Geometry of the space of complex cubic polynomials in one variable. Discrimanant. The space of unordered triples of complex numbers. Relation to the trefoil.

The group of trefoil as a braid group. Artin braid groups: geometric braids, their equivalence, multiplication.
Braid groups as the fundamental groups and as the group of isotopy classes
of homeomorphisms of disk fixed on the boundary and preserving a finite set
of interior points.
The group of a knot. The groups of torus links.
Stratification of the space of unordered *n*-tuples of pairwise distinct
points and the standard presentation of the braid group.
The isotopy classes of homeomorphisms corresponding to the standard generators of the braid group.
Dehn twists.

Homomorphism of a braid group onto symmetric group. Pure braids. Their interpretations via configuration spaces of ordered collections of points and isotopy classes of homeomorphisms. Relation to the braid groups and symmetric groups. Centers of the braid groups. Coverings related to the sequences. Higher homotopy groups of the configuration spaces. Eilenberg-MacLane spaces. Homology of the braid groups.

Topological Classification of 1-manifolds.
Mapping class groups of the line and circle. The fundamental group
of *Homeo(S ^{1})*.

Topology as the only field in Mathematics which hesitates its
own finite objects.
How many points are needed for a non-trivality of the fundamental
group?

Digital circle and digital line. The fundamental group of the digital
circle.
Digital plane and digital Jordan Theorem.

Topological structure as the set of all continuous maps from the space to the connected pair of points. Axioms of topological structure in terms of the set of maps to the connected pair of points.

The space of simplices of a simplicial space. Recovering of the simplicial space out of its space of simplices. Weak homotopy equivalence between a simplicial space and its space of simplices. The space of cells of a polyhedron.

*T _{0}*-equivalence of points in a topological space.
Preordering of points.

Any finite topological space is weak homotopy equivalent to a compact polyhedron.

Baricentric subdivision of a triangulated space. Baricentric subdivision
of a poset. Baricentric subdivision turns any *T _{0}* finite
topological space to the space of simplices of a compact simplicial space.

** Toric varieties.**
Convex cones in Euclidean spaces. Semigroup ring of integer points in a
cone. Affine toric variety defined by a cone. Generators and relations
of the semigroup and the corresponding embedding into an Affine space.
Examples: Affine space, quadratic cone in 3-space, hyperbola (= punctured
line).
The toric variety defined by a convex polyhedron (gluing affine toric
varieties). Examples: the projective plane, affine plane with blown up
origin.

Algebraic torus *(k\{0}) ^{n}*. The action of the
torus in a toric variety. Examples. The orbit space.

Mapping of the complex projective plane onto the triangle with vertices
*(0,0), (1,0), (0,1)*. Preimages of points.
Moment maps of a toric variety to the corresponding convex polyhedron.

Klein's map of the set of imaginary points of the projective plane onto the 2-sphere.

Topology of complex projective lines and complex plane projective conic.

Projective duality. Dual plane curves. Duality between an inflection point and a cusp. Surface dual to a spatial curve.

The Newton polyhedron of a polynomial. The hypersurface of a toric variety defined by a polynomial. Singular points of a hypersurface.

Real toric surfaces. Topology of a real toric surface. Orientability of a real toric surface.