Mathematics

This advanced topics course covers reinforcement learning, search, and test-time scaling of large language models that are expected to drive the next generation of AI systems. 

Topics include: Basics of RL (Markov Decision Process and Policy evaluation), Basics RL (Imitation learning, Deep policy gradient methods), Basics of RL (Deep Q-Learning, Rainbow DQN); Symmetric alternating Markov games, Monte Carlo tree search, expert iteration, and AlphaGo; Imperfect information games, Counerfactural regret minimization, and Pluribus; NLP basics (RNN, beam search, tokenizers); NLP basics (Transformers, encoder-decoder architectures); Instruction fine-tuning, Scaling laws of LLM pre-training; Reinforcement learning with human feedback, direct policy optimization, Group Relative Policy Optimization (GRPO); Chain of thought, Process reward models, Prover-verifier games; In-context learning, Scaling LLM Test-Time Compute; DeepSeek-R1. 

This course discusses differential geometry of curves and surfaces in Euclidian Space: curves in 2- and 3-dimensional spaces, local and global theory of surfaces, special classes of surfaces, discrete curves and surfaces.

This course explores the classical theory of games involving concepts of dominance, best response, and equilibria, where it proves Nash’s Theorem on the existence of equilibria in games. Students learn the concept of when a game is termed zero-sum and prove the related Von Neumann’s Minimax Theorem. The course explores cooperation in games and investigates the interesting Nash bargaining solution which arises from reasonable bargaining axioms. Students also explore the concept of a congestion game, often applied to situations involving traffic flow, where they see the counterintuitive Braess paradox emerge and prove Nash’s theorem in another context. 

This course looks at definition of a curve, arc length, curvature, and torsion of a curve, Frenet-Serret equations. It also looks at definition of a surface patch, first and second fundamental forms, isometries, conformal maps, area, Gaussian curvature, mean curvature, principal curvatures, Gauss map, geodesics, and Theorema Egregium.

Quantitative finance remains one of the fastest growing areas in modern finance. Alternative names are financial engineering, mathematical finance, or financial mathematics. This is an application-based course on the mathematical and computational aspects of derivative pricing. It lies at the heart of mathematics, computing, finance, and economics. Both theory and numerical techniques are presented, with computer simulations performed on MS Excel. If you are interested in technical finance and have wondered what Brownian Motion is, or how Monte Carlo methods are used to price options; then this module is precisely what you are looking for – covering Itô Calculus, Black-Scholes world and Monte Carlo simulations. This is not a theorem-proof based course, but all results are derived.

This course provide students with basic training on modern financial mathematics methods, which covers an overview of data analysis, principles of actuarial modelling and financial transactions, the understanding of real and nominal interest rates, the time value of money methods, bond pricing methods, assets replication methods, the equation of value methods, and project appraisals methods. This course focuses on applying the above methods to the mathematical modelling of financial markets.

This course examines the theory of systems of ordinary differential equations. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behavior. The applications are from biology, physics, chemistry, and engineering, including population dynamics, epidemics, chemical reactions, and simple mechanical systems.

The course introduces rings, subrings, homomorphisms, ideals, quotients, and isomorphism theorems. It includes integral domains, unique factorization domains, principal ideal domains, Euclidean domains, Gauss' lemma and Eisenstein's criterion. Fields, field of quotients, field extensions, the tower law, ruler and compass constructions, construction of finite fields.  Students state the definitions of concepts and prove their main properties, describe fields and rings and perform computations in them. Students discuss the theoretical results covered in the course and outline their proofs. They perform and apply the Euclidean algorithm in a Euclidean domain, giving examples of sets for which some of the defining properties of fields. They focus on proving the tower law, and use it to prove the impossibility of some classical ruler and compass geometric constructions. Students learn to identify concepts as particular cases of fields, rings, and modules (e.g. functions on the real line as a ring, abelian groups, and vector space).

This course covers number theory. Topics include integers on a ring: completely closed rings, quadratic bodies, norm, trace, discriminant in the case of extensions of bodies. Example of cyclotomic bodies of degree p-1; Dedekind rings: Noetherian property; application to integer elements, fractional ideals, fraction rings, localization, group of fractional ideals, norm of an ideal, multiplicativity; decomposition of ideals in an extension: prime ideal, discriminant and ramification, quadratic and cyclotomic bodies of degree p-1, quadratic reciprocity law; class group and unit theorem: networks, canonical folding, statement and proof of the finiteness of the class group, statement of the unit theorem, illustration in the case of quadratic bodies, Fermat cases (or other Diophantine equations); analytical opening (Riemann zeta function, Dirichlet L-functions, Dedekind zeta functions, link to counting prime numbers and ideals).
 

This course introduces students to more advanced topics in Probability Theory and Statistical Inference. The first part is devoted to investigating mathematical aspects of probability, with a special emphasis on multivariate distributions and limiting theorems. In the second part, students are guided through the methodological core of point estimation (both from a frequentist and Bayesian perspective) and hypothesis testing. These theoretical aspects are complemented by an in-depth presentation of elementary simulation and computational techniques that are routinely used within most popular statistical procedures. Prerequisites: Solid knowledge of calculus and of basic programming tools in R.