-1
votes
0answers
21 views

AI / Machine Learning related to high/modern/front mathematics [on hold]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
2
votes
2answers
62 views

A comment on a proof of equivalent knots

The following theorem and its proof is from A First Course in Algebraic Topology By Czes Kosniowski pp. 219-220 ...
6
votes
1answer
92 views

Airy differential equation and Galois group

Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over ...
0
votes
1answer
52 views

Soft Question: Geometry of Rings

So.. I was thinking about something last night and literally have no idea where to start. Can the ring axioms be connected to topology? If so, how? Basically, the motivation for my question has to ...
0
votes
1answer
35 views

invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
6
votes
1answer
60 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
6
votes
1answer
69 views

Describe all the quotient groups of $\mathbb{Z} \oplus \mathbb{Z}$

I'm thinking about a problem in algebraic topology of how to determine all the $k$-fold covers of $\mathbb{T}^2$, where $k$ is a certain integer. In thinking about the problem, I came upon the ...
0
votes
1answer
39 views

Question on Hamming distance

Let V_n be n-dimentional vector space over GF(q). E is k-dimentional vector subspace which is a linear q-ary (n,m,d) code and also consider the radius e = [(d-1)/2]. Assume that E is not a perfect ...
1
vote
1answer
29 views

A question about subexponential growth about group

Recall a group $\Gamma$ is said to have subexponential growth if lim sup$|E^{n}|^{1/n}=1$ for every finite subset $E\subset \Gamma. (E^{n}=\{g_{1}g_{2}...g_{n}: g_{i}\in E\}.)$ My question is: Can we ...
1
vote
0answers
35 views

Real Spectrum of a Commutative Ring

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance. $\text{Spec}_{r}(A)$
0
votes
1answer
61 views

Relative singular chains basis

If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
1
vote
0answers
26 views

What is the topological relation between $ C^\infty (E, S^1) $ and $ E $

What is the topological relation between $ C^\infty (E, S^1) $ and $ E $? Here, $ C^\infty (E, S^1) $ is the algebra of all continuous functions from $ S^1 $ to $ E $. $ E $ is a four dimensional ...
2
votes
1answer
51 views

$\mathrm{Homeo}(S^1)$ and the Mapping Class Group

Is there a full description of $\mathrm{Homeo}(S^1)$ (i.e. the group of self-homeomorphisms of the circle)? By full description I mean a presentation/list of subgroups ect. Basically anything ...
1
vote
0answers
23 views

Reference: Topology on Ind-object

In some articles I've recently seen authors mention that pro-finite groups or pro-finite algebras possess a topology, but they do not explicitly describe it. I was wondering how is the topology ...
4
votes
0answers
133 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
4
votes
1answer
77 views

Inverse question: Recognizing when a phenomenon behaves like an algebra

I might be phrasing this question incorrectly, but my students asked me about it and I did not have a good answer. I am in statistics and I look at all sorts of social data, network data, climate ...
2
votes
3answers
107 views

finding a topological group with specific conditions

I have a question, it sounds difficult. The question is the following: Let $X$ be a topological group such that the binary operation defined on it is $*$. For any two points $a$ and $b$ in $X$ ...
0
votes
2answers
69 views

Question on relative homology

i have this: where $|\tau|$ is the support of the chain $\tau$, i don't understand the first part why $[\sigma]=0$ in $H_{m-1}(\phi^{c+\varepsilon},\emptyset)$ ??? Please, thank you.
2
votes
1answer
63 views

On a coalgebra structure of simplicial homology

Are there any results on the homology group of an abstract simplicial complex with coefficients in a field $k$ being a $k$-coalgebra? Are there any assumptions and restrictions on the topological or ...
2
votes
1answer
61 views

Basic constructions for graded algebras.

I'm reading about the Weil algebra of a Lie group and it involves some constructions I'm not very familiar with, for instance the "free graded-commutative graded algebra on $a_1...a_n$ with degrees ...
1
vote
1answer
82 views

relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
3
votes
5answers
96 views

Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
0
votes
1answer
70 views

Why call them cycles and boundaries?

I have a small question. Why we have this designation: $n$-cycles for $Z_n$ and $n$-boundaries for $B_n$ ? Why they are called cycles and boundaries ? ...
3
votes
0answers
76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
1
vote
3answers
107 views

Symmetric monoidal products that preserve limits and colimits

Are there common examples of a symmetric monoidal product $\otimes$ that preserves both limits and colimits in each variable? This question is worded incorrectly, I now realize: (A) I originally ...
3
votes
1answer
83 views

Surjective inclusions in Van Kampen's Theorem

Let $X = U \cup V$ be an open cover. Assume $U,V$ and $U \cap V$ path-connected. We have the inclusions $u : U \cap V \to U$ and $v : U \cap V \to V$ with induced maps $u_* : \pi_1(U \cap V) \to \pi_1 ...
3
votes
1answer
75 views

Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$

Let $F_n$ denote the free group on $n$ elements. Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$. I noted that the wedge product of 13 copies of $S^1$ is a 3 fold covering ...
3
votes
3answers
91 views

Is $\langle abab^{-1}\rangle$ a normal subgroup of $\langle a,b|\varnothing\rangle$?

Let $G=\langle a,b | \varnothing\rangle$ and let $H\leq G$ s.t $H=\langle abab^{-1}\rangle$. Is $H\triangleleft G$? I'm asking this question in order to understand the fundamental group of the Klein ...
4
votes
2answers
158 views

definition of convergence of a spectral sequence

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant and ...
2
votes
1answer
78 views

filtered modules (LNAT, Davis & Kirk)

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written: Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a ...
5
votes
1answer
113 views

Showing path connected matrices of a group $G$ is a normal subgroup

Let $G$ be a subgroup of $GL_n(\Bbb{R})$. Define $$H = \biggl\{ A \in G \ \biggl| \ \exists \ \varphi:[0,1] \to G \ \text{continuous such that} \ \varphi(0)=A , \ \varphi(1)=I\biggr\}$$ Show that $H$ ...
4
votes
2answers
180 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
1
vote
1answer
77 views

filtration on the (co)homology of a space from the filtration of a space

Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let ...
-1
votes
1answer
62 views

Isomorphism $\left(\mathbb{C}^{n}\setminus\{0\}\right)/\mathbb{Z}$ with $S^{1} \times S^{2n-1}$

I have to prove that there is this isomorphism: $$\frac{\mathbb{C}^{n}{\setminus\{0\}}}{ \mathbb{Z}} \simeq S^{1} \times S^{2n-1},$$ where there is this equivalence relation in the left side: $(w_1, ...
3
votes
2answers
77 views

free groups and bouquet of circles

For any free group $F$ generated by the set $S$, one can construct a graph specifically a bouquet of circles $X$ s.t $\pi_1(X)=F$. My question is: Does this mean free groups are isomorphic to a free ...
0
votes
1answer
46 views

Monotonic in sigma algebra

Please help me prove that if $A \subseteq B$, then $m(A) \leq m(B)$ (That $m$ is monotonic). How would you prove this? Can we say $m^*(A \cup B) \leq m^*(A) + m^*(B)$ where $m^*(A \cup B) + m^*(A) = ...
4
votes
1answer
236 views

Defining multiplication on a Koszul complex

Let $R$ be a Noetherian commutative ring and $x$ and $y$ two elements in $R$. We construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes: $$ C_2=0\to ...
14
votes
1answer
370 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
4
votes
2answers
412 views

Definition of a filtration on a ring, module, algebra

In most books, a graded ring/module/algebra means either a $\mathbb{N}$- or $\mathbb{Z}$-graded ring/module/algebra. But often, different gradings appear: doubly graded (spectral sequences) = ...
2
votes
1answer
113 views

Confusion about commutative differential algebra

Here we read the following Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ and $\dim H^p(A,d)<\infty$ for each $p$. There exists then a ...
6
votes
2answers
864 views

Prerequisites for Algebraic Topology

I'd like to self-study Munkres' Topology. I'm already comfortable with point-set topology, so the first part of the book will serve as a nice review with some new theorems every now and then. My main ...
1
vote
1answer
71 views

Fixed-point problem (Weyl group)

Let $G=U(n)$ a compact Lie group and $T$ a maximal torus in $G$ (subgroup of diagonal matrix). We define $W=N(T)/T$ the Weyl group where $N(T)$ is the normalizer of $T \in G$. I have to prove that $W ...
2
votes
1answer
126 views

Maximal torus and Lie group

Let $T$ be a maximal torus of a compact Lie group $G$. We define the Weyl group of $G$ as the quotient space $N(T)/T$ where $N(T)$ is the normalizer of $T$ in $G$. I have to prove that if $G$ is ...
2
votes
0answers
107 views

Bruhat decomposition of flag variety.

Let $GL_{n}/B_{n}$ be the complete flag variety (where $B_{n}$ is the group of upper-triangular nonsingular matrices). Let $W$ be the Weyl group of $U(n)$ (that in this contest is a permutation ...
1
vote
0answers
46 views

Direct limits and germs of continuous functions

Consider the germs of continuous functions about some real number; say 0 for simplicity. Is there a nice way of quantifying the germs, in the sense of putting them into a bijection with a simpler set? ...
4
votes
0answers
191 views

Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
4
votes
2answers
187 views

Finitely generated singular homology

Let G be a finitely generated abelian group and M a compact manifold, I want to prove that $H_r(M,G)$ is finitely generated for $r\ge 0 $. First I was thinking if I could do induction over $r$ ...
1
vote
1answer
86 views

Convolution algebra on a compact group - unital?

Let $G$ be a compact group. Let $C(G)$ denote the set of all continuous functions $G\to \mathbb{C}$ and let $\mu$ denote the normed Haar measure on $G$. Convolution on $G$, $*:C(G)\times C(G)\to ...
83
votes
1answer
2k views

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
4
votes
1answer
231 views

Why isn't this free product of groups abelian?

I'm trying to prove that the free group $A=A_1*A_2$, where $A_1, A_2\neq 1$ is not abelian. Following the hints below: Let $x,y\in A_1*A_2$, where $x\neq y$. Suppose now $A_1=F(S)$ and $A_2=F(T)$, ...