# 数学代写|表示论代写Representation theory代考|MTH4107

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## 数学代写|表示论代写Representation theory代考|Notes and Exercises

The concept of topological persistence came to the fore in early 2000 with the paper by Edelsbrunner, Letscher, and Zomorodian [152] though the concept was proposed in a rudimentary form (e.g., zero-dimensional homology) in other papers by Frosini [162] and Robins [266]. The persistence algorithm as described in this chapter was presented in [152] which has become the cornerstone of topological data analysis. The original algorithm was described without any matrix reduction, which first appeared in [105]. Since then various versions of the algorithm have been presented. We have already seen that persistence for filtrations of simplicial 1-complexes (graphs) with $n$ simplices can be computed in $O(n \alpha(n))$ time. Persistence for filtrations of simplicial 2manifolds also can be computed in an $O(n \alpha(n))$ time algorithm by essentially reducing the problem to computing persistence on a dual graph. In general, for any constant $d \geq 1$, the persistence pairs between $d$ – and $(d-1)$-simplices of a simplicial $d$-manifold can be computed in $O(n \alpha(n))$ time by considering the dual graph. If the manifold has a boundary, then one has to consider a “dummy” vertex that connects to every dual vertex of a $d$-simplex adjoining a boundary $(d-1)$-simplex.

For efficient implementation, clearing and compression strategies as described in Section $3.3 .2$ were presented by Chen and Kerber [94]. We have given a proof based on matrix reduction that the same persistence pairs can be computed by considering the anti-transpose of the boundary matrix. This is termed the cohomology algorithm first introduced in [114]. The name is justified by the fact that, considering cohomology groups and the resulting persistence module that reverses the arrows (Fact 2.14), we obtain the same barcode. The anti-transpose of the boundary matrix indeed represents the coboundary matrix filtered reversely. These tricks are further used by Bauer for processing Rips filtration efficiently in the Ripser software [19]; see also [304]. Boissonnat et al. [41, 42] have suggested a technique to reduce the size of a given filtration using the strong collapse of Barmak and Minian [17]. The collapse on the complex can be efficiently achieved through only simple manipulations of the boundary matrix.

The concept of bottleneck distance for persistence diagrams was first proposed by Cohen-Steiner et al. [101] who also showed the stability of such diagrams in terms of bottleneck distances with respect to the infinity norm of the difference between functions generating them. This result was extended to Wasserstein distance, though in a weaker form in [103] which got improved recently [278]. The more general concept of interleaving distance between persistence modules and the stability of persistence diagrams with respect to them was presented by Chazal et al. [77]. The fact that the bottleneck distance between persistence diagrams is not only bounded from above by the interleaving distance but is indeed equal to it was shown by Lesnick [221] which was further studied by Bauer and Lesnick [23] later. Also, see [54] for more generalization at algebraic level.

The use of the reduced Betti numbers for the lower link of a vertex to quantify its criticality was originally introduced in [150] for a PL-function defined on a triangulation of a $d$-manifold. Our PL-criticality considers both the lower link and upper link for more general simplicial complexes. As far as we know, the relations between such PL-critical points and homology groups of sublevel sets for the PL-setting have not been stated explicitly elsewhere in the literature. The concept of homological critical values was first introduced in [101], and the more general concept of “levelset critical values” (and levelset tame functions) was originally introduced in [63].

The idea of using union-find data structure to compute the zeroth persistent homology group was already introduced in the original persistence algorithm paper [152]. In this chapter, we present a modification for the PL-function setting.

## 数学代写|表示论代写Representation theory代考|Stability of Towers

Just like in the previous chapter, we define the stability with respect to the perturbation of the towers themselves, forgetting the functions that generate them. This requires a definition of the distance between towers at simplicial (space) levels and homology levels.

It turns out that it is convenient and sometimes appropriate if the objects (spaces, simplicial complexes, or vector spaces) in a tower are indexed over the positive real axis instead of a discrete subset of it. This, in turn, requires one to spell out the connecting map between every pair of objects.

Definition 4.1. (Tower) A tower indexed in an ordered set $A \subseteq \mathbb{R}$ is any collection $\mathrm{T}=\left{T_a\right}_{a \in A}$ of objects $T_a, a \in A$, together with maps $t_{a, a^{\prime}}: T_a \rightarrow$ $T_{a^{\prime}}$ so that $t_{a, a}=\mathrm{id}$ and $t_{a^{\prime}, a^{\prime \prime}} \circ t_{a, a^{\prime}}=t_{a, a^{\prime \prime}}$ for all $a \leq a^{\prime} \leq a^{\prime \prime}$. Sometimes we write $\mathrm{T}=\left{T_a \stackrel{t_{a, a^{\prime}}^{\longrightarrow}}{\longrightarrow} T_{a^{\prime}}\right}_{a \leq a^{\prime}}$ to denote the collection with the maps. We say that the tower $\mathrm{T}$ has resolution $r$ if $a \geq r$ for every $a \in A$.

When $\mathrm{T}$ is a collection of topological spaces connected with continuous maps, we call it a space tower. When it is a collection of simplicial complexes connected with simplicial maps, we call it a simplicial tower, and when it is a collection of vector spaces connected with linear maps, we call it a vector space tower.

Remark 4.1. As we have already seen, in practice, it may happen that a tower needs to be defined over a discrete set or more generally an index set A that is only a subposet of $\mathbb{R}$. In such a case, one can “embed” A into $\mathbb{R}$ and convert the input to a tower according to Definition $4.1$ by assuming that for any $a<$ $a^{\prime} \in A$ with $\left(a, a^{\prime}\right) \notin A$ and for any $a \leq b<b^{\prime}<a^{\prime}, t_{b, b^{\prime}}$ is an isomorphism.
Definition 4.2. (Interleaving of simplicial (space) towers) Let $X=\left{X_a \stackrel{x_{a, a^{\prime}}}{\longrightarrow}\right.$ $\left.X_{a^{\prime}}\right}_{a \leq a^{\prime}}$ and $y=\left{Y_a \stackrel{y_{a, a^{\prime}}}{\longrightarrow} Y_{a^{\prime}}\right}_{a \leq a^{\prime}}$ be two towers of simplicial complexes (resp. spaces) indexed in $\mathbb{R}$.

# 表示论代考

## 数学代写|表示论代写Representation theory代考|Notes and Exercises

Cohen-Steiner 等人首先提出了持久图瓶颈距离的概 念。 [101] 他们还展示了此类图在瓶颈距离方面的稳定 性，相对于生成它们的函数之间差异的无穷范数。这个 结果被扩展到 Wasserstein 距离，尽管在 [103] 中的形 式较弱，最近得到改进 [278]。Chazal 等人提出了持久 性模块之间的交错距离和持久性图相对于它们的稳定性 的更一般概念。[77]。Lesnick [221] 展示了持久性图之 间的瓶颈距离不仅从上方受交错距离限制，而且实际上 等于交错距离，这一事实后来由 Bauer 和 Lesnick [23] 进一步研究。还，
[150] 最初在 [150] 中针对在 $a$ 的三角剖分上定义的 PL 函数引入了对顶点的下部链接使用减少的 Betti 数来量 化其临界性 $d$-歧管。我们的 PL-criticality 考虑了更一般 的单纯复形的下链㢺和上链接。据我们所知，此类 $P L$ 临界点与 PL 设置的子水平集的同源群之间的关系尚末 在其他文献中明确说明。同源临界值的概念首先在 [101] 中引入，更一般的概念”水平集临界值”（和水平集 驯服函数）最初在 [63] 中引入。

## 数学代写|表示论代写Representation theory代考|Stability of Towers

Imathrm ${T}=V$ left{T_alright $}_{-}{a$ lin $A}$
Imathrm{T}=\left{T_a\right } } { a \backslash \text { in A } } \text { 对象的 } T _ { a } , a \in A \text { , }

$t_{a^{\prime}, a^{\prime \prime}} \circ t_{a, a^{\prime}}=t_{a, a^{\prime \prime}}$ 对所有人 $a \leq a^{\prime} \leq a^{\prime \prime}$. 有时我们
Imathrm ${T}=| l$ eft $\left{T_{-} a\right.$ Istackrel $\left{t_{-}{a, a \wedge{1 p r i m e}} \wedge{\right.$ Vlongrigh

$\mathrm{y}=$ Veft $\left{Y_{-} a\right.$ Istackrel $\left{y_{-}\left{a, a^{\wedge}{{\right.\right.$ prime $\left.}}\right}{$ Vlongrightarrow $} Y_{-}$

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