The partition of the 3-sphere to fibers of this map was considered.
We cosiderd also a similar Hopf fibration *S ^{3}->S^{2}=CP^{1}:
(z,w)|->(z:w).*

Then we discussed geometry of the space of complex cubic polynomials in
one variable. After factoring out a punctured plane (by dividing by the
coefficient at the cubic term) and a plane (by translating of the plane
of roots killing the term of degree 2), we identified the space of
polynomials *X ^{3}+pX+q* with

Then the space of unordered triples of
points in the plane *C* considered up to translations and dilations was identified with
the 3-sphere plus a point. (The point corresponds to the triple in which
all 3 points coincide.) The triples in which 2 of 3 points coincide and the
third one is different correspond under this identification to a trefoil.

The fundamental group of unordered triples of pairwise distinct points
is the 3 string braid group *B _{3}*.
Therefore the group of 3 string braids is identified with the group of a
trefoil.

**Lecture 2, on Wednesday January 28**, started with revisiting
and generalizations of theorems presented in the first lecture.
Trefoil was partly replaced by torus knots and links.

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.

**Lecture 3, Monday February 2**.

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.
Short exact sequence

**Lecture 4, on Wednesday February 4**,
was devoted to the **little ones of Topology**.

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.

**Lecture 5, Monday February 9.**

**Lost Chater of General Topology.**

Topological structure can be interpreted as the set of all continuous
maps from the space to the connected pair of points.

Axioms of topological structure then mean that the set of maps is closed with respect to operations of taking supremum, minimum and contains constant maps.

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.

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

**Lecture 6, Wednesday February 11.**

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.

**Lecture 7, Monday February 16.**

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.

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

**Lecture 8, Wednesday February 18.**

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.

Moment maps of a toric variety to the corresponding convex polyhedron.

** Lecture 9, Monday February 23.**

The Newton polyhedron of a polynomial. The hypersurface of a toric variety defined by a polynomial. Singular points of a hypersurface. Is a generic hypersurface singular? If one considers all polynomials of fixed degree, then yes. If the Newton polyhedron is fixed, then in affine or projective space the answer may be no. In what sense a generic toric variety is non-singular in the toric variety defined by its Newton polyhedron.

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