2024–25–B

Ordinary differential equations of first order, existence and uniqueness theorems, linear equations of order n and the Wronskian, vector fields and autonomous equations, systems of linear differential equations, nonlinear systems of differential equations and stability near equilibrium

Topological spaces and continuous functions (product topology, quotient topology, metric topology). Connectedness and Compactness. Countabilty Axioms and Separation Axioms (the Urysohn lemma, the Urysohn metrization theorem, Partition of unity). The Tychonoff theorem and the Stone-Cech compactification. Metrization theorems and paracompactness.

• Complex numbers. Analytic functions, Cauchy–Riemann equations.
• Conformal mappings, Mobius transformations.
• Integration. Cauchy Theorem. Cauchy integral formula. Zeroes, poles, Taylor series, Laurent series. Residue calculus.
• The theorems of Weierstrass and of Mittag-Leffler. Entire functions. Normal families.
• Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.

• Fields: basic properties and examples, the characteristic, prime fields
• Polynomials: irreducibility, the Eisenstein criterion, Gauss’s lemma
• Extensions of fields: the tower property, algebraic and transcendental extensions, adjoining an element to a field
• Ruler and compass constructions
• Algebraic closures: existence and uniqueness
• Splitting fields
• Galois extensions: automorphisms, normality, separability, fixed fields, Galois groups, the fundamental theorem of Galois theory.
• Cyclic extensions
• Solving polynomial equations by radicals: the Galois group of a polynomial, the discriminant, the Cardano-Tartaglia method, solvable groups, Galois theorem
• Roots of unity: cyclotomic fields, the cyclotomic polynomials and their irreducibility
• Finite fields: existence and uniqueness, Galois groups over finite fields, primitive elements

Embedded differentiable manifolds with boundary in Euclidean space. The tangent space, normal, vector fields. Orientable manifolds, the outer normal orientation. Smooth partitions of unity. Differential forms on embedded manifolds, the exterior derivative. Integration of differential forms and the generalized Stokes theorem. Classical formulations (gradient, curl and divergence and the theorems of Green, Stokes and Gauss). Closed and exact forms. Conservative vector fields and existence of potentials. Application to exact ordinary differential equations. Introduction to differential geometry: curvature of curves and surfaces in 3 dimensional space, the Gauss map, the Gauss-Bonnet theorem (time permitting).

Ordinary differential equations of first order, existence and uniqueness theorems, linear equations of order $n$ and the Wronskian, vector fields and autonomous equations, systems of linear differential equations, nonlinear systems of differential equations and stability near equilibrium

1. Preliminaries: floating point arithmetic, round-off errors and stability. Matrix norms and the condition number of a matrix.
2. Introduction to numerical solutions for ODE’s:initial value problems, Euler’s method, introduction to multistep methods. Boundary value problems.
3. Numerical solution of linear equations: Gauss elimination with pivoting, LU decomposition. Iterative techniques: Jacobi, Gauss-Seidel, conjugate gradient. Least squares approximation.
4. Numerical methods for finding eigenvalues: Gershgorin circles. The power method. Stability considerations in Gram-Schmidt: Hausholder reflections and Givens rotations. Hessenberg and tridiagonal forms. QR decomposition and the QR algorithm.

The purpose of the course is to provide students with the ability to deal with mathematical problems in a variety of subjects by becoming familiar with common strategies for solving mathematical problems. The course requires active participation of the students during class and includes both group and individual work. The meetings will be conducted as a seminar where initially a classical problem and its solution will be presented. The strategy for solving problems arising from the solution will be discussed and then the participants will be challenged to use this strategy with specific examples. In addition, problems/riddles given as weekly homework will be discussed. We will cover a variety of techniques for solving problems: exploiting parity, pigeonhole principle, checking extreme cases, double counting, the method of geometric transformations in dealing with sophisticated geometric problems, methods of Dynamic programming, the principle of induction and Fermat’s descent method for treating Diophantine equations. The method of generating functions.. Probabilistic considerations and their uses.

Deep Learning, often imprecisely called “Artificial Intelligence”, has become a hugely successful discipline recently. At its core are mathematical tools in the fields of Linear algebra, Optimization, Probability and Statistics. The main objective of this class is to prepare students for advanced courses in Deep learning by introducing them to the mathematical tools on which Deep Learning is based. We will also consider simple examples of Deep Learning to motivate the subject as well as to see how the acquired techniques are being used. We will also use computer demonstrations using the python based SAGE computer algebra system, serving as an introduction to the use of python in advanced Deep Learning classes. The course will focus on Linear Algebra and Optimization. It is thereore recommended to supplement it with a Probability and/or Statistics class. Textbook: trang - Linear algebra and learning from data Prerequisites: Two linear algebra and one calculus class in the Mathematics, Computer Science or Electrical Engineering departments. Students in other departments who would like to take the class will be accepted on a case by case basis.

In the 1980s A. Grothendieck suggest a project for developing a tame topology that will not suffer from the many counter-examples and pathologies known in classical topology. Nowadays many view the notion of o-minimality as successful fulfilment of this program: in o-minimal fields all (unary) functions are piecewise differentiable (and therefore infinitely differentiable at almost every point); unary functions are piecewise monotone, connectedness is the same as path connectedness and the axiom of choice holds for definable sets. In the o-minimal setting most of the classical differential calculus can be developed, and so are large portion of the theory of Lie groups, algebraic topology and much more. O-minimality plays a key role in real geometry and in recent years had a crucial role in important breakthroughs in Diophantine geometry and in Hodge theory. In the course we will define o-minimality and develop its basic theory. We will show that real closed fields are o-minimal and discuss – time permitting – some applications.

• Short review of prerequisites from algebraic geometry
• Definition, examples and first properties of linear algebraic groups
• Derivations, differentials and Lie algebras
• Parabolic subgroups, Borel subgroups and solvable groups
• Weyl groups, roots and root datum
• Reductive groups and their structure
• Advanced topics

Symbolic dynamics is a branch of mathematics that deals with sequences of characters letters or “symbols” form the point of view of dynamical systems. The basic guiding philosophy is that sometimes it is possible to code and understand complicated systems by a sequence of discrete samples. The decimal expansion of real numbers is a simple example of this kind of procedure. Techniques and ideas from symbolic dynamics have found significant applications in data storage and transmission as well as other parts of mathematics. In this course we will introduce basic notions and results in symbolic dynamics, via interesting examples. We will illustrate relations to other fields and relate to the more general frameworks of topological dynamics and ergodic theory.

“Branching processes” are a family of discrete valued random processes, characterized by a very strong correlation between the state of the process and the transition probabilities.

The original model was derived by Galtoe & Watson during the late 19th century, in order investigate the dynamics of the aristocrat families in Victorian Britain, but soon found numerous applications in many scientific disciplines, such as computational biology, nuclear engineering, epidemiology and economics. In the course, we will introduce the basic concepts in branching processes, underlining the mathematical formalism, and discuss two applications: epidemiology (and the COVID pandemic in particular), and fission chain models in nuclear engineering.