diff --git a/README.md b/README.md index ef2ac9b..4f40b50 100644 --- a/README.md +++ b/README.md @@ -3,7 +3,7 @@ These are my notes for Part II and Part III of Mathematics at the University of Cambridge. There are many hyperlinks, marked in blue, to help both learning and revising a topic: I recommend using a PDF viewer with a back function (such as Skim) to make navigation more convenient. -If you would like help - especially diagrams in Ti*k*Z - feel free to get in touch with me; especially if you're producing open-source notes. +If you would like $\LaTeX$ help - especially diagrams in Ti*k*Z - feel free to get in touch with me; especially if you're producing open-source notes. ## Part III notes ### Michaelmas diff --git a/ii/lent/algebraic_geometry.tex b/ii/lent/algebraic_geometry.tex index cece83a..fe385eb 100644 --- a/ii/lent/algebraic_geometry.tex +++ b/ii/lent/algebraic_geometry.tex @@ -245,7 +245,7 @@ \subsection{Basic notions} \begin{equation*} Y_1 = Y_2 \cup Y_2',\quad Y_2 = Y_3 \cup Y_3',\quad \dotsc \end{equation*} - and so we get an infinite chain of affine varities $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$. + and so we get an infinite chain of affine varieties $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$. But each $Y_i = Z(I_i)$ for some ideals $I_i$. Let \begin{equation*} W = \bigcap Y_i = Z\left(\sum I_i\right) = Z(I) @@ -939,7 +939,7 @@ \section{Projective space} \begin{equation*} \begin{tikzcd} \left\{\parbox{2cm}{\centering closed subvarieties of $\A^n$}\right\} \rar[leftrightarrow]& - \left\{\parbox{5.5cm}{\centering closed subvarities $\overline{Z}$ of $\proj^n$ such that no \hyperlink{def:reducible}{irreducible} component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\ + \left\{\parbox{5.5cm}{\centering closed subvarieties $\overline{Z}$ of $\proj^n$ such that no \hyperlink{def:reducible}{irreducible} component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\ Z \rar[mapsto] &\overline{Z} = \text{closure of } \iota(Z) \text{ in } \proj^n \end{tikzcd} \end{equation*} @@ -949,7 +949,7 @@ \section{Projective space} A projective variety is a closed subvariety of $\proj^n$, some $n$ \end{defi} Recall an \hyperlink{def:affVar}{affine variety} is $k[X] = k[x_1, \dotsc, x_n]/I$, $I = \sqrt{I}$. -\begin{defi}[Quasivarities] +\begin{defi}[Quasivarieties] A \textbf{quasi-affine variety} is an open subvariety of an \hyperlink{def:affVar}{affine variety}. A \textbf{quasi-projective variety} is an open subvariety of a \hyperlink{def:projVar}{projective variety}. \end{defi} @@ -1153,8 +1153,8 @@ \section{Smooth points, dimension, Noether normalisation} \end{enumerate} \end{eg} -\begin{defi}[Transcendance dimension]\hypertarget{def:trdim} - Now we can define $\trdim X$, the \textbf{transcendance dimension} of extension $k \subseteq \hyperlink{def:k(x)}{k(X)}$. +\begin{defi}[Transcendence dimension]\hypertarget{def:trdim} + Now we can define $\trdim X$, the \textbf{transcendence dimension} of extension $k \subseteq \hyperlink{def:k(x)}{k(X)}$. \end{defi} It is not hard to see $\hyperlink{def:trdim}{\trdim} k(x_1, \dotsc, x_n)/k = \trdim \A^n$ = n. Generally, \begin{thm} @@ -1249,7 +1249,7 @@ \section{Smooth points, dimension, Noether normalisation} % new lecture To recap: Suppose we have affine varieties $X$ and $Y$ with a morphism \begin{equation*}k[X] = k[Y][t]/f(t) \leftarrow k[Y].\end{equation*} -We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varities $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$. +We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varieties $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$. \begin{defi}[Integral extension]\hypertarget{def:intExt} $B \subseteq A$ is an \textbf{integral ring extension} if $\forall a \in A$, $\exists$ a monic polynomial $f \in B[t]$ with $f(a) = 0$. @@ -1290,7 +1290,7 @@ \section{Smooth points, dimension, Noether normalisation} and indeed any projection onto a line other than the $x$ or $y$ axis will work. \end{eg} \begin{thm} - If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varities, and $X$, $Y$ irreducible. + If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varieties, and $X$, $Y$ irreducible. \begin{enumerate}[label=(\alph*)] \item $\overline{\varphi(X) = Y} \iff$ algebra homomorphism $k[Y] \to k[X]$ is injective. \item Suppose $\overline{\varphi(X) = Y}$. Then @@ -1335,7 +1335,7 @@ \section{Smooth points, dimension, Noether normalisation} \item $x_1, \dotsc, x_n$ generate $A$ \item for each $i > d$, $x_i$ satisfies a monic irreducible polynomial $F_i$ with coefficients in $k[x_1, \dotsc, x_{i-1}]$. \end{enumerate} -Moreoever, if $k$ is perfect, then $F_i$ can be chosen to be separable. +Moreover, if $k$ is perfect, then $F_i$ can be chosen to be separable. \begin{defi}[Perfect] A field $k$ is perfect if $\chara k = p > 0$ and $x \mapsto x^p$ is a surjection. \end{defi} @@ -1511,7 +1511,7 @@ \section{Algebraic Curves} =0 \quad \text{when } X = (x_{ij}) \tag{$*$} \label{eq:13star} \end{equation*} Now recall $X \text{adj}(X) = \det(X) I$ where $\text{adj}(X)$ is the matrix of determinants of minors. - Mulitply \eqref{eq:13star} by $\text{adj}(I-X)$, get $d m_i = 0$ for $i = 1, \dotsc, n$ where $d = \det(I-X) = 1+r$ with $r \in J$ by expanding out the determinant, use $J$ ideal i.e.\ $(1+r) M = 0$. + Multiply \eqref{eq:13star} by $\text{adj}(I-X)$, get $d m_i = 0$ for $i = 1, \dotsc, n$ where $d = \det(I-X) = 1+r$ with $r \in J$ by expanding out the determinant, use $J$ ideal i.e.\ $(1+r) M = 0$. (ii) is immediate from (i), by applying (i) to $M/N$. \end{proof} @@ -1535,7 +1535,7 @@ \section{Algebraic Curves} \end{remark} % ... \begin{proof} - $X$ is a curve, $\alpha$ defined on an open subset of $X$, so it is defined eexcept possibly at a finite set of points. + $X$ is a curve, $\alpha$ defined on an open subset of $X$, so it is defined except possibly at a finite set of points. SO it is enough to show $\alpha$ is defined at $p$. $Y$ is projective, $Y \subseteq \proj^m$ for some $m$, enough to prove this for $Y = \proj^m$. %(why?) \begin{equation*} @@ -1578,7 +1578,7 @@ \section{Algebraic Curves} \begin{equation*} k \hookrightarrow k(Y) \hookrightarrow k(X) \end{equation*} - but $k \hookrightarrow k(Y)$ has transcedance dimension 1, and $k \hookrightarrow k(X)$ has transcedance dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension. + but $k \hookrightarrow k(Y)$ has transcendence dimension 1, and $k \hookrightarrow k(X)$ has transcendence dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension. \end{enumerate} \end{proof} \begin{eg} @@ -1619,7 +1619,7 @@ \section{Differentials} \begin{ex} \leavevmode \begin{enumerate}[label=(\alph*)] - \item let $X$ be an affine algebraic vairety, $x \in X$ and consider the ring homomorphism $\text{ev}_x: k{X} \to k$ given by $f \mapsto f(x)$. + \item let $X$ be an affine algebraic variety, $x \in X$ and consider the ring homomorphism $\text{ev}_x: k{X} \to k$ given by $f \mapsto f(x)$. Show \begin{equation*} \text{Hom}_{k[X]}(\Omega^1_{k[X]/k]}, k) \xrightarrow{\sim} \text{Der}(k[X], k) = T_x X diff --git a/ii/lent/analysis_of_functions.tex b/ii/lent/analysis_of_functions.tex index f9a0e1d..3486f20 100644 --- a/ii/lent/analysis_of_functions.tex +++ b/ii/lent/analysis_of_functions.tex @@ -263,7 +263,7 @@ \subsection{Lebesgue integration} \begin{equation*}\int_E f_k \, d\mu \xrightarrow{k \to \infty} \int_E f \, d\mu.\end{equation*} \end{thm} \begin{proof} - Reduce to $E = X$ by consdering $f_k \chi_E, f \chi_E$. + Reduce to $E = X$ by considering $f_k \chi_E, f \chi_E$. Then $\left(\int_X f_k \, d\mu\right)_{k \geq 1}$ is a sequence in $[0, \infty]$, non-decreasing. By monotonicity, $f_k \nearrow f$, so $\int_X f_k \, d\mu \leq \int_X f \, d\mu$. diff --git a/ii/lent/coding_and_cryptography.tex b/ii/lent/coding_and_cryptography.tex index 2330414..976bdf4 100644 --- a/ii/lent/coding_and_cryptography.tex +++ b/ii/lent/coding_and_cryptography.tex @@ -616,7 +616,7 @@ \subsection{Huffman coding} \begin{remark} Not all \hyperlink{def:optCode}{optimal} codes are \hyperlink{def:huffmanCode}{Huffman}. For instance, take $m=4$, and probabilities $0.3, 0.3, 0.2, 0.2$. An optimal code is given by $00, 01, 10, 11$, but this is not Huffman. - But, the previous result says that if we have a prefix-free optimal code with word lengths $s_1, \dotsc, s_m$ and associated probabilites $p_1, \dotsc, p_m$, $\exists$ a Huffman code with these word lengths. + But, the previous result says that if we have a prefix-free optimal code with word lengths $s_1, \dotsc, s_m$ and associated probabilities $p_1, \dotsc, p_m$, $\exists$ a Huffman code with these word lengths. \end{remark} \begin{defi}[Joint entropy]\hypertarget{def:jointEntropy} @@ -1972,7 +1972,7 @@ \subsection{Reed-Muller Codes} \subsubsection*{Algebraic aside} A \textbf{ring} $R$ is a set with two operations, $+$ and $\times$. $(R, +)$ is an additive group and $\times$ is distributive over addition i.e.\ $a(b+c)=ab+ac$. Think of $\Z$ or $\Z_{10}$. -An \textbf{ideal} $I \unlhd R$ is an additive subgroup, closed under external mutliplication, i.e.\ if $a \in I$ and $r \in R$ then $ra \in I$. Think of $2\Z \unlhd Z$. +An \textbf{ideal} $I \unlhd R$ is an additive subgroup, closed under external multiplication, i.e.\ if $a \in I$ and $r \in R$ then $ra \in I$. Think of $2\Z \unlhd Z$. \begin{nlemma}\label{lem:2.27} Let $I$ be an ideal in ring $R$ and let $q: R \to R/I$ be the quotient map. Then there is a bijection between the set of ideals $J \subseteq R$ containing $I$ and the set of ideals in $R/I$. @@ -2304,7 +2304,7 @@ \subsubsection*{Decoding BCH codes} &= \sum_{j=1}^\infty e(\alpha^j) X^j \\ \implies w(X) &= \left(\sum_{j=1}^\infty e(\alpha^j) X^j \right) \sigma(X). \end{align*} - By defnition of $C$ we have $c(\alpha^j) = 0$ for $1 \leq i \leq \delta-1$ so for $1 \leq j \leq 2t$. + By definition of $C$ we have $c(\alpha^j) = 0$ for $1 \leq i \leq \delta-1$ so for $1 \leq j \leq 2t$. So $r(\alpha^j) = e(\alpha^j)$ for $1 \leq j \leq 2t$. Thus \begin{equation*} \sigma(X) \sum_{j=1}^{2t} r(\alpha^j) X^j \equiv w(X) \pmod{X^{2t+1}}. @@ -2326,7 +2326,7 @@ \subsubsection*{Decoding BCH codes} \end{proof} \subsection{Shift registers} \begin{defi}[General feedback shift register]\hypertarget{def:fsr} - A \textbf{general feedback shift register} is a fucntion $f:\F_2^d \to \F_2^d$ of the form + A \textbf{general feedback shift register} is a function $f:\F_2^d \to \F_2^d$ of the form \begin{equation*} f(x_0, \dotsc, x_{d-1}) = (x_1, \dotsc, x_{d-1}, c(x_0, \dotsc, x_{d-1})) \end{equation*} diff --git a/ii/lent/test.tex b/ii/lent/test.tex index 7d44476..39810ae 100644 --- a/ii/lent/test.tex +++ b/ii/lent/test.tex @@ -244,7 +244,7 @@ \subsection{Basic notions} \begin{equation*} Y_1 = Y_2 \cup Y_2',\quad Y_2 = Y_3 \cup Y_3',\quad \dotsc \end{equation*} - and so we get an infinite chain of affine varities $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$. + and so we get an infinite chain of affine varieties $Y \supsetneq Y_1 \supsetneq Y_2 \supsetneq \dotsb$. But each $Y_i = Z(I_i)$ for some ideals $I_i$. Let \begin{equation*} W = \bigcap Y_i = Z\left(\sum I_i\right) = Z(I) @@ -938,7 +938,7 @@ \section{Projective space} \begin{equation*} \begin{tikzcd} \left\{\parbox{2cm}{\centering closed subvarieties of $\A^n$}\right\} \rar[leftrightarrow]& - \left\{\parbox{5.5cm}{\centering closed subvarities $\overline{Z}$ of $\proj^n$ such that no irreducible component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\ + \left\{\parbox{5.5cm}{\centering closed subvarieties $\overline{Z}$ of $\proj^n$ such that no irreducible component of $\overline{Z}$ is contained in $\proj V \setminus \A^n = \proj H$ }\right\} \\ Z \rar[mapsto] &\overline{Z} = \text{closure of } \iota(Z) \text{ in } \proj^n \end{tikzcd} \end{equation*} @@ -948,7 +948,7 @@ \section{Projective space} A projective variety is a closed subvariety of $\proj^n$, some $n$ \end{defi} Recall an \hyperlink{def:affVar}{affine variety} is $k[X] = k[x_1, \dotsc, x_n]/I$, $I = \sqrt{I}$. -\begin{defi}[Quasivarities] +\begin{defi}[Quasivarieties] A \textbf{quasi-affine variety} is an open subvariety of an \hyperlink{def:affVar}{affine variety}. A \textbf{quasi-projective variety} is an open subvariety of a \hyperlink{def:projVar}{projective variety}. \end{defi} @@ -1221,7 +1221,7 @@ \section{Smooth points, dimension, Noether normalisation} % new lecture Suppose we have affine varieties $X$ and $Y$ with morphism $k[X] = k[Y][t]/f(t) \leftarrow k[Y]$. -We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varities $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$. +We noticed that if $f \in k[Y][t]$ is a monic polynomial, then the map of algebraic varieties $X \xrightarrow{\varphi} Y$ is surjective with finite $\varphi^{-1}(y)$ $\forall y \in Y$. \begin{defi} $B \subseteq A$ is an integral ring extension if $\forall a \in A$, $\exists$ a monic polynomial $f \in B[t]$ with $f(a) = 0$. @@ -1260,7 +1260,7 @@ \section{Smooth points, dimension, Noether normalisation} and indeed any projection onto a line other than the $x$ or $y$ axis will work. \end{eg} \begin{thm} - If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varities, and $X$, $Y$ irreducible. + If $k = \overline{k}$, and $\varphi: X \to Y$ is a morphism of algebraic varieties, and $X$, $Y$ irreducible. \begin{enumerate}[label=(\alph*)] \item $\overline{\varphi(X) = Y} \iff$ algebra homomorphism $k[Y] \to k[X]$ is injective. \item Suppose $\overline{\varphi(X) = Y}$. Then @@ -1546,7 +1546,7 @@ \section{Algebraic Curves} \begin{equation*} k \hookrightarrow k(Y) \hookrightarrow k(X) \end{equation*} - but $k \hookrightarrow k(Y)$ has transcedance dimension 1, and $k \hookrightarrow k(X)$ has transcedance dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension. + but $k \hookrightarrow k(Y)$ has transcendence dimension 1, and $k \hookrightarrow k(X)$ has transcendence dimension $1$, therefore $k(Y) \hookrightarrow k(X)$ has $\trdim = 0$, i.e. is an algebraic extension. \end{enumerate} \end{proof} \begin{eg} diff --git a/ii/mich/algebraic_topology.tex b/ii/mich/algebraic_topology.tex index 8c8e9e8..3c58fc6 100644 --- a/ii/mich/algebraic_topology.tex +++ b/ii/mich/algebraic_topology.tex @@ -1021,7 +1021,7 @@ \subsection{The Galois correspondence} \end{defi} \begin{defi} - If $p: \overline{X} \to X$ is a covering space, a covering transofmation or deck transformation is a self-homoemorphism $\phi: \overline{X} \to \overline{X}$ such that $p \circ \phi = p$. + If $p: \overline{X} \to X$ is a covering space, a covering transformation or deck transformation is a self-homeomorphism $\phi: \overline{X} \to \overline{X}$ such that $p \circ \phi = p$. In particular, $\phi$ is a lift of $p$ along itself, and so if $\overline{X}$ is path connected, $\phi$ is determined by its value at a point. \begin{equation*} @@ -1063,7 +1063,7 @@ \subsection{The Galois correspondence} The next tool is another action of $\pi_1(X, x_0)$. Let $p : (\widetilde{X}, \widetilde{x}_0) \to (X, x_0)$ be a universal cover. For each $[\gamma] \in \pi_1(X, x_0)$, we can consider $\widetilde{x}_0 \cdot [\gamma]$ as a different choice of basepoint for $\widetilde{X}$. -The Lifting Criterion gives a (ubique) covering transofmation $\phi_{[\gamma]} : \widetilde{X} \to \widetilde{X}$. +The Lifting Criterion gives a (unique) covering transformation $\phi_{[\gamma]} : \widetilde{X} \to \widetilde{X}$. \begin{lemma} This defines a left action of $\pi_1(X, x_0)$ on $\widetilde{X}$ by covering transformation. diff --git a/ii/mich/galois_theory.tex b/ii/mich/galois_theory.tex index 946e8ce..c942098 100644 --- a/ii/mich/galois_theory.tex +++ b/ii/mich/galois_theory.tex @@ -63,7 +63,7 @@ \subsection{Course overview} % Galois' papers have been studied by Peter Neumann: % The math writings of Evariste Galois, European Math Soc % Different books: I. Steward Galois Theory, (something) and Hall -% contains a historcal introduction and covers almost all the syllabus. +% contains a historical introduction and covers almost all the syllabus. % Artin Galois Theory % Van der Waerden Modern Algebra (covers a lot more than Galois theory) % Lang Algebra (late editions are preferred, covers a lot of algebra) @@ -322,7 +322,7 @@ \subsection{Digression on (Non-)Constructibility} Schedules mention `other classical problems' and we are now in a position to tackle some of these using \cref{cor:1.11}. A classical question from Greek geometry concerns the existence or otherwise of constructions using ruler and compasses (where a ruler refers to a single unmarked straight edge). -If you're an expert you can divide a line betwen 2 points into arbitrarily many equal segments, you can bisect an angle, or you can produce parallel lines. +If you're an expert you can divide a line between 2 points into arbitrarily many equal segments, you can bisect an angle, or you can produce parallel lines. Given a polygon you can produce a square of the same area or double the area. However, \begin{enumerate} \item You cannot duplicate the cube using ruler and compasses (given a cube you can't produce a cube of double the volume) @@ -1812,7 +1812,7 @@ \subsection{Galois Theory of Finite Fields} \begin{remark}[About \cref{thm:3.13}] We'll discover in Number Fields that $\Gal(\overline{f}) \hookrightarrow \Gal(f)$ if $f(t) \in \Z[t]$. We factorised $\overline{f}(t) = \overline{g_1}(t) \dotsm \overline{g_s}(t)$, a product of irreducibles. - We know from \cref{lem:3.6} that the orbits of $\Gal(\overline{f})$ correspond to the factorsiation. + We know from \cref{lem:3.6} that the orbits of $\Gal(\overline{f})$ correspond to the factorisation. We now know $\Gal(\overline{f})$ is cyclic generated by the Frobenius map, which must have cycle type $(n_1, \dotsc, n_s)$ where $n_j = \deg \overline{g_j}(t)$ \end{remark} @@ -2604,7 +2604,7 @@ \subsection{Algebraic closure} \begin{equation*}\tilde{f}(t) = f(t) - \prod_{j=1}^{\deg g} (t - X_{f,j}) \in K[X_s: s \in \mathcal{S}][t].\end{equation*} Let $I \lhd K[X_s : s \in \mathcal{S}]$ generated by all the coefficients of all the $\tilde{f}(t)$. - Denote the coefficents of $\tilde{f}(t)$ by $a_{f,l}$ for $0 \leq l \leq \deg f$. + Denote the coefficients of $\tilde{f}(t)$ by $a_{f,l}$ for $0 \leq l \leq \deg f$. Claim: $I \neq K[X_s : s \in \mathcal{S}]$. Proof: Suppose $1 \in I$ and aim for a contradiction. diff --git a/ii/mich/galois_theory_def.tex b/ii/mich/galois_theory_def.tex index 76afbe6..56c853c 100644 --- a/ii/mich/galois_theory_def.tex +++ b/ii/mich/galois_theory_def.tex @@ -58,7 +58,7 @@ \subsection{Course overview} % Galois' papers have been studied by Peter Neumann: % The math writings of Evariste Galois, European Math Soc % Different books: I. Steward Galois Theory, (something) and Hall -% contains a historcal introduction and covers almost all the syllabus. +% contains a historical introduction and covers almost all the syllabus. % Artin Galois Theory % Van der Waerden Modern Algebra (covers a lot more than Galois theory) % Lang Algebra (late editions are preferred, covers a lot of algebra) diff --git a/ii/mich/galois_theory_thm.tex b/ii/mich/galois_theory_thm.tex index 40d8549..25a92be 100644 --- a/ii/mich/galois_theory_thm.tex +++ b/ii/mich/galois_theory_thm.tex @@ -58,7 +58,7 @@ \subsection{Course overview} % Galois' papers have been studied by Peter Neumann: % The math writings of Evariste Galois, European Math Soc % Different books: I. Steward Galois Theory, (something) and Hall -% contains a historcal introduction and covers almost all the syllabus. +% contains a historical introduction and covers almost all the syllabus. % Artin Galois Theory % Van der Waerden Modern Algebra (covers a lot more than Galois theory) % Lang Algebra (late editions are preferred, covers a lot of algebra) diff --git a/ii/mich/galois_theory_thp.tex b/ii/mich/galois_theory_thp.tex index 7aab857..4b7ac42 100644 --- a/ii/mich/galois_theory_thp.tex +++ b/ii/mich/galois_theory_thp.tex @@ -58,7 +58,7 @@ \subsection{Course overview} % Galois' papers have been studied by Peter Neumann: % The math writings of Evariste Galois, European Math Soc % Different books: I. Steward Galois Theory, (something) and Hall -% contains a historcal introduction and covers almost all the syllabus. +% contains a historical introduction and covers almost all the syllabus. % Artin Galois Theory % Van der Waerden Modern Algebra (covers a lot more than Galois theory) % Lang Algebra (late editions are preferred, covers a lot of algebra) @@ -1757,7 +1757,7 @@ \subsection{Algebraic closure} \begin{equation*}\tilde{f}(t) = f(t) - \prod_{j=1}^{\deg g} (t - X_{f,j}) \in K[X_s: s \in \mathcal{S}][t].\end{equation*} Let $I \lhd K[X_s : s \in \mathcal{S}]$ generated by all the coefficients of all the $\tilde{f}(t)$. - Denote the coefficents of $\tilde{f}(t)$ by $a_{f,l}$ for $0 \leq l \leq \deg f$. + Denote the coefficients of $\tilde{f}(t)$ by $a_{f,l}$ for $0 \leq l \leq \deg f$. Claim: $I \neq K[X_s : s \in \mathcal{S}]$. Proof: Suppose $1 \in I$ and aim for a contradiction. diff --git a/ii/mich/linear_analysis.tex b/ii/mich/linear_analysis.tex index be5a568..a593add 100644 --- a/ii/mich/linear_analysis.tex +++ b/ii/mich/linear_analysis.tex @@ -2798,7 +2798,7 @@ \subsection{Application: Boundary Value Problem} \begin{fact}[Example sheet 4] Let $X,Y$ be \hyperlink{def:banach}{Banach spaces} and $D \subset X$ a \hyperlink{def:dense}{dense} subspace. - Then a bounded (\hyperlink{def:compact}{compact}) operator $T:D \to Y$ extends uniquely to a bounded (compact) operator $T:X \to Y$ with the same norm (BLT thoerem). + Then a bounded (\hyperlink{def:compact}{compact}) operator $T:D \to Y$ extends uniquely to a bounded (compact) operator $T:X \to Y$ with the same norm (BLT theorem). \end{fact} \begin{cor} $K$ extends uniquely to a \hyperlink{def:compact}{compact} \hyperlink{def:normalMap}{self-adjoint} operator $K: H \to H$. diff --git a/iii/lent/introduction_to_approximate_groups.tex b/iii/lent/introduction_to_approximate_groups.tex index 9f7cab9..0cd553b 100644 --- a/iii/lent/introduction_to_approximate_groups.tex +++ b/iii/lent/introduction_to_approximate_groups.tex @@ -1103,7 +1103,7 @@ \section{Nilpotent Groups} &\supset [G_{i+1},G] &&\text{by induction} \\ &=G_i &&\text{by \cref{prop:8.2}}. \end{align*} - We also have $Z_0(G) > H_{r+1}$ by definition, so we may assume $i > 0$ and, by inudction, that $H_{r+2-i} \subset Z_{i-1}(G)$. + We also have $Z_0(G) > H_{r+1}$ by definition, so we may assume $i > 0$ and, by induction, that $H_{r+2-i} \subset Z_{i-1}(G)$. But that \begin{equation*} G/Z_{i-1}(G) = \frac{G/H_{r+2-i}}{Z_i(G)/H_{r+2-i}} diff --git a/iii/mich/algebraic_topology.tex b/iii/mich/algebraic_topology.tex index df73439..37e21d6 100644 --- a/iii/mich/algebraic_topology.tex +++ b/iii/mich/algebraic_topology.tex @@ -353,7 +353,7 @@ \subsection{Singular (co)chains} Then $f_*(d\alpha) = 0 = d(f_* \alpha) \implies f_*(\alpha) \in D_i$ is a cycle in the $D_*$-chain complex, and hence defines an element in $H_i(D_*) = \frac{\ker(d:D_i \to D_{i-1})}{\im(d:D_{i+1}\to D_i)}$. Call this element $b$ and set $f_*(a) = b$. This is well-defined. - If $\alpha'$ also repesents $a \in H_i(C_*)$, then $\alpha-\alpha'$ is a boundary, i.e.\ $\alpha-\alpha_i = d_{i+1}(\gamma)$ for some $\gamma \in C_{i+1}$. + If $\alpha'$ also represents $a \in H_i(C_*)$, then $\alpha-\alpha'$ is a boundary, i.e.\ $\alpha-\alpha_i = d_{i+1}(\gamma)$ for some $\gamma \in C_{i+1}$. Then $f_*(\alpha) - f_*(\alpha') = f_*(d_{i+1}\gamma) = d_{i+1} f_*(\gamma)$ so $f_*(\alpha')$ and $f_*(\alpha)$ differ by a boundary, so define some element $b \in H_i(D_*)$. It is an easy exercise to check that this map $f_*:H_i(C_*) \to H_i(D_*)$ is indeed a homomorphism of groups. \end{proof} diff --git a/iv/lent/connections_between_model_theory_and_combinatorics.tex b/iv/lent/connections_between_model_theory_and_combinatorics.tex index 0c5b682..3d6eb31 100644 --- a/iv/lent/connections_between_model_theory_and_combinatorics.tex +++ b/iv/lent/connections_between_model_theory_and_combinatorics.tex @@ -368,7 +368,7 @@ \subsection{Characterisation of stability in terms of types} \end{eg} % lecture 4 \begin{defi} - Let $b \in M$, $A \subseteq M$. Then the type of $b$ over $A$ is the collecitno of all formulas with parameters in $A$ that are satisfied by $b$: + Let $b \in M$, $A \subseteq M$. Then the type of $b$ over $A$ is the collection of all formulas with parameters in $A$ that are satisfied by $b$: \begin{equation*} \tp_\varphi(b/A) \coloneqq \{\varphi(x,a) \mid a \in A, \models \varphi(b,a)\}. \end{equation*}