__Outline:__

A brief history and motivation;

Monotone and maximal monotone operators, and their elementary properties ;

Some examples of monotone and maximal monotone operators;

Main properties of monotone and maximal monotone operators [Debrunner–Flor lemma, local boundedness and Rockafellar theorems, the convexity of D(T) and R(T) (or convexity of their interiors or closures), Libor Vesel'y theorem];

Subdiﬀerentials of convex functions (the most basic class of maximal monotone operators) and related examples, Sum formula for subdifferential, Rockafellar’s maximal monotonicity theorem for subdiﬀerentials;

Cyclically monotone operators;

Fitzpatrick function (as an essential tool in modern monotone operator theory),
Sum theorem, and Minty surjectivity theorem
.

__Sets:__

Basic concepts and examples, Convex combinations and (closed) convex hulls, Best approximation properties (projection theorem), Separation theorems and applications, Extreme points and Krein-Milman theorem.

__Cones:__

Convex cones (properties and illustrations), Generalized interiors (core, relative interior, quasi-relative interior, and their relations), Polar, dual and bipolar cones, Tangent (Bouligand) and normal cones, Recession, asymptotic and barrier cones.

__Functions:__

Basic definitions and examples of convex functions; Lower semicontinuous (Closed ) functions, Closed convex functions, Convex hull and closed convex hull of a function; Aﬃne minorant of a closed convex function; Local and global behavior of a convex function, Support functions, Infimal convolution, Coercive function; Fréchet and Gâteaux derivative, Subdiﬀerentinal of Convex Functions, Convex conjugate of a function, Conjugacy, coercivity and convexity.

Our primary reference is the following book:

**Walter Rudin**, *Real and Complex Analysis*, (3rd ed., 1987, 416 pp.).

We consider the first 8 chapters of the book for two seasons (6 months). Indeed, we teach the first 4 chapters of the book in the first season (3 months) and Chapters 5-8 for the second season.

Our main references books are as follows (depending on the conditions, we choose one of them for teaching):

1) __ Haim Brezis:__ ,

We consider the first 7 chapters for two seasons (6 months). Actually, 4 chapters for the first season (3 months) and Chapters 5-7 for the second season.

2) __ John B. Conway:__

*A Course in Functional Analysis* (2007).

We consider the first 8 chapters for two seasons (6 months). Practically, 4 chapters for the first season (3 months) and Chapters 5-8 for the second season.

Our primary reference is the following book:

Mokhtar S. **Bazaraa**, Hanif D. **Sherali**, C. M. **Shetty**:

*Nonlinear Programming: Theory and Algorithms* (3rd ed, 2006).

The main topics we focus on it in this course are:

Convex analysis with a debate of properties of convex sets, separation and support of convex sets, polyhedral sets, extreme points and extreme directions of polyhedral sets, and linear programming, Optimality conditions, and duality with coverage of the nature, interpretation, and value of the classical Fritz John (FJ) and the Karush-Kuhn-Tucker (KKT) optimality conditions; the interrelationships between various proposed constraint qualifications; and Lagrangian duality and saddle point optimality conditions.

Our main reference book is:

**James Munkres**: *Topology* (2nd ed., 2000).

The main topics we focus on it in this course are:

Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness.