Semester 1: Foundations of Pure Mathematics
Advanced Calculus I (Real Analysis)

Set theory, relations, and functions
Construction of real numbers (Dedekind cuts, Cauchy sequences)
Sequences and limits in metric spaces
Bolzano-Weierstrass theorem
Continuity and uniform continuity
Differentiability and Mean Value Theorem
Advanced treatment of Riemann integrals
Sequences of functions: pointwise vs. uniform convergence
Abstract Algebra I

Group theory: cyclic groups, Lagrange's theorem, normal subgroups
Homomorphisms, isomorphisms, cosets, and quotient groups
Fundamental Theorem of Finitely Generated Abelian Groups
Ring theory: integral domains, ideals, and quotient rings
Fields: subfields, extensions, and finite fields
Introduction to Galois theory (depending on pace)
Linear Algebra (Theoretical Approach)

Abstract vector spaces and linear transformations
Inner product spaces and orthogonalization (Gram-Schmidt)
Eigenvalues, eigenvectors, and diagonalization
Jordan canonical form and spectral theorems
Tensor products and multilinear algebra (intro)
Matrix decompositions: SVD, LU, QR
Mathematical Logic and Set Theory

Propositional and first-order logic
Axiomatic set theory (Zermelo-Fraenkel, Axiom of Choice)
Ordinals and cardinals
Russell’s Paradox, Cantor’s Theorem
Gödel’s completeness and incompleteness theorems (overview)
Model theory and ultraproducts (intro)
Introduction to Topology

Topological spaces and continuous maps
Bases, subspaces, and product topology
Connectedness, compactness, and Tychonoff theorem
Quotient topology and fundamental group (intro)
Metric spaces and convergence
Applications to real analysis