Here you will find (rather extensive) information regarding the material typically covered in some of the courses I have taken while at University. As in the University material section, I have divided the courses into five main sections.
Graduate Courses
Operator Theory (Evgenios Kakariadis, Fall 2025) Essentially covered the first three chapters of Murphy's C*-algebras and Operator Theory, as well as a few elements of positive functionals on matrix algebras and the spectrum (point, approximate and compression spectrum) of an operator in Banach and Hilbert spaces.
Stochastic Processes (Dimitrios Cheliotis, Fall 2025) Essentially covered paragraphs 4.1-4.8 from Durrett's Probability: Theory and Examples, as well as an introduction to elements of Stochastic Calculus, Brownian motion and Random Graphs (for the last we used Frieze-Karoński's Random Graphs and Networks: A First Course, Blum-Hopcroft-Kannan's Foundations of Data Science and Roch's Modern Discrete Probability)
Analysis
Real Analysis - Theory of Metric Spaces (Irene Deligianni, Spring 2025) Metric spaces, convergence of sequences, topology of metric spaces: open and closed sets, closure and interior, accumulation points and boundary of a set, dense sets and separable spaces, continuous functions between metric spaces, Lipschitz functions, isometries, homeomorphisms, Uryshon's Lemma, partitions of unity, oscillation of a function, completeness and Cantor's theorem, the Baire category theorem and Osgood's theorem, the completion of a metric space, Banach's fixed point theorem, compactness and sequential compactness, continuous functions in compact sets, the Cantor set, function series, uniform and pointwise convergence, Dini's theorem, the Weierstrass approximation theorem.
Introduction to Topology (Irene Deligianni, Fall 2025) Elementary notions of Topology, bases and subbases, the product topology, topological subspaces, T1, T2, T3, T3 1/2, T4 spaces, and basic properties, metrizability of the product of topological spaces and connection to the separation axioms, Uryshon's Lemma, the Embedding Lemma, the Uryshon metrizability theorem, first and second countable spaces, quotient spaces and the quotient topology, compactness, Tychonoff's theorem, sequences, nets and subnets in topological spaces, connected topological spaces, the Tietze extension theorem.
Measure Theory (Marina Iliopoulou, Fall 2025) Vitali set, σ-algebras, Dynkin systems, measures, complete measures, outer measures, construction of Lebesgue outer measure, (Lebesgue) measurable sets, inner and outer measure, Carathéodory's extension theorem, regularity of Lebesgue measure, transformations of Lebesgue measure, the Cantor set and the Cantor-Lebesgue function, simple and measurable functions, the Lebesgue integral, indefinite integral and integrable functions, Fatou's lemma and the basic convergence theorems (dominated, bounded and monotone convergence), Fubini's and Tonelli's theorems.
Algebra
Galois Theory (Mihalis Maliakas, Spring 2025) Field extensions, algebraic extensions, splitting fields, the Galois group, the fundamendal theorem of Galois theory, cyclotomic polynomials and constructible polygons, solvable groups, polynomials solvable with radicals, finite fields. Some applications (symmetric functions, the fundamendal theorem of Algebra, the inverse Galois problem, Kummer extensions) of the material were also covered.
Basic Abstract Algebra (Mihalis Maliakas, Fall 2025) Basic number theory, rings, polynomials, homomorphisms and ideals, quotient rings and isomorphism theorems, the Chinese Remainder Theorem, ring characteristic and finite fields, groups and symmetric groups, Cayley's theorem, subgroups and Lagrange theorem, odd and even permutations, multiplicative group of a finite field, group homomorphisms, structure of cyclic groups, normal subgroups, Cauchy's theorem for abelian groups, group (inner product) and group isomorophism theorems.
Topics in Algebra and Geometry I (Maria Papatriantafillou, Fall 2025) Elements of topology, topological groups, neighborhoods of unity, subgroups-products-quotients of topological groups, open and closed subgroups, separability of topological groups, connected topological groups, left and right invariant metrics in topological groups. Topological vector spaces, balanced-convex-absorbing sets and neighborhoods of 0, continuous linear maps in topological vector spaces, seminorms and convex neighborhoods of 0, Minkowski functional, locally convex spaces and the Hahn-Banach theorem, Kolmogorov's theorem in topological vector spaces.
Commutative Algebra and Applications (Konstantinos Karagiannis, Fall 2025) Commutative rings, ideals, factorization in integral domains, unique factorization domains, basic elements of Algebraic Geometry, modules, exact sequences, free modules, Noetherian and Artinian rings and modules, Hilbert's basis theorem, composition series, Jacobson radical and properties, primitive ideals, Noether normalization, integral dependence, Hilbert's Nullstellensatz.
Homological Algebra and Categories (Ioannis Emmanouil, Fall 2025) Basic notions of Category Theory, functors and natural transformations, Yoneda's Lemma, the category of modules, additive functors, exact sequences, Hom-functors, projective and injective modules, Eilenberg's trick, chain complexes and homology, homotopy and projective/injective resolutions, Ext functors and applications.
Number Theory
Analytic Number Theory (Nikolaos Skarmogiannis, Fall 2025) Elements of basic number theory, arithmetic functions and Dirichlet product, Möbius inversion formula, derevative of an arithmetic function and Selberg's identity, Abel summation, Euler-Maclaurin formula, Dirichlet hyperbola method and applications to the partial sum of the divisor function, the first and the second Chebyshev function, Mertens' theorems, the prime number theorem and its correlation with θ(x), ψ(x), Bertrand's postulate (Ramanunjan's and Erdős proofs), Perron's formula, analytic continuation of the Riemman zeta function, upper and lower bounds for the Riemann zeta function near σ=1, the Euler product, the analytic proof of the prime number theorem (following the ideas of the original independent proof by Hadamard and de la Vallée Poussin).
Geometry
Projective Geometry (Maria Papatriantafillou, Spring 2025) The associated affine plane, the projective plane, completion of an affine plane and decompletion of a projective plane, the duality principle, projectivities and perspectivities, projective plane morphisms, (central and axial) collineations, homologies and elations, algebraic study of the real projective plane (morphisms, homologies, elations), the projective plane as the completion of the usual real plane, Desargues projective planes (algebraic and geometric perspective), projectivities in Desargues planes, Pappian planes, projective planes and division rings, the division ring of a Desargues plane, construction of a projective plane from a divisiron ring and vice-versa, Pascal's and Brianchon's theorems.