Research

My Master’s Thesis

In my master’s thesis, I investigate a conjecture appearing in Hesselholt’s paper Galois Cohomology of Witt Vectors of Algebraic Integers.

Let $K$ be a complete discrete valuation field of zero characteristic with residue field $k_K$ of characteristic $p>0$ , and $L/K$ a finite Galois extension with Galois group $G$ . Suppose the extension $k_L/k_K$ of residue fields is separable. Recall that the morphisms in the category of projective systems of Abelian groups (pro-Abelian groups) indexed by the positive integers, are defined by $\hom(A_\cdot,B_\cdot):=\varprojlim_m(\varinjlim_n\hom(A_n,B_m))$ , and that the zero pro-Abelian group is the pro-Abelian group where each group $A_n$ is zero. Let $\mathbb{W}_{n}(\mathcal{O}_L)$ denote the ring of $p$ -typical Witt vectors of length $n$ over the ring of integers $\mathcal{O}_L$ of $L$ , and observe $\mathbb{W}_{n}(\mathcal{O}_L)$ has the structure of a $\mathbb{Z}[G]$ -module via the natural $G$ -action given by $\sigma\cdot(a_0,a_1,\ldots,a_{n-1})=(\sigma(a_0),\sigma(a_1),\ldots,\sigma(a_{n-1}))$ for all $\sigma\in G$ . Then we have a pro-Abelian group $H^1(G,\mathbb{W}_{\cdot}(\mathcal{O}_L))$ , where the group homomorphisms are induced from the (composed) restriction maps $R^m:\mathbb{W}_{n}(\mathcal{O}_L)\rightarrow\mathbb{W}_{n-1}(\mathcal{O}_L)\rightarrow\cdots\rightarrow\mathbb{W}_{n-m}(\mathcal{O}_L)$ . Hesselholt’s conjecture states:

Conjecture The pro-Abelian group $H^1(G,\mathbb{W}_{\cdot}(\mathcal{O}_L))$ is isomorphic to zero.

By using the inflation-restriction sequence (for the first cohomology group) and considering the quotients $G_i/G_{i+1}$ of the ramification groups of $G$ , Hesselholt’s paper reduces the conjecture to the case where $L/K$ is totally ramified and cyclic of degree $p$ . Let $\sigma$ be a generator of $G$ , and let $\pi_L$ be a uniformizer for $\mathcal{O}_L$ . Let us define $t:=v_L(\sigma(\pi_L)/\pi_L-1)=v_L(\sigma(\pi_L)-\pi_L)-1$ . Hesselholt shows that the conjecture holds under the hypothesis that $t>\frac{e_K}{p-1}$ , where $e_K:=v_K(p)$ denotes the absolute ramification index. Thus, in particular, Hesselholt shows that the conjecture holds when $K=\mathbb{Q}_p$ , the field of $p$ -adic numbers.

In 2010, Hogadi and Pisolkar proved Hesselholt’s conjecture in their paper On the Cohomology of Witt Vectors of p-adic Integers and a Conjecture of Hesselholt. I have provided a simplifed proof of the conjecture in my paper A Simplified Proof of Hesselholt’s Conjecture on Galois Cohomology of Witt Vectors of Algebraic Integers, which appears in the Bulletin of the Australian Mathematical Society.

Other Research – Graph Rigidity Theory

During the 2008-2009 summer vacation period, I undertook a research project in Graph Rigidity Theory at the Australian National University, Canberra. The main result of my research is outlined below.

Literature Review

Basically a graph is rigid iff for almost all embeddings, any continuous movement of its vertices subject to its edge constraints preserves the distance between every pair of vertices. The following definitions will make this idea precise.

A representation of an $m$ -dimensional ( $m=2,3$ ) graph $G=(V,E)$ is an injective function $r:V\rightarrow \mathbb{R}^m$ .

We define the distance between two representations $r_1$ and $r_2$ of the same graph by $d(r_1,r_2)=\max\limits_{i\in V} |r_1(i)-r_2(i)|$

A distance set $D$ for $G$ is a set of distances $d_{ij}>0$ , defined for all edges $ij\in E$ . A distance set $D$ is realizable if there exists a representation $r$ of the graph for which $|r(i)-r(j)|=d_{ij}$ for all $ij\in E$ . Such a representation is then called a realization of $D$ . It follows any representation $r$ is a realization of the distance set defined by $d_{ij}=|r(i)-r(j)|$ for all $ij\in E$ .

A realization $r$ of a distance set $D$ is rigid if there exists $\epsilon>0$ such that for all realizations $r'$ of $D$ satisfying $d(r,r')<\epsilon$ , there holds $|r'(i)-r'(j)|=|r(i)-r(j)|$ for all $i,j\in V$ .

Definition (Rigid graph) A graph is rigid if almost all (i.e. an open dense set of) its realizations are rigid.

Remark By definition, the trivial graph is rigid.

Example The ‘triangle’ $C_3$ is rigid, but the ‘rectangle’ $C_4$ is not.

Definition (Minimally rigid graph) A rigid graph is minimally rigid iff no edge can be removed without losing rigidity.

Theorem (Laman’s theorem) A non-trivial graph $G=(V,E)$ is minimally rigid iff $|E|=2|V|-3$ and for all non-empty $E'\subseteq E$ , $|E'|\leq 2|V(E')|-3$ , where $V(E')$ is the set of vertices incident to the edges of $E'$ .

We can perform vertex addition and edge-splitting operations to construct minimally rigid graphs. These operations are as follows:

Performing the operation of vertex addition on $G=(V,E)$ yields $G'=(V\cup \{v\},E\cup\{vi,vj\})$ , $i,j\in V$ .

Performing the operation of edge-splitting on $G=(V,E)$ yields $G'=(V\cup \{v\},E\cup\{vi,vj,vk\} \setminus \{e\})$ , $i,j,k\in V$ with at least two of $i,j,k$ adjacent, and $e$ one of $ij, ik, jk$ .

Theorem A graph obtained by performing a vertex addition or edge-splitting operation on a minimally rigid graph is minimally rigid.

Main Result

Definition (Split) A split of $G=(V,E)$ is a partition of $V$ into two subsets, with the distance constraints (edges) between the two subsets suppressed.

Definition (Connected split) A connected split is a split where both components (the subgraphs induced by the subsets of $V$ in the split) are connected.

We call a connected split of a graph on $n$ vertices into connected components of orders $i$ and $n-i$ a $(i,n-i)$ connected split.

Definition (Connected-splittable) We call $G=(V,E)$ connected-splittable if for all $i=1,2,\ldots,|V|-1$ , $G$ has a $(i, |V|-i)$ connected split.

Proposition All rigid graphs are connected-splittable.

I have provided a constructive proof to the above proposition, via a new algorithm for finding connected-splits. The proposition together with its proof, and the algorithm, appears in the paper Splitting Rigid Formations.

Wilson Ong's Blog

Research

My Master’s Thesis

Other Research – Graph Rigidity Theory

Literature Review

Main Result

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