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\title{Motivation and Definition of Manifolds}
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\begin{document}
	\maketitle
	\section{Introduction: Finding the Tangent Line With or Without Calculus}
		Consider the classic question on an ellipse: Finding the tangent line of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at $(m,n)$, where $m^2< a^2$ and $n^2 < b^2$.
		\begin{proof}[What a high schooler will do (no calculus)]
			Write the desired tangent line as
			\[
				y=r(x-m)+n.
			\]
			Inserting it into the equation of ellipse, we have
			\[
				\frac{x^2}{a^2}+\frac{[r(x-m)+n]^2}{b^2}-1=0.
			\]
			Since the line and the curve share only one point, the equation above should have only one solution. This is to say that the discriminant ($\Delta = b^2-4ac$) should be $0$. But do you want to solve this manually? I don't.
		\end{proof}
		\begin{proof}[What a college student will do (with calculus)]
			Consider the function
			\[
				f(x,y)=\frac{x^2}{a^2}+\frac{y^2}{b^2}-1.
			\]
			It follows that $f(m,n)=0$. By implicit function theorem, there is a neighbourhood $W \subset \mathbb{R}$ around $n$, and a smooth function $g:W \to \mathbb{R}$ such that $g(y)=x$ (in particular, $g(b)=a$) and
			\[
				f(g(y),y)=0, \quad y \in W,
			\]
			and
			\[
				g'(n) = -\frac{f_y}{f_x}\vert_{(x,y)=(m,n)}.
			\]
			Around this neighbourhood, we only need to consider the function $x=g(y)$. If we work out the tangent line of this function then we are done. It follows that
			\[
				g'(n) = -\frac{2n/b^2}{2m/a^2}
			\]
			Therefore the tangent line is
			\[
				x=g'(n)(y-n)+m = -\frac{2n/b^2}{2m/a^2}(y-n)+m.
			\]
		\end{proof}
		The most challenging thing in the second solution is evaluating two partial derivatives. Calculus saved our life. Note the neighbourhood $W$ around $n$, together with the function $g$, actually determine a neighbourhood $U$ around $(m,n)$, which can be considered as a small fraction of the ellipse curve. In $U$, to every $y \in W$, there corresponds a unique $x$ such that $(x,y) \in U$ and $f(x,y)=0$.

		What's the crucial difference? The high schooler version is making use of the whole curve (by using the equation). Everywhere on it is considered. But the college student version is not doing that: only a small neighbourhood is concerned. We don't even care about how big that neighbourhood look like. On the small neighbourhood of the curve, we also pretend we are not working on a curve, but on the straight line (let's say, the Euclidean space).

		This idea shall be a life-saver. For example, if you want to compute the tangent line of the curve $(x+\sin{x},x-\tan{x})$, then working with discriminant is not feasible at all: there is no quadratic polynomials to work with. But with much more calculus it becomes possible: maybe we can still use implicit function theorem (in fact we do need implicit function theorem and inverse function theorem extensively).

		Our ultimate goal is quite ambitious: we want to study geometry by doing calculus on \textit{many} objects, not just Euclidean spaces, although we mainly rely on it. However, as we have seen, working in a classic way (equation, discriminant, etc.) is not a good way. We need a more modern approach. The example above enlightens us: maybe what really matters is local structures or the information around a point, and perhaps we don't even need to \textit{decode} it, we only need to know that they are there.

		\section{Generalisation of Curves and Surfaces in Higher Dimensionals}
		What is a curve or a surface anyway? Maybe the ellipse is a curve, but how do we validate it? Because its area is as small as possible? What is a surface? The unit ball sphere $S^2=\{(x,y,z):x^2+y^2+z^2=1\}$ is a surface, but how? Is it because it is as thin as possible?

		We cannot define mathematical objects like this. The definition can be extracted from intuition but not everything about intuition. This is not cool. However, we have good understanding of $\mathbb{E}$ and $\mathbb{E}^2$, so why don't we good use of it?
		\begin{definition}
			A curve is a \textbf{topological space} where every point has an open neighbourhood homeomorphic to some open subset of $\mathbb{E}$. Similarly, a surface is a \textbf{topological space} where every point has an open neighbourhood homeomorphic to some open subset of $\mathbb{E}^3$.
		\end{definition}
		This definition is quite intuitive: with this we can say with absolute certainty that an ellipse yields a curve, a unit ball sphere is a surface.. However, we cannot make sure that every object we study is curve or surface. It may be a higher dimensional thing (for example, what about the unit ball sphere in $4$ dimensional space?). So we have to deliver a natural generalisation.
		\begin{definition}
			A topological space $M$ is called \textbf{locally Euclidean of dimension $n$} if every point of $M$ has a open neighbourhood homeomorphic to some open subset of $\mathbb{E}^n$.
		\end{definition}
		Here we have some good local information: locally, it \textit{looks like} a Euclidean space. What really matters is that such information exist. You don't need to write down every open set of the unit circle and write down every homeomorphism. So what about the unit ball sphere in $4$ dimensional space? The set is written as
		\[
			S^3 = \{(x_0,x_1,x_2,x_3):x_0^2+x_1^2+x_2^2+x_3^2=1\}.
		\]
		If we fix $x_1,x_2,x_3$, then $x_0$ is also determined (up to plus or minus). Every point on this set can be determined by three variables. Around one fixed point we have a neighbourhood that is determined by three variables, which can be considered as a subset of $\mathbb{E}^3$, though we don't care about the explicit function. We cannot imagine such a thing in our mind but we can still attack it with calculus.

		With the concept of locally Euclidean space, we have many objects to study, much beyond curves and surfaces. But, we want to study calculus everywhere, but this concept only gives continuity (local homeomorphism), and differentiation is not even touched. We may want to study the $4$-dimensional ball alone, not treat it as a subset of $\mathbb{E}^4$. This is to say we have some other job to work on.

		\section{Topological Manifold, differentiable structure and differentiable manifold}
		We are pretty close to our main goal but we shall not rush. Let us sum up what is going on. We want to study calculus on many objects, so we are looking for good structures. However we did not introduce differentiation in a natural sense yet.
			\subsection{Topological Manifold}
			\begin{definition}
				A locally Euclidean space $M$ of dimension $n$ is called an \textbf{$n$-dimensional manifold (French: varieté)} if it is Hausdorff and second countable.
			\end{definition}
			Yes we only added two restrictions on being locally Euclidean: Hausdorff and second countable. This is to make our life easier and to exclude some weird cases. Being Hausdorff makes sure that there are enough open sets. In geometry if we cannot separate two points in a natural sense then we are so powerless (we don't want to study trivial topology here, for example). Being second countable makes sure that there will few enough open sets. It cannot be explained easily here, but second countability gives \textbf{paracompactness}, which gives \textbf{partition of unity}, without which \textbf{integration} becomes impossible. In the following lectures you will be convinced that our \textit{ad hoc} restrictions are quite reasonable. If possible we will study Prüfer's surface introduced by Radó in 1925, which is a manifold that is not second countable and  so weird that we don't want to include them in our general study of manifolds. Algebraic geometers are interested in Zariski's topology, which is coarser than Euclidean topology and is not Hausdorff, but at this moment this is not of our concern.

			Another important thing is that we want the dimension of $M$ to be consistent. It cannot be both $1$ and $2$ at the same time. The union of a curve and a surface cannot be a manifold - you cannot study the tangent line and plane at the same time.

			Study of topological manifolds alone can be a big thing and is of great importance. To study deeper one may need many knowledge of algebraic topology, which is not easy at all.
			\subsection{Differentiable structure and Differentiable manifold}
			On a topological manifold we can work with continuous functions with ease. But this is far from enough. We are looking for a topological manifold with \textbf{differentiable structure}. We want to, for instance, study $S^3$ extensively with differentiation (and integration in the future), so we need to grant it with a bunch of information to \textit{differentiate}.

			To begin with we first recall our definition of differentiability in $\mathbb{R}^n$.

			\begin{definition}
				Let $U \subset \mathbb{R}^n$ be open, and let $f:U \to \mathbb{R}$. If $\alpha = (\alpha_1,\dots,\alpha_n)$ is a $n$-tuple of non-negative integers, we write
				\[
					[\alpha]= \sum_{i=1}^{n}\alpha_i, \quad \frac{\partial^\alpha}{\partial r^\alpha} = \frac{\partial^{[\alpha]}}{\partial r_1^{\alpha_1} \cdots r_n^{\alpha_n}}
				\]

				We say that $f$ is differentiable of class $C^k$ on $U$ if $\frac{\partial^\alpha}{\partial r^\alpha}f$ exist and are continuous on $U$ for all $[\alpha] \le k$. In particular, $f \in C^0$ if $f$ is continuous. \\
				If $f:U \to \mathbb{R}^n$, then $f$ is differentiable of class $C^k$ if each of the component function $f_i = r_i \circ f$ is $C^k$. We say $f$ is $C^\infty$ if it is $C^k$ for all $k \ge 0$.
			\end{definition}

			This doesn't go any further from multivariable calculus. We are much more interested in the case when $k=\infty$, when everything is called \textbf{smooth} on top of differentiable. We are now ready to introduce the concept of differentiable structure.

			\begin{definition}
				A \textit{differentiable structure} $\mathscr{F}$ of class $C^k$ on a $n$-dimensional topological manifold $M$ is a collection of pairs $\{(U_\alpha,\varphi_\alpha):\alpha \in A\}$ where $U_\alpha \subset M$ is a connected open subset and $\varphi_\alpha$ is a homeomorphism of $U_\alpha$ onto a open subset of $\mathbb{R}^n$ with the following properties:
				\begin{enumerate}
					\item $\bigcup_{\alpha \in A}U_\alpha = M$. \label{fst}
					\item $\varphi_\alpha \circ \varphi_\beta^{-1}$ is $C^k$ for all $\alpha, \beta \in A$. \label{snd}
					\item The collection $\mathscr{F}$ is maximal with respect to \ref{snd}. If $(U,\varphi)$ is another pair such that $\varphi \circ \varphi_\alpha^{-1}$ and $\varphi_\alpha \circ \varphi^{-1}$ are $C^k$ for all $\alpha \in A$, then $(U,\varphi) \in \mathscr{F}$.
				\end{enumerate}
			\end{definition}

			\begin{definition}
				An $n$-dimensional differentiable manifold of class $C^k$ is an $n$-dimensional manifold $M$ together with a differentiable structure $\mathscr{F}$.
			\end{definition}
			One may write $(M,\mathscr{F})$ to denote a differentiable manifold, just like writing $(X,\tau)$ to denote a topological space. When the context is clear and the differentiable structure needs not to be emphasised, we simply use $M$ to denote a differentiable manifold.

			But wait. Have we gone too far? Is the definition of manifolds coming from nowhere? The answer is no. What we have done is nothing but a generalisation of curves and surfaces in higher dimensions, with the potential to work with differentiation. Let's recall our second solution of finding the tangent line. We were working on a open neighbourhood of the point $(m,n)$ on the ellipse curve. What's the difference between $(U,g)$ there and the pairs here? Chances are very little.

			Let's have some remarks on the definition. The pair $(U,\varphi)$ is generally called charts or \textbf{coordinate system}. The reason is clear: we cannot assign coordinate on $M$, but we can do it on $\mathbb{R}^n$. The coordinate system allows us to assign coordinates locally. For this reason we also call $\varphi$ here the coordinate map, the $i$-th components $x_i = r_i \circ \varphi$ are called \textbf{coordinate functions}. The requirement in \ref{snd} may look confusing. But in fact that's where the differentiable structure comes from. Note $\varphi_\alpha \circ \varphi_\beta^{-1}$ is a map of $\mathbb{R}^n \to \mathbb{R}^n$. Since these $\{U_\alpha\}$ cover $M$, there may be overlaps among them. We require that within these overlaps, the interchange of coordinate maps are differentiable of class $C^k$. For example we tend to discuss the differentiation on $S^1$ but not on the edge of a rectangle.

			Since $\varphi_\alpha \circ \varphi_\beta^{-1}$ is a $\mathbb{R}^n \to \mathbb{R}^n$ map, the Jacobian matrix is $n \times n$. Further, if the determinant is positive everywhere, then we say that this manifold is \textbf{oriented}. We will return to orientation much later.

			Another remark is, in practice, one rarely sees this bulky collection of data explicitly. Suppose we want to study a point $x \in M$. This lies in at least one $U_\alpha$ together with $\varphi_\alpha$, then we choose one. When doing calculations around this point, we tend to reduce ourselves to the classic analysis. What if we pick another coordinate system? \ref{snd} ensures that the change of systems is of $C^k$, and we can send it back of $C^k$. So in general the result is independent of the coordinate system we pick.

			Now consider the following two subsets on $\mathbb{R}^2$:
			\[
				E_1 = \mathbb{R} \times \{0\}, \quad E_2 = \mathbb{R} \times \{1\}.
			\]
			Put $F=E_1 \cup F_2$ and define an equivalence relation $\sim$ by
			\[
				(x,a)\sim(y,b) \iff \text{$(x,a)=(y,b)$ or $x=y<0$}
			\]
			Then $X=F/\sim$ can be visualised as a forked real line, and is not Hausdorff. We can grant $X$ a differentiable structure by
			\[
				\begin{aligned}
					&U_1 = E_1, \quad \varphi_1((x,0))=x, \\
					&U_2 = \{(x,0):x<0\} \cup \{(y,1):y \ge 1\}, \quad \varphi_2((x,0))=x, \varphi_2((y,1))=y.
				\end{aligned}
			\]
			Then $X$ is a smooth manifold but is too weird to study in a normal sense. We exclude this kind of manifolds by excluding non-Hausdorff cases.
		\section{Standard Differentiable structure and More Examples}
		It seems not too many restrictions was given onto the differentiable structure. We have no idea whether the differentiable structure is the one we want or not. Sometimes we want the differentiable structure to contain some specific coordinate systems. For example, for $\mathbb{R}^n$ itself, the standard topology is the Euclidean topology. We also want a standard differentiable structure on $\mathbb{R}^n$, which should at the very least contain $(\mathbb{R}^n,i)$ where $i$ is the identity map.
		\begin{theorem}
			Let $\mathscr{F}_0$ be any collection of coordinate systems that satisfies \ref{fst} and \ref{snd}, then there is a unique differentiable structure $\mathscr{F} \supset \mathscr{F}_0$.
		\end{theorem}
		\begin{proof}
			Let
			\[
				\mathscr{F} = \{(U,\varphi):\text{$U\subset M$ is open and connected, $\varphi \circ \varphi_\alpha^{-1}$ and $\varphi_\alpha \circ \varphi^{-1}$ are $C^k$ for all $\varphi_\alpha \in \mathscr{F}_0$}\}
			\]
			Then $\mathscr{F} \supset \mathscr{F}_0$ by definition. Since all satisfied coordinate systems are included, $\mathscr{F}$ is maximal, hence it is a differentiable structure. Besides, suppose $\mathscr{F}'$ is another differentiable structure containing $\mathscr{F}_0$, then for any $(U,\varphi) \in \mathscr{F}'$, we have $U \subset M$ is open and connected, $\varphi \circ \varphi_\alpha^{-1}$ and $\varphi_\alpha \circ \varphi^{-1}$ are $C^k$ for all $\varphi_\alpha \in \mathscr{F}_0$. But this is essentially to say that $(U,\varphi) \in \mathscr{F}$, hence $\mathscr{F}' \subset \mathscr{F}$, which proves the uniqueness of $\mathscr{F}$.
		\end{proof}
		With this theorem being proved, we can construct standard differentiable structures of given topological manifolds much easily.
		\begin{enumerate}
			\item The standard differentiable structure on Euclidean space $\mathbb{R}^n$ is obtained by taking $\mathscr{F}$ to be the maximal collection containing $\{(\mathbb{R}^n,i)\}$. This can be extended to all finite dimensional real and complex vector spaces.
			\item The $n$-sphere is the set
			\[
				S^n = \{x \in \mathbb{R}^{n+1}:x_0^2+\dots+x_n^2=1\}.
			\]
			Let $N=(0,\dots,0,1)$ and $S = (0,\dots,0,-1)$. Then the standard differentiable structure on $S^n$ is obtained by taking $\mathscr{F}$ to be the maximal collection containing $(S^n-N,p_N)$ and $(S^n-S,p_S)$, where $p_N$ and $p_S$ are stereographic projections from $N$ and $S$ respectively. Unit sphere is the most \textit{simple but not trivial} example in geometry and topology.
			\item If $(M,\mathscr{F}_M)$ is a differentiable manifold, then an open subset $U$ is also a differentiable (sub)manifold with differentiable structure
			\[
				\mathscr{F}_U = \{(U_\alpha \cap U, \varphi_\alpha|_{U_\alpha \cap U}):(U_\alpha,\varphi_\alpha) \in \mathscr{F}_M\}
			\]
			We will always be using this submanifold structure unless specified otherwise.
			\item As an example of submanifold, consider $GL(n,\mathbb{R})$, the general linear group. Naturally the set of $n \times n$ real matrices can be identified as $\mathbb{R}^{n^2}$ (which is a differential manifold), and the determinant function $\det$ can be considered as a homogeneous polynomial of degree $n$ with $n!$ terms. Therefore $GL(n,\mathbb{R})$ can be identified as $\det^{-1}((-\infty,0) \cup (0,\infty))$, which is open. Hence $GL(n,\mathbb{R})$ is a differential manifold.
			\item For some reasons we may also consider $O(n)$, $SO(n)$, $SL(n)$, $PSL(n)$. They are subgroups of $GL(n)$, and none of them are open subsets. However it can be proved that they are submanifolds of $GL(n)$. We will have different techniques to work on them. Those matrix groups are examples of \textbf{Lie groups}. If a differential manifold $M$ of class $C^\infty$ is also a group where the product and inverse are of $C^\infty$, then we say that $M$ is a Lie group. This class of groups have many applications in various branches of mathematics, physics and engineering. We will return to them when we are ready.
			\item Let $(M_1,\mathscr{F}_1)$ and $(M_2,\mathscr{F}_2)$ be two differential manifolds of dimensions $d_1$ and $d_2$ respectively. Then $M_1 \times M_2$ has a canonical differentiable structure $\mathscr{F}$ making it a $d_1+d_2$-dimensional differential manifold as follows:
			\[
				\mathscr{F} \supset \mathscr{F}_1 \times \mathscr{F}_2 = \{(U_\alpha \times V_\beta,\varphi_\alpha \times \psi_\beta):(U_\alpha,\varphi_\alpha)\in \mathscr{F}_1,(V_\beta,\psi_\beta) \in \mathscr{F}_2\}
			\]
		\end{enumerate}
	\section{References}
		\begin{enumerate}
			\item Frank W. Warner, \textit{Foundations of Differentiable Manifolds and Lie Groups} (The main book we will be using; we will go through chapter 1 to 4 at our best.), G.T.M. vol. 94.
			\item Marcle Berger, Bernard Gositaux, \textit{Differential Geometry: Manifolds, Curves and Surfaces} (French to Chinese edition can be found in our library but I strongly recommend against reading it. It was poorly translated. Try French to English edition instead. We use this book for more examples to compute.) G.T.M. vol. 115.
			\item Serge Lang, \textit{Fundamentals of Differential Geometry} (This book is written by an algebraist and is way too abstract. But we will use this book when we need some deeper insights.) G.T.M. vol. 191.
		\end{enumerate}
\end{document}


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\section{Variable Assignment}

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\item Do Exercise 2.1, and for each bullet point jot down a note about what you can/can't do

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\item Exercise 2.2.1:  The volume of a sphere with radius r is $\frac{4}{3}\pi r^3$.  What is the volume of a sphere with radius $5$?  Write out your code below:

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\item Exercise 2.2.2: Suppose the cover price of a book is \$24.95, but bookstores get a 40\% discount.  Shipping costs \$3 for the first copy and 75 cents for each additional copy.

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\underline{Variable}

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\underline{Variable Name}

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\underline{Variable Value}

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Write out the equation for the Total Wholesale Cost \textbf{using only the variable names}:

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Now, use the values given in the problem: TWC = \$\rule{3cm}{0.15mm}
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\item Exercise 2.2.3:  If I leave my house at 6:52 am and run 1 mile at an easy pace (8:15 per mile), then 3 miles at tempo (7:12 per mile) and 1 mile at easy pace again, what time do I get home for breakfast?  Sketch your plans in a notebook, then draft out your code:

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Worksheet 4 - Scaffolded ch2 book exercises - ch 2.tex


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\section{Monstrosity}
Monstrosity is a numeric representation of how close to the void any character is. It can be raised or lowered by: certain class-specific actions; spells; drinking Elixirs; taking a Long Rest; and entering or exiting areas of intense Eldritch activity.

Monstrosity (MST) can be thought of as a separate health bar, with the difference being that lower MST is better. MST has two values for the player to manage: the first is a 20-point scale, against which the player must make checks. For example, a player with 10 MST would have a DC 10 saving throw on MST. All normal rules concerning saving throws apply to MST. As the character approaches 20 MST, they become increasingly attuned to their Eldritch heritage, and act accordingly. This rule is left intentionally vague so that individual DMs and players can decide for themselves how they wish to roleplay or integrate MST scaling into gameplay.

When a creature receives Monstrosity damage, the player associated with them rolls a d20. If that number is greater than 10 + 1/2 your current Monstrosity, your Monstrosity value does not change. If it is less than that number, the creature's Monstrosity increases by the amount specified in the attack details.

The second aspect of Monstrosity tracks a character's long-term relationship with the Void. If the player ever goes above 20 MST, they gain a level in this system. Gaining a prestige level in MST means that the character has gone so far into the void that it is impossible to return to normalcy afterwards. The character manifests a new, permanent Eldritch-influenced feature and they are subsumed into their Eldritch Manifestation for 1d6 days, after which point their MST drops to 0.

For each prestige level up to 3, the player receives a set negative effect, outlined by the 'Prestige Effects' table below.

\header{Prestige Effects}
\begin{dndtable}[cX]
\textbf{Level} & \textbf{Effect} \\
1              & -1 to all Skills \\
2              & -1 to all Ability Scores \\
3              & MST damage taken is doubled \\
4+             & d20 (DC 10) to stay sane \\
\end{dndtable}

When a player gains a prestige level (up to level 3), the DM selects three traits (either randomly or by choice) from the 'Prestige Traits' table below. The player must then select one of them to add to their character.

A number of the below traits are sourced from official Dungeons \& Dragons materials; others are our own creation.

\header{Prestige Traits}
\begin{dndlongtable}[ccp{14em}]
\textbf{d20} & \textbf{Trait} & \textbf{Effect} \\
\rownumber             & Hard-skinned    & Gain resistance to Bludgeoning and Slashing attacks \\
\rownumber             & Amphibious    & Breathe, move, and fight underwater as if you were on land \\
\rownumber             & The Eye         & Gain Blindsight \\
\rownumber             & Vapor Form       & You can move through a space as narrow as 1 inch wide without squeezing. You move at half your normal speed and cannot attack while in this form \\
\rownumber             & Seer            & Once every long rest you may peek up to 1 hour into the future \\
\rownumber             & Vampiric        & During combat, damage dealt heals you for up to 10\% of your max HP every turn\\
\rownumber             & Killer Instinct & Gain advantage on melee attack rolls against Eldritch enemies \\
\rownumber             & Ravager         & Your unarmed strike damage becomes (Strength Modifier) d4 \\
\rownumber             & Heated Skin     & You gain resistance to Cold damage and do not need a fire to stay warm in cold environments \\
\rownumber             & Shadow Form  & While in dim light or darkness, you may turn Invisible as a bonus action \\
\rownumber             & Eldritch Nature  & You no longer require air, food, drink, or sleep. For all checks, you count as an Eldritch creature \\
\rownumber             & Sanguine Thirst  & You gain advantage on melee attack rolls against any Bloodied creature \\
\rownumber             & Chameleon Skin  & While you remain motionless, you blend into the background. If a creature comes within 10 ft of you, make a Stealth check to stay hidden. \\

\rownumber             & Eldritch Phobia & Rather than making a DC 10 monstrosity check to determine whether you gain a point of monstrosity, you now make a DC 5 check. However, the first time you encounter an eldritch creature after a long rest, you must make a DC 10 wisdom saving throw or become Frightened of that creature for a turn. \\
\rownumber             & Treasure Sense  & You can pinpoint, by scent, the location of
precious metals and stones, such as coins and gems, within 60 feet of you.\\
\rownumber             & Spider Climb    & You can climb difficult surfaces, including upside down on ceilings, without needing to make an ability check. \\
\rownumber             & Bewitching Eyes & Once per week, you can attempt to charm one sentient creature through supernatural guile. The creature must make a DC 10 + the player's Wisdom bonus Wisdom saving throw or is Charmed until this feature is used again. On a successful save the creature is aware of the attempt and is permanently immune to this effect.\\
\rownumber             & Inscrutable     & You are immune to any effect that would sense your emotions or read your thoughts, as well as any divination spell targeted within a 30ft area surrounding you unless you choose to allow it. Insight checks made to ascertain your intentions or sincerity have disadvantage. \\
\rownumber             & Grave Conversationalist & You can perceive and speak with the spirits who linger on the mortal plane. \\
\rownumber             & Soul Anchor   &  Your soul has become irrevocably tied to an object in your possession at the time of transformation. At will, you can teleport to the location of that object, and upon death you reform beside that object after 1d4 days. However, if the object is destroyed, your character's soul is destroyed irrevocably.\\

\end{dndlongtable}

% \dndfeat[]{Test}{feat test text}

After level 3 Monstrosity, the player receives no new Traits or Effects. Instead, on reaching >20 MST the player rolls a d20. If the number is greater than 10, the player succeeds the check and returns to humanity after 1d6 days. If the number is less than 10, the player fails the check and is permanently rendered their Eldritch Manifestation. This is effectively character death.

\end{document}