Advanced Mathematics

\usepackage{amsmath}        % Essential math enhancements
\usepackage{amssymb}        % Additional symbols
\usepackage{amsthm}         % Theorem environments
\usepackage{mathtools}      % Extensions to amsmath
\usepackage{tensor}         % Tensor notation
\usepackage{mathrsfs}       % Script letters
\usepackage{bbm}            % Blackboard bold
\usepackage{dsfont}         % Double stroke fonts

% Group theory
G = \langle a, b \mid a^2 = b^3 = (ab)^2 = e \rangle

% Quotient groups
G/H \cong \mathbb{Z}_n

% Ring homomorphism
\phi: R \to S, \quad \phi(xy) = \phi(x)\phi(y)

% Ideals
\mathfrak{p} \triangleleft R

% Galois groups
\text{Gal}(K/F) = \text{Aut}(K/F)

% Field extension degree
[K : F] = \dim_F K

% Algebraic closure
\overline{\mathbb{Q}}

% Splitting field
K = F(\alpha_1, \ldots, \alpha_n)

% Minimal polynomial
m_{\alpha,F}(x) = \text{irr}(\alpha, F)

% Holomorphic function
f: \mathbb{C} \to \mathbb{C} \text{ holomorphic}

% Cauchy-Riemann equations
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

% Residue theorem
\oint_{\gamma} f(z) \, dz = 2\pi i \sum_{k} \text{Res}(f, z_k)

% Laurent series
f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n

% Complex integral
\int_{\gamma} f(z) \, dz = \int_a^b f(\gamma(t)) \gamma'(t) \, dt

% Winding number
n(\gamma, z_0) = \frac{1}{2\pi i} \oint_{\gamma} \frac{dz}{z - z_0}

% Branch cuts
\log z = \ln|z| + i\arg(z), \quad -\pi < \arg(z) \leq \pi

% Open sets
\tau = \{U \subseteq X : U \text{ is open}\}

% Closure and interior
\overline{A} = \bigcap\{F : A \subseteq F, F \text{ closed}\}
A^{\circ} = \bigcup\{U : U \subseteq A, U \text{ open}\}

% Continuity
f^{-1}(V) \in \tau_X \text{ for all } V \in \tau_Y

% Compactness
X = \bigcup_{i \in I} U_i \implies X = \bigcup_{j=1}^n U_{i_j}

% Fundamental group
\pi_1(X, x_0) = \{[\gamma] : \gamma \text{ loop at } x_0\}

% Homology groups
H_n(X) = \ker(\partial_n) / \text{im}(\partial_{n+1})

% Euler characteristic
\chi(X) = \sum_{i=0}^{\infty} (-1)^i \text{rank}(H_i(X))

% Covering spaces
p: \tilde{X} \to X \text{ covering map}

% Category
\mathcal{C} = (\text{Ob}(\mathcal{C}), \text{Mor}(\mathcal{C}))

% Morphisms
f: A \to B \in \text{Hom}_{\mathcal{C}}(A, B)

% Functors
F: \mathcal{C} \to \mathcal{D}

% Natural transformation
\eta: F \Rightarrow G

% Commutative diagram
\begin{tikzcd}
A \arrow[r, "f"] \arrow[d, "g"'] & B \arrow[d, "h"] \\
C \arrow[r, "k"'] & D
\end{tikzcd}

% Ring of integers
\mathcal{O}_K = \{x \in K : x \text{ integral over } \mathbb{Z}\}

% Norm and trace
N_{K/\mathbb{Q}}(\alpha) = \prod_{i=1}^n \sigma_i(\alpha)
\text{Tr}_{K/\mathbb{Q}}(\alpha) = \sum_{i=1}^n \sigma_i(\alpha)

% Class number
h_K = |\text{Cl}(K)|

% Dedekind zeta function
\zeta_K(s) = \sum_{\mathfrak{a}} \frac{1}{N(\mathfrak{a})^s}

% Riemann zeta function
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}}

% Dirichlet L-functions
L(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}

% Prime number theorem
\pi(x) \sim \frac{x}{\ln x}

% Möbius function
\mu(n) = \begin{cases}
1 & n = 1 \\
(-1)^k & n = p_1 \cdots p_k \\
0 & \text{otherwise}
\end{cases}

% Tangent space
T_p M = \{v : C^{\infty}(M) \to \mathbb{R} \mid v \text{ derivation at } p\}

% Differential forms
\omega \in \Omega^k(M)

% Exterior derivative
d\omega = \sum_{i_0 < \cdots < i_k} \sum_{j} \frac{\partial f_{i_0\ldots i_k}}{\partial x^j} dx^j \wedge dx^{i_0} \wedge \cdots \wedge dx^{i_k}

% Lie derivative
\mathcal{L}_X \omega = d(i_X \omega) + i_X(d\omega)

% Metric tensor
g_{ij} = \langle \frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j} \rangle

% Christoffel symbols
\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left(\frac{\partial g_{jl}}{\partial x^i} + \frac{\partial g_{il}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^l}\right)

% Riemann curvature tensor
R^l_{ijk} = \frac{\partial \Gamma^l_{jk}}{\partial x^i} - \frac{\partial \Gamma^l_{ik}}{\partial x^j} + \Gamma^l_{im}\Gamma^m_{jk} - \Gamma^l_{jm}\Gamma^m_{ik}

% Geodesic equation
\frac{d^2 x^i}{dt^2} + \Gamma^i_{jk} \frac{dx^j}{dt} \frac{dx^k}{dt} = 0

% In preamble
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{note}[theorem]{Note}

\begin{theorem}[Fermat's Last Theorem]
\label{thm:fermat}
For $n > 2$, there are no three positive integers $a$, $b$, and $c$ 
that satisfy the equation $a^n + b^n = c^n$.
\end{theorem}

\begin{proof}
The proof is beyond the scope of this document. 
See Wiles (1995) for details.
\end{proof}

\begin{lemma}
\label{lem:helper}
Every finite integral domain is a field.
\end{lemma}

By Theorem~\ref{thm:fermat} and Lemma~\ref{lem:helper}, we conclude...

% Tensor products
V \otimes W, \quad \bigotimes_{i=1}^n V_i

% Direct sums
V \oplus W, \quad \bigoplus_{i=1}^n V_i

% Wedge products
\alpha \wedge \beta

% Cup and cap products
\alpha \cup \beta, \quad \alpha \cap \beta

% Hom and End
\text{Hom}(V, W), \quad \text{End}(V)