2025–26–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

Graphs and sub-graphs, trees, connectivity, Euler tours, Hamilton cycles, matching, vertex and edge colorings, planar graphs, introduction to Ramsey theory, directed graphs, probabilistic methods and linear algebra tools in Graph Theory.

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.

An introduction to applications of algebra and number theory in the field of cryptography. In particular, the use of elliptic curves in cryptography is studied in great detail.

• Introduction to cryptography and in particular to public key systems, RSA, Diffie-Hellman, ElGamal.
• Finite filelds, construction of all finite fields, efficient arithmetic in finite fields.
• Elliptic curves, the group law of an elliptic curve, methods for counting the number of points of an elliptic curves over a finite field: Baby-step giant step, Schoof’s method.
• Construction of elliptic curves based cryptographic systems.
• Methods for prime decomposition, the elliptic curves method, the quadratic sieve method.
• Safety of public key cryptographic methods.

• 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

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.

 Graph rigidity theory studies the structural stability of frameworks, which are embeddings of graphs in d-dimensional space. These frameworks can be viewed as mechanical structures built from rigid rods (edges) connected by joints or hinges (the vertices), around which they are free to rotate. The basic question in rigidity theory is: Is a given structure rigid or flexible? That is, can the vertices be continuously moved in a way that preserves all edge lengths, other than by trivially translating or rotating of the entire framework? The theory of rigidity lies at the intersection of combinatorics, geometry, and algebra, and its history can be traced back to the foundational work of Maxwell and Cauchy in the 19th century. This theory has found applications in diverse fields, such as biology (the structure of proteins), control theory (formation control of autonomous vehicle systems), and structural engineering. In this course, we will explore various classical problems and results, as well as more recent developments in rigidity theory, providing an introduction to key concepts, techniques, and open questions in the field.

 The course aims to introduce students to the topic of convexity, which underlies a lot of mathematics and applications of mathematics. We will start in the finite-dimensional setting and cover the basics of convexity theory, such as separation, duality, and extreme points. After we have some of the basics, we will discuss examples in convex optimization. In particular, we will see what LMI domains are and what they are useful for. The last part of the course will extend what we have learned to the infinite-dimensional setting. In particular, we will prove Choquet’s theorem and see some of its applications. We will also see Edward?s separation theorem and what it entails. Lastly, I will discuss infinite-dimensional simplices and their application to dynamics.

 he course is intended to introduce the students to the field of applied mathematics. It is concerned with the construction, analysis, and interpretation of mathematical models that shed light on significant problems in the natural sciences. It is aimed to provide material of interest to students in mathematics, science, and engineering at the upper undergraduate and graduate level. Nature of applied mathematics. Deterministic systems and Ordinary Differential equations (ODEs). Poincare perturbation theory. Random processes and Partial Differential equations (PDEs). Heat Flow and Fourier Analysis. Advanced Fourier Analysis and Fourier Transform. Fundamental applied mathematics procedures illustrated on ODEs. Introduction to theories of continuous fields: motion of a bar, the continuous medium, field equations of continuum mechanics, inviscid fluid flow, elements of potential theory.

 The course is a continuation of the course: Modern Algebraic Geometry 1, that was given in the first semester. It is a course on algebraic varieties over an algebraically closed field, using the technique of sheaves. Algebraic schemes will be mentioned briefly. We shall cover most of the standard material, with some additional glances into more advanced or specialized topics. The content of the course will be adapted to the background and capabilities of the registered students. The topics listed below are for both semesters.

Topics:

1. Categories and functors (interspersed among other topics).
2. Topological spaces equipped with sheaves of rings of functions, including non-algebraic examples.
3. Recalling commutative algebra.
4. Affine algebraic varieties.
5. Projective algebraic varieties.
6. Separability and algebraic varieties.
7. Survey of algebraic schemes.
8. Sheaves of modules.
9. Vector bundles.
10. Line bundles and the Picard group.
11. Types of maps between varieties.
12. Enumerative geometry and the Bezout Theorem.
13. Sheaf cohomology.
14. Algebraic groups.