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A metric space is a set $X$ together with a function $$ \operatorname{dist}: X \times X \to \R $$ called a metric such that the following laws are satisfied: • (Positivity) $\operatorname{dist}(x,y) \ge…

A homotopy $h : p \simeq q$ between maps $p,q : X \to Y$ is a continuous map \[ h: X \times I \to Y \] such that \[ h(x,0) = p(x), h(x,1) = q(x) \] where $I = [0,1]$ the unit interval.

An equivalence class is a subset of a larger set containing elements that are considered "equivalent" to eachother in the context of a equivalence relation. An equivalence relation is a binary operation denoted $a \s…

Let $$ \gamma_0(t) = (t,0), \quad \gamma_1(t) = (t,t), \quad t \in [0,1]. $$ Both paths go from $(0,0)$ to $(1,1)$. Define a homotopy $H : [0,1] \times [0,1] \to \mathbb{R}^2$ by $$ H(t,s) = (t, s t). $$ Proof th…

A loop is a path \[ f : I \to X \] such that $f(0)=f(1)$

Paths being homotopic defines an equivalence class. Depends on and

Let $f$ be a loop. Let $[f]$ denote the equivalence class under homotopy for $f$. We define $\pi_1(X,x)$ to be the set of equivalence classes of loops that start and end at $x$. We will show that…

The Fundamental Group is indeed a group.

For a path $a : x \to y$, define \[ \gamma[a] : \pi_1(X,x) \to \pi_1(X,y) \] by \[ \gamma[a][f] = [a \cdot f \cdot a^{-1}] \] This is a homomorphism of groups.

Prove that the map $\gamma[a]$ as defined in is indeed a group homomorphism.

A representation of a group $G$ on a vector space $V$ over a field $k$ is a group homomorphism \[ \rho : G \to GL(V). \] The dimension $\dim_k V$ is called the degree of the representation.

A subspace $W \subseteq V$ is a subrepresentation (or $G$-invariant subspace) if $\rho(g)(W) \subseteq W$ for all $g \in G$. This notion relies on the setup of .

A representation $(\rho, V)$ (in the sense of ) is irreducible (or simple) if $V \neq 0$ and the only subrepresentations (see ) of $V$ are $\{0\}$ and $V$ itself.

Let $(\rho, V)$ and $(\sigma, W)$ be irreducible representations () of $G$ over an algebraically closed field $k$. • Any $G$-module homomorphism $\phi : V \to W$ is either zero or an isomorphism. • $\mathrm…

Let $G$ be a finite group and $k$ a field with $\mathrm{char}(k) \nmid |G|$. Then every representation () of $G$ over $k$ is completely reducible, i.e.\ decomposes into irreducibles ().

The regular representation of $G$ is the action of $G$ on the group algebra $k[G]$ by left multiplication (a special case of ): \[ \rho(g) \cdot \sum_{h \in G} a_h \, h \;=\; \sum_{h \in G} a_h \, (gh). \] By , over a…

Over $\mathbb{C}$, the number of (isomorphism classes of) irreducible representations of $G$ equals the number of conjugacy classes of $G$. This follows from and .

For a finite group $G$ over $\mathbb{C}$, combining and , \[ \sum_{\chi \in \hat{G}} (\dim V_\chi)^2 = |G|. \]

Using and , compute the complete character table of the symmetric group $S_3$. Verify the column orthogonality relations: \[ \sum_{\chi \in \hat{G}} \chi(g)\,\overline{\chi(h)} \;=\; |C_G(g)| \cdot \delta_{[g],[h]}…

The category of $kG$-modules and the category of representations over a group $G$ $Rep_k (G)$ are equivalent.

The triple $(\Omega, \mathcal{F}, P)$ is a probability space if • $\Omega$ is the sample space, that is some possibly abstract set. • $\mathcal{F}$ is a $\sigma$-algebra of sets - the measurable subsets o…

A random variable $X$ is a measurable function from the sample space $\Omega$ to $\R$ $$ X : \Omega \to \R $$ that is, the inverse of any Borel Set in $\R$ is $\mathcal{F}$-measurable: $$ X^{-1} (A) = \{\omega : X(…

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Given a seperating class $\mathcal{H}$ : \[ d_\mathcal{H}(F,G) = \operatorname{sup}_{h \in \mathcal{H}} | \mathbb{E}(h(F)) - \mathbb{E}(h(G)) | \]

A real valued random variable $N$ has the standard Gaussian distribution if and only if for every test function $f : \mathbb{R} \to \mathbb{R}$ that is differentiable with $f' \in \mathcal{L}'(\gamma)$, the expe…

Let $F$ be a random variable such that \[ \mathbb{E}(f'(F)-Ff(f)) \approx 0 \] for a large class of test functions $f$ we want to say that this is close to the standard Gaussian. \[ L(F) \approx N(0,…

Let $\gamma$ be the standard Gaussian Measure \[ \gamma(dx) = \frac{1}{} \]

Let $N \sim N(0,1)$. Let $h : \mathbb{R} \to \mathbb{R}$ be a Borel function such that \[ \mathbb{E}(|h(N)|) < \infty \] or in other words \[ h \in \mathcal{L}(\gamma) \] The Stein Equation a…

All solutions of Stein's Equation are of the form \[ f(x) = Ce^{x^2/2} + e^{x^2/2}\int_{-\infty}^x [h(y)-\mathbb{E}(h(N))]e^{y^2/2} dy \] In particular, denote by \[ f_h(x) = e^{x^2/2}\int_{-\i…

Let $\mathcal{H}$ be a seperating class and let $h \in \mathcal{H}$. Let $f_h$ be the solution to the Stein Equation associated with $h$. Then \[ f'_h(x)-xf_h(x) = h(x)-\mathbb{E}(h(N)) \] Let $…

The total variation metric is is defined as follows: \[ d_\operatorname{TV}(F,G) = \sup_{B \in \mathcal{B}(\mathbb{R})} | \mathbb{P}(F \in B) - \mathbb{P}(G \in B) \] where $\mathcal{B}(\mathbb{R})$ are the Borel …

The Kolmogorow Metric metrizes the space of probability distributions given the following definition: \[ d_{\operatorname{Kol}}(F,G) = \sup | P(F \le z) - P(G\le z) | \]

We take the seperating class that defined the total variation metric : \[ \mathcal{H}_{TV} = \{\mathbb{1}_B : B \in \mathcal{B}(\mathbb{R})\} \] The proposition is as follows: Let $h…

A group is a pair $(G,\cdot)$ consisting of a set $G$ together with a binary operation \[ \cdot : G \times G \to G \] such that: • Associativity: For all $a,b,c \in G$, $(a \cdot b)\cdot c = a \cdot (b \cdot c)…

A group is a group object in the category $\mathbf{Set}$. That is, a set $G$ equipped with morphisms: \[ m : G \times G \to G, \quad e : 1 \to G, \quad i : G \to G \] such that the following identities hold: • As…

Let $(G, \cdot)$ and $(H, \ast)$ be groups in the sense of Definition~. A map $\phi : G \to H$ is a group homomorphism if \[ \phi(a \cdot b) = \phi(a) \ast \phi(b) \quad for all a, b \in G. \] If $\phi$ is also a bij…

For a homomorphism $\phi : G \to H$ (Definition~), the kernel and image are \[ \ker \phi := \{ g \in G \mid \phi(g) = e_H \}, \qquad \operatorname{im} \phi := \{ \phi(g) \mid g \in G \}. \]

Let $G$ be a group (Definition~). A subset $H \subseteq G$ is a subgroup, written $H \leq G$, if it is closed under multiplication and inverses and contains the identity. It is normal, written $H \trianglelefteq G$, if …

For any homomorphism $\phi : G \to H$ (Definition~), $\ker \phi \trianglelefteq G$ and $\operatorname{im} \phi \leq H$ in the sense of Definition~, where $\ker\phi$ and $\operatorname{im}\phi$ are as in Definition~.

Let $N \trianglelefteq G$ be a normal subgroup (Definition~). The quotient group $G/N$ is the set of left cosets $\{ gN \mid g \in G \}$ equipped with the operation $(aN)(bN) := (ab)N$.

Let $\phi : G \to H$ be a group homomorphism (Definition~). Then the quotient group (Definition~) by the kernel (Definition~) satisfies \[ G / \ker\phi \;\cong\; \operatorname{im}\phi. \]

A left action of a group $G$ (Definition~) on a set $X$ is a map $G \times X \to X$, written $(g, x) \mapsto g \cdot x$, satisfying \[ e \cdot x = x \quad and \quad g \cdot (h \cdot x) = (gh) \cdot x \quad for all g,…

Let $G$ act on $X$ (Definition~). For $x \in X$, the orbit and stabilizer of $x$ are \[ G \cdot x := \{ g \cdot x \mid g \in G \}, \qquad G_x := \{ g \in G \mid g \cdot x = x \}. \] Note that $G_x \leq G$ is a subgr…

For any $x \in X$ with orbit and stabilizer as in Definition~, \[ |G \cdot x| = [G : G_x], \] where $[G : G_x]$ is the index of the subgroup $G_x \leq G$ (Definition~), equivalently the cardinality of the quotient $G/…

The direct product $G \times H$ of two groups (Definition~) is the Cartesian product with componentwise operations. More generally, given a normal subgroup $N \trianglelefteq G$ and a subgroup $H \leq G$ (Definition~) w…

A topological space is a pair $(X, \mathcal{T})$ where $\mathcal{T}$ is a collection of subsets of $X$ (the open sets) satisfying: $\emptyset, X \in \mathcal{T}$; arbitrary unions of open sets are open; finite intersect…

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces (Definition~). A map $f : X \to Y$ is continuous if $f^{-1}(U) \in \mathcal{T}_X$ for every $U \in \mathcal{T}_Y$. If $f$ is a continuous bijection…

A topological space $X$ (Definition~) is connected if it cannot be written as a disjoint union of two nonempty open sets. It is path-connected if for every $x, y \in X$ there exists a continuous path $\gamma : [0,1] \to…

Path-connectedness implies connectedness, but not conversely. For Lie groups the two notions coincide: every connected Lie group is path-connected.

A path-connected space $X$ (Definition~) is simply connected if every loop $\gamma : [0,1] \to X$ with $\gamma(0) = \gamma(1)$ can be continuously contracted to a point, i.e., the fundamental group $\pi_1(X) = 0$.

A topological space $X$ (Definition~) is compact if every open cover of $X$ has a finite subcover. A subset $A \subseteq \mathbb{R}^n$ is compact if and only if it is closed and bounded (Heine--Borel).

A topological $n$-manifold is a Hausdorff, second-countable topological space $M$ (Definition~) such that every point $p \in M$ has a neighbourhood homeomorphic (Definition~) to an open subset of $\mathbb{R}^n$.

A topological $n$-manifold $M$ (Definition~) is a smooth manifold if it is equipped with a maximal smooth atlas: a collection of charts $(U_\alpha, \varphi_\alpha)$ covering $M$ such that all transition maps $\varphi_\b…

For $p \in M$ a smooth manifold (Definition~), the tangent space $T_pM$ is the vector space of derivations on smooth functions near $p$. Concretely in a chart, it is spanned by $\partial/\partial x^i|_p$. The tangent bu…

A map $f : M \to N$ between smooth manifolds (Definition~) is smooth if its coordinate representations are smooth. It is a diffeomorphism if it is a smooth bijection with smooth inverse; in particular a diffeomorphism i…

A smooth map $f : M \to N$ (Definition~) is an immersion if the differential $df_p : T_pM \to T_{f(p)}N$ (Definition~) is injective for all $p$. It is an embedding if it is additionally a homeomorphism (Definition~) ont…

This distinction matters for Lie subgroups: closed subgroups are always embedded (Definition~), but immersed subgroups (e.g.\ a dense winding on a torus) need not be.

A continuous map $p : \widetilde{X} \to X$ (Definition~) between topological spaces (Definition~) is a covering map if every point $x \in X$ has an open neighbourhood $U$ such that $p^{-1}(U)$ is a disjoint union of ope…

If $G$ is a connected (Definition~) Lie group, its universal cover $\widetilde{G}$ (Definition~) carries a unique Lie group structure such that the covering map $p : \widetilde{G} \to G$ is a Lie group homomorphism (Def…

A (linear) representation of a group $G$ (Definition~) on a vector space $V$ over a field $k$ is a group homomorphism (Definition~) \[ \rho : G \to \mathrm{GL}(V). \] We say $(V, \rho)$ is a $G$-representation, or sim…

Given a $G$-representation $(V, \rho)$ (Definition~), a subspace $W \subseteq V$ is $G$-invariant (a subrepresentation) if $\rho(g)w \in W$ for all $g \in G$, $w \in W$. The restriction $\rho|_W$ then defines a represen…

A representation $(V, \rho)$ (Definition~) is irreducible (or simple) if $V \neq 0$ and its only subrepresentations (Definition~) are $\{0\}$ and $V$ itself.

A linear map $T : V \to W$ between $G$-representations (Definition~) is a $G$-equivariant map (or intertwiner) if \[ T \circ \rho_V(g) = \rho_W(g) \circ T \quad for all g \in G. \] An invertible intertwiner is an iso…

Let $(V, \rho)$ and $(W, \sigma)$ be irreducible $G$-representations (Definition~) over an algebraically closed field, and let $T : V \to W$ be an intertwiner (Definition~). • Either $T = 0$ or $T$ is an isomorphism.…

Given $G$-representations $(V, \rho)$ and $(W, \sigma)$ (Definition~), their direct sum is $(V \oplus W, \rho \oplus \sigma)$ where $(\rho \oplus \sigma)(g)(v, w) = (\rho(g)v, \sigma(g)w)$.

A representation $V$ (Definition~) is completely reducible (semisimple) if it decomposes as a direct sum (Definition~) of irreducible subrepresentations (Definition~, Definition~).

Every finite-dimensional continuous representation (Definition~) of a compact Lie group (Definition~) over $\mathbb{R}$ or $\mathbb{C}$ is completely reducible (Definition~).

The character of a finite-dimensional representation $(V, \rho)$ (Definition~) is the function $\chi_V : G \to k$ defined by $\chi_V(g) = \mathrm{tr}(\rho(g))$. Characters are class functions: $\chi_V(hgh^{-1}) = \chi_V…

Isomorphic representations (Definition~) have equal characters (Definition~). For compact groups (Definition~), the converse holds: two representations are isomorphic if and only if their characters are equal.

Let $\rho_1 : G \to \mathrm{GL}(V_1)$ and $\rho_2 : G \to \mathrm{V_2}$ be two linear representations of $G$ (Definition ), and let $\chi_1$ and $\chi_2$ be their characters (Definition ). Then: • The character …

The inner product of two characters (Definition ) of representations of a finite group $G$ (Definition is \[ \langle \chi_1, \chi_2 \rangle = \frac{1}{|G|}\sum_{g \in G} \chi_1 (g) \overline{\chi_2(g)} \] …

A Lie Group is a group (Definition ) that is also a finite dimensional smooth differentiable manifold (Definition ), with the added condition that the group operations of multiplication and inversion are smooth maps…

The unit circle in $\mathbb{C}$ denoted $$S^1 = \{e^{i\theta} \colon \theta \in [0,2\pi)\} = \{z \in \mathbb{C}\colon |z|= 1\}$$ endowed with group multiplication as \[ e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha+…

$S^1$ (Example ) is a Lie group and is isomorphic to $SO(2)$ the rotation group.

A representation of a Lie group (Definition ) is the exact same as a representation of a finite group (Definition ), however with the added condition that the representation must be a continuous map.

A category $\mathcal{C}$ consists of the following data: • A collection of objects, denoted $\operatorname{Ob}(\mathcal{C})$. • For every pair of objects $X,Y \in \operatorname{Ob}(\mathcal{C})$, a set $$ \opera…

Let $\mathcal{C}$ be a category as in Definition~. If $f \in \Hom_{\mathcal{C}}(X,Y)$ we write $$ f : X \to Y. $$ The object $X$ is called the domain of $f$ and $Y$ the codomain. Composition of morphisms is written…

Let $\mathcal{C}$ be a category (Definition~). A morphism $$ f : X \to Y $$ is called an isomorphism if there exists a morphism $$ g : Y \to X $$ such that $$ g \circ f = \operatorname{id}_X \qquad f \circ g = \oper…

Let $\mathcal{C}$ be a category (Definition~). If a morphism $f : X \to Y$ is an isomorphism (Definition~), then its inverse is unique.

Let $\mathcal{C}$ be a category (Definition~). The relation $$ X \cong Y $$ defined via isomorphisms (Definition~) is an equivalence relation on $\operatorname{Ob}(\mathcal{C})$.

Let $\mathcal{C}$ be a category (Definition~). A subcategory $\mathcal{D}$ of $\mathcal{C}$ consists of • a collection of objects $$ \operatorname{Ob}(\mathcal{D}) \subseteq \operatorname{Ob}(\mathcal{C}) $$ • fo…

Let $\mathcal{C}$ be a category (Definition~). The opposite category $\mathcal{C}^{op}$ is defined as follows. • Objects: $$ \operatorname{Ob}(\mathcal{C}^{op}) = \operatorname{Ob}(\mathcal{C}) $$ • Morphisms: $$…

Let $\mathcal{C}$ be a category (Definition~). Any statement about $\mathcal{C}$ has a dual statement obtained by replacing • morphisms $f : X \to Y$ by $f : Y \to X$ • compositions $g \circ f$ by $f \circ g$ …

A functor $F$ is a map of categories that preserves commutative diagrams. In particular, given two categories, $\mathcal{C}$, $\mathcal{D}$, a functor $F : \mathcal{C} \to \mathcal{D}$ satisfies the following prope…

Let $$ F : \mathcal{C} \to \mathcal{D} $$ be a functor (Definition~). If $$ f : X \to Y $$ is an isomorphism in $\mathcal{C}$ (Definition~), then $$ F(f) : F(X) \to F(Y) $$ is an isomorphism in $\mathcal{D}$. Refe…

Let $$ F : \mathcal{C} \to \mathcal{D}, \qquad G : \mathcal{D} \to \mathcal{E} $$ be functors (Definition~). The composition of functors $$ G \circ F : \mathcal{C} \to \mathcal{E} $$ is defined as follows. On objects…

The composition of functors is a functor. References: Definition~, Definition~.

Let $\mathcal{C}$ be a category. The identity functor $$ \operatorname{Id}_{\mathcal{C}} : \mathcal{C} \to \mathcal{C} $$ is defined by Objects $$ \operatorname{Id}_{\mathcal{C}}(X) = X $$ Morphisms $$ \operator…

The identity functor is a functor. References: Definition~, Definition~.

Functor composition is associative. References: Definition~.

Let $$ \mathbf{Grp} $$ be the category of groups and group homomorphisms, and $$ \mathbf{Set} $$ the category of sets. Define a functor $$ U : \mathbf{Grp} \to \mathbf{Set} $$ as follows. Objects $$ U(G) = the…

Let $$ F,G : \mathcal{C} \to \mathcal{D} $$ be functors (Definition~). A natural transformation $$ \eta : F \Rightarrow G $$ consists of morphisms $$ \eta_X : F(X) \to G(X) $$ for every object $X$ of $\mathcal{C…

Let $\Omega$ be a nonempty set. A sigma-algebra (or $\sigma$-algebra) on $\Omega$ is a collection $\mathcal{F} \subseteq 2^\Omega$ of subsets satisfying: • $\Omega \in \mathcal{F}$. • If $A \in \mathcal{F}$, then $…

A probability space is a triple $(\Omega, \mathcal{F}, \mathbb{P})$ where: • $(\Omega, \mathcal{F})$ is a measurable space (Definition~), with $\Omega$ the sample space. • $\mathbb{P} : \mathcal{F} \to [0,1]$ is a …

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~). A random variable is a measurable function $X : \Omega \to \mathbb{R}$, meaning $X^{-1}(B) \in \mathcal{F}$ for every Borel set $B \subseteq …

Let $X$ be a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~). The expectation of $X$ is \[ \mathbb{E}[X] := \int_\Omega X(\omega)\, d\mathbb{P}(\omega), \] provided the integral exists. For $p \ge…

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~). • Events $A, B \in \mathcal{F}$ are independent if $\mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B)$. • Random variables $X, Y$ (Defin…

Let $X \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~) and let $\mathcal{G} \subseteq \mathcal{F}$ be a sub-$\sigma$-algebra (Definition~). The conditional expectation $\mathbb{E}[X \mid \mathcal{G}]$ is the $\m…

The Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ on $\mathbb{R}$ is the smallest $\sigma$-algebra (Definition~) containing all open subsets of $\mathbb{R}$. More generally, for a topological space $X$, the Borel $\s…

A measure space is a triple $(\Omega, \mathcal{F}, \mu)$ where $(\Omega, \mathcal{F})$ is a measurable space (Definition~) and $\mu : \mathcal{F} \to [0, \infty]$ satisfies $\mu(\emptyset) = 0$ and countable additivity.…

Let $(\Omega, \mathcal{F}, \mu)$ be a measure space (Definition~) and let $0 \leq f_1 \leq f_2 \leq \cdots$ be a non-decreasing sequence of non-negative measurable functions with $f_n \to f$ pointwise. Then \[ \lim_{n…

Let $(\Omega, \mathcal{F}, \mu)$ be a measure space (Definition~). Suppose $f_n \to f$ pointwise $\mu$-a.e.\ and $|f_n| \leq g$ $\mu$-a.e.\ for all $n$, where $g \in L^1(\mu)$. Then $f \in L^1(\mu)$ and \[ \lim_{n \to…

Let $(\Omega, \mathcal{F})$ be a measurable space (Definition~) and let $\mu, \nu$ be $\sigma$-finite measures (Definition~) with $\nu \ll \mu$ (i.e.\ $\mu(A) = 0 \Rightarrow \nu(A) = 0$). Then there exists a non-negati…

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~) and let $T \subseteq [0, \infty)$. A stochastic process indexed by $T$ is a collection $\{X_t\}_{t \in T}$ of random variables (Definition~) o…

A filtration on $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~) is an increasing family $(\mathcal{F}_t)_{t \geq 0}$ of sub-$\sigma$-algebras of $\mathcal{F}$: \[ s \leq t \implies \mathcal{F}_s \subseteq \mathcal{F…

Let $\{X_t\}_{t \geq 0}$ be an adapted process (Definition~) with $X_t \in L^1$ (Definition~) for all $t$. The process is a martingale with respect to $(\mathcal{F}_t)$ if \[ \mathbb{E}[X_t \mid \mathcal{F}_s] = X_s \…

Let $(\mathcal{F}_t)_{t \geq 0}$ be a filtration (Definition~). A random variable $\tau : \Omega \to [0, \infty]$ is a stopping time with respect to $(\mathcal{F}_t)$ if \[ \{\tau \leq t\} \in \mathcal{F}_t \quad for …

A standard Brownian motion (or Wiener process) is a stochastic process $\{B_t\}_{t \geq 0}$ (Definition~) on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~) satisfying: • Initial value: $B_0 = 0…