# Education

The geometry department provides geometric education for all specialties: mechanics, mathematics, applied mathematics, computer science.

The first year students are invited for of
*analytical geometry* course (semesters 1 and 2), which includes the following topics: vector algebra, vector equation of the trajectory of a moving point, equations of lines and planes, the mutual positions of lines and planes, curves of order 2 and their canonical equation, surfaces of the 2nd order and their canonical equations, the invariants of the equation of the curve and the surface of the 2nd order with respect to orthogonal transformations of coordinates, rectilinear elements on the surface of the 2nd order.

For the second-year students we offer the differential geometry course (semesters 3 and 4), in frame of which we study the properties of curves on a plane and in the space, give the mechanical interpretation of geometric characteristics of a curve, apply the Serret-Frenet formulas to describe the motion of a rigid body and to find the vector of instantaneous rotation (the Darboux vector), use the geometric methods to derive the equations of motion of a material point. In the study of the geometry of surfaces, the students are introduced to various types of curvilinear coordinates, geometric analysis of motion of a point on a curved surface, the basics of tensor analysis.

We offer the course of
**Analytical geometry** (semester 1 and 2), the main stream of which is the systematic usage of method of coordinates in application to the Euclidean and affine geometry, the methods of vector and linear algebra in solution of geometric problems. In particular, we study the problem of the mutual disposition of points, lines and planes, generalize the concepts of line and plane for the case of multidimensional space, describe and study analytically the convex sets, analytically describethe motions (isometries) of the plane and space (the Chasles' theorems). The second semester is devoted to analytical study of curves and surfaces of the 2nd order, their canonical equations and the reduction to the canonical form methods. We introduce the concept of invariant and use of invariants in solving the problem of classification of Euclidean, affine and projective quadrics.

The course of
**Topology** (3rd semester) at the first stage concentrate on general topology concepts, such as: topological and metric space, countable base space, separation axioms, sequence and its limit in general topological space, continuous mapping of topological spaces, homeomorphism and the topological invariant. We define the operations over the topological spaces, such as sum, gluing, product of the topological spaces. We introduce the notion of a subspace and a quotient space of the topological space. We consider the fundamental properties of topological spaces, such as compactness, connectedness and linear connectedness, the corresponding topological invariants. At the second stage we introduce the concept manifold, the connected sum of manifolds, the Euler characteristic and the orientation of two-dimensional manifold, the topological classification of compact two-dimensional closed manifolds. We study some concepts of the algebraic topology: homotopy of paths and the fundamental group, singular and cellular homology of two-dimensional manifolds. We consider the embedding and immersion theorems for compact manifolds.

In the course of
**Differential geometry** (4th and 5th semesters) we systematically study the geometry of embedded curves and surfaces in the 3-dimensional Euclidean space (the classical differential geometry). In the "Theory of curves" we define the curvature and torsion of a curve as a complete set of invariants that uniquely define a curve up to the isometry, discuss some aspects of the geometry of curves in the large and analytically analyze some problems of the theory of envelopes. In the "Theory of surfaces," we focus on invariants a surface parameterization as the ones that belong to the geometry of a surface and do not dependent on its representation. The course includes the Gauss Egregium theorem, the Gauss and Codazzi equations, the Bonnet theorem, the Gauss-Bonnet formula, the Gauss integral formula.

In the development of the theory of plane curves, we discuss the theory of curves on curved surfaces. We consider some special types of curves on surfaces: the lines of curvature (the principal lines), asymptotic and geodesic lines. We use the elements of calculus of variations to study the extremal properties of the geodesic lines and the minimal surfaces.

The course includes the elements Riemannian geometry, such as the covariant differential and the covariant derivative of a vector field, the Levi-Civita parallelism, the Laplacian and the gradient of a smooth function on a manifold, the basics of tensor analysis including the curvature tensor of the Riemannian connection.

The disciplines of
** further specialization** in geometry (6-10 semesters) include:

At the
*Bachelor’s * level -- classical problems of the geometry in the large, Riemannian geometry, the geometry of submanifolds, differential geometry with MAPLE, algebraic topology, geometry of Lie groups, the basics of the affine differential geometry;

At the
*Master’s* level -- the homogeneous and symmetric spaces, the mean curvature flows, the vector bundles, the Ricci flows, the global Riemannian geometry, the geometric foundations of physics.

The students study the
**Algebra and Geometry**** course** (semester 1 and 2). The Department provides the geometric part of the course. In frame of this part we teach algebra and basic theory of linear spaces, study the equations of lines and planes in the 3-dimensional and multi-dimensional Euclidean space. As applications of the method of vector algebra, we discusses the analytical description of convex sets, the Hahn-Banach theorem on the separability of convex sets. As application of the method of coordinates, we consider the Chasles' theorem on the classification of isometries of the plane and space. The linear algebra methods are used for the classification of Euclidean quadrics and study their general properties.

The course of
**Topology** (3rd semester) at the first stage concentrate on general topology concepts, such as: topological and metric space, countable base space, separation axioms, sequence and its limit in general topological space, continuous mapping of topological spaces, homeomorphism and the topological invariant. We define the operations over the topological spaces, such as sum, gluing, product of the topological spaces. We introduce the notion of a subspace and a quotient space of the topological space. We consider the fundamental properties of topological spaces, such as compactness, connectedness and linear connectedness, the corresponding topological invariants. At the second stage we introduce the concept manifold, the connected sum of manifolds, the Euler characteristic and the orientation of two-dimensional manifold, the topological classification of compact two-dimensional closed manifolds.

In the course of
**Differential geometry** (4th and 5th semesters) we study fundamental and applied problems of differential geometry of curves and surfaces in the 3-dimensional Euclidean space. In the "Theory of curves" we define the curvature and torsion of the curve, discuss the problem of description of the motion of a material point, the envelopes of wave fronts and caustics of plane curves. In the "Theory of surfaces," in addition to the fundamental questions of the geometry of surfaces, such as the Gaussian and mean curvature, the Bonnet theorem, the Gauss-Bonnet formula, the integral formula of Gauss we consider the geometric properties of certain classes of surfaces which have practical usage: ruled, tubular, conical and cylindrical surfaces, transfer surface, surface of revolution, minimal surfaces, surfaces of constant curvature, equidistant surfaces. The course also contains the same elements of tensor analysis.

The disciplines of
** further specialization** in geometry (6-10 semesters) include:

fundamentals of mathematical calculations in the "Matehematica" system, raster and vector graphics, Riemannian geometry with MAPLE, computer tomography, computer algorithms for geometry, analytical methods of geometric modeling, geometry of Lie groups (qualification level "

*bachelor*");

Creation of packages and add-ons to expand the engineering of computer graphics systems, basics of mathematical modeling and computational experiment, computer graphics and CAD, the mean curvature flow and image processing, Ricci flows, global Riemannian geometry, geometric image processing techniques (qualification level

*"master*")

The department provides the course "
**Algebra and Geometry**" (1st and 2nd semester). The course includes: the basic algebraic structures - fields, rings, groups; the fundamentals of linear algebra - matrices, determinants, systems of linear equations; the elements of analytic geometry - lines, planes, curves and surfaces of the 2nd-order, the invariants; complex numbers and polynomials; linear spaces, bilinear functionals and forms; Euclidean and unitary linear space; linear operators in affine, Euclidean and unitary spaces; the affine and projective transformations.

*Further** geometrical education* the students can get symbolic computation, basics of differential geometry, mathematical foundations of computer tomography, mathematical methods of image processing.