ...
user avatar
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 ...
Jing Zhang's user avatar
  • 3,138
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 ...
Davide Cesare Veniani's user avatar
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.)...
Gary McGuire's user avatar
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 ...
JSCB's user avatar
  • 1,610
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 ...
Valery Isaev's user avatar
  • 4,410
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 ...
Stefan Witzel's user avatar
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 $\mathcal{B}(F)$ ...
Schüler's user avatar
  • 724
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: $$A\in\mathbb{R}^{m\times n}:\ \sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{m}a_{ij}=0$$ ...
Manfred Weis's user avatar
  • 12.6k
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 ...
H1ghfiv3's user avatar
  • 1,225
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 ...
smapers's user avatar
  • 338
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 ...
Asvin's user avatar
  • 7,648
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 $NS(X)$?
Gerard's user avatar
  • 181
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 $C$ ...
DBB's user avatar
  • 51
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 ...
Mohammad Golshani's user avatar
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(...
Dylon Chow's user avatar
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 ...
J Cameron's user avatar
  • 551
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 + \...
Christopher King's user avatar
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^{...
user avatar
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 ...
user avatar
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 ...
kneidell's user avatar
  • 993
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 ...
Gerard's user avatar
  • 181
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) $...
user avatar
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 ...
Max Muller's user avatar
  • 4,485
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 ...
Ali Taghavi's user avatar
-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 ...
Ning Bao's user avatar
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 ...
Camille's user avatar
  • 41
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 \...
Vlad's user avatar
  • 51
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?
user avatar
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}^...
SMD's user avatar
  • 480
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 \...
Jeremy's user avatar
  • 401
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^{-...
Harry Reed's user avatar
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 ...
Ben Sprott's user avatar
  • 1,281
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....
Ludwig's user avatar
  • 2,682
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 $(B \setminus 0) \times F$. ...
Lawrence Jack Barrott's user avatar
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\...
Student's user avatar
  • 1,154
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\...
Singh's user avatar
  • 179
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
Gérard Lang's user avatar
  • 2,617
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?
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