Modern Mathematics

Abstract

The introduction The aim of this course is to teach the probabilistic techniques and concepts from the theory of stochastic processes required to understand the widely used financial models. In particular concepts such as martingales, stochastic integration/calculus, which are essential in computing the prices of derivative contracts, will be discussed. The specific topics include: Brownian motion (Gaussian distributions and processes, equivalent definitions of Brownian motion, invariance principle and Monte Carlo, scaling and time inversion, properties of paths, Markov property and reflection principle, applications to pricing, hedging and risk management, Brownian martingales), martingales in continuous time, stochastic integration (including It\^{o}'s formula), stochastic differential equations (including Feynman-Kac interested in learning from data. Students with other backgrounds such as life sciences are also welcome, provided they have maturity in mathematics. The mathematical content in this course will be linear algebra, multilinear algebra, dynamical systems, and information theory. This content is required to understand some common algorithms in data science. I will start with a very basic introduction to data representation as vectors, matrices, and tensors. Then I will teach geometric methods for dimension reduction, also known as manifold learning (e.g. diffusion maps, t-distributed stochastic neighbor embedding (t-SNE), etc.), and topological data reduction (introduction to computational homology groups, etc.). I will bring an application-based approach to spectral graph theory, addressing the combinatorial meaning of eigenvalues and eigenvectors of their associated graph matrices and extensions to hypergraphs via tensors. I will also provide an introduction to the application of dynamical systems theory to data including dynamic mode decomposition and the Koopman operator. Real data examples will be given where possible and I will work with you write code implementing these algorithms to solve these problems. The methods discussed in this class are shown primarily for biological data, but are useful in handling data across many fields. A course features several guest lectures from industry and government. This course introduces the measure theory and other topics of real analysis for advanced math master’s students, and AIM and non-math graduate students. The main focus will be on Lebesgue measure theory and integration theory. Tentative topics included: Lebesgue measure, measurable functions, Lebesgue integral, convergence theorems, metric spaces, topological spaces, Hilbert and Banach spaces. This course has some overlaps with MATH 597, but covers about 1/2 of the content and proceeds at a slower pace. bifurcations transcritical, subcritical, supercritical, Hopf), unstable dissipative attractors, logistic period-doubling, renormalization, Lyapunov fractals, Hausdorff Lorenz nonlinear oscillations, quasiperiodicity, Hamiltonian systems, integrability, resonance, KAM homoclinic intersections, Melnikov's Physiological systems are typically modeled by differential equations representing the state of their components, for example , Hodgkin and Huxley’s mathematical description of the electrical activity of neurons. Black box methods using machine learning have recently had remarkable success in predicting physiological state in some settings, for example, in scoring sleep from wearables. This course will explore the differences between these two approaches and new techniques using both mechanistic differential equation models and machine learning. Topics include backpropagation methods for learning in artificial neuronal networks and biophysical neuronal networks, methods for filtering physiological signals (e.g., autoencoders) to serve as inputs to physiological models, machine learning methods to solve differential equations and learn dynamics, and new methods to study noise in biological systems. Final projects and teamwork will allow students to apply these techniques to their own choice of systems to study. Reed-Muller, and Reed-Solomon codes. We will further discuss linear codes and cyclic codes and give fundamental asymptotic bounds on coding efficiency. This is about solving linear systems numerically, finding eigenvalues and singular values, and solving linear least squares problems. We will discuss condition numbers, numerical stability, QR factorization, Cholesky, SVD, and the QR algorithm as well as iterative methods (GMRES, Arnoldi, Conjugate Gradients, Lanczos). The following applications are included: KKT conditions, convergence of the perceptron, and back propagation networks. The homework assignments will use either Python or Matlab, with the choice left to the student. resulting sparse linear systems conjugate gradients, preconditioning). Multistep, Runge-Kutta methods for initial value problems. Absolute stability, stiff problems, and A-stability. Barrier theorems. Explicit and implicit finite difference schemes for parabolic equations. Stability and convergence analysis via the maximum principle, energy methods, and the Fourier transform. Operator splitting techniques, the alternating direction implicit method. Advection equation. Lax-Wendroff, upwind methods, the CFL condition. Hyperbolic systems, initial boundary value problems. an introduction to topology. We topological spaces, continuous functions and homeomorphisms, the separation axioms, the quotient and product topology, compactness, connectedness, and metric spaces. We some topics in algebraic topology, time permitting. This is a topics course in commutative algebra which assumes as background an introductory course. Topics will include the structure theory of complete local rings, an introduction to homological methods, including a treatment of the functors Tor and Ext, Koszul homology, Grothendieck groups, regular sequences and depth, Cohen-Macaulay rings and modules, and the theory of flat homomorphisms, with applications to the method of proof by reduction to positive characteristic. Many open questions will be discussed. lecture notes will be provided. Monte Carlo simulations – the most general computational method for probabilistic equations. This method is based on generating a large number of paths of the underlying stochastic processes, in order to approximate the expectations of certain functions of these paths (which, e.g., may determine prices, portfolio weights, default probabilities, etc.). In addition to the standard Monte Carlo algorithms, we will study the variance reduction techniques, which are often necessary to obtain accurate results. The computational methods presented in this course will be illustrated using the popular models of equity markets (e.g. Black-Scholes, Heston), fixed income (e.g. Vasicek, CIR, Hull-White, Heath-Jarrow-Morton) and credit risk (e.g. Merton, Black-Cox, reduced-form models). In particular, we will cover certain deep learning algorithms for option pricing and introduce FBSDEs. and related topics. we Goals: Dynamical systems is the study of the long-time behavior of diffeomorphims and flows. We will discuss ergodicity and mixing, and natural invariants such as entropy. This theory is particularly successful and interesting in the presence of a lot of hyperbolicity, i.e., when the flow or diffeomorphism expands and contracts tangent vectors exponentially fast. For these systems, we will then develop some of the crucial tools in Pesin theory such as unstable manifolds and Lyapunov exponents. We will illustrate the general theory by particular examples from geometry and dynamical systems on homogeneous spaces. I will apply these ideas to study some more general group actions, e.g., actions of higher rank abelian and semi simple groups. As was discovered in the last decade, these actions show remark- able rigidity properties, and sometimes can even be classified. These results have had important applications in other areas, e.g., geometry and number theory. While the material in the first part of the course is fundamental for many investigations in dynamics, geometry, several complex variables. the second half of the semester will bring us right to the forefront of current research. Books: No text is the following are recommended Description: By now, deep learning has become a tool that permeates every scientific field capacity, although recent advances in theory for deep learning is classroom. In this class, we will study deep learning through the lenses of mathematics and acquire practical know-how through hands-on tutorials. From the theoretical side, we will study recent approximation theorems and convergence results pertaining to deep neural networks, using tools from numerical analysis and probability theory. Other topics include: sampling strategies, optimisation, interpretability, convolutional neural networks, generative adversarial networks, encoder-decoders, reinforcement learning, physics informed neural networks. From the practical side, you will learn a well-known framework (pytorch) with focus on topics of reproducibility, stability, reliability and convergence, emphasizing good development practices. By the end of the course, you will have had the chance to develop an end-to-end deep learning code. A strong mathematical background in calculus and probability is expected. Experience in python is required. Previous experience in machine learning is not required but might be helpful. presentation.