Spring Semester 2012, Place: GC 273 A, Time: 10:00 a.m.  4:00 p.m.
Fall Semester 2011, Place: SIPA 125 (actually, DM 413A), Time: 10:00 a.m.  2:00 p.m.
 Structure of the Seminar and Topics to be covered
 The Seminar will consist of four parts covering topics from the four Independent Study classes
I am teachig this semester: Finite Dimensional Algebras, Homological Algebra, Algebraic Number Theory,
and Master's Research respectively. The time schedule is 10:00  11:20, 11:30  12:50, 13:00  14:20,
and 14:30  15:50 for the parts listed above respectively.
Finite Dimensional Algebra: we continue studying these algebras by the book of Drozd and Kirichenko with
the same title, Chapters 36.
Homological Algebra: we are studying the basics of the theory by the book of Aluffi "Algebra: chapter 0',
Chapters VIII and IX, and then, for the theory of cohomology of Lie algebras, by the book of Weibel
"Homological Algebra". We will be discussing topological applications of the theory as well.
Algebraic Number Theory: This is an introduction to the area by the classical textbook of Borevich and
Shafarevich "Number Theory", Chapters 1, 2, 3.
Master's Research: This is a continuation of the fall semester's class with the same title. The main topic
will be HoriVafa's method for constructing mirror partners for Homological Mirror Symmetry.
 Meeting on March 2nd
 We are finishing with the group cohomology. The link to cohomology of Lie algebras will be given too.
In the Number Theory session, we are discussing exercises from BorevichShafarevich: Section 1, 812, Section 2, 913,
Section 3, 812.
 Meeting on February 24th
 We continue, in the first two sessions, with the group cohomology. Number Theory and Master's research are
following their proper schedules.
 Meeting on February 17th
 We follow the format of the previous meeting. The first two sessions will be devoted to applications of
group cohomology to theory of groups (explaining the role of the first 34 cohomology groups of a group G).
There will be a hint toward cohomology of Lie algebras as well. The last two sessions are devoted to Algebraic
Number Theory and Master's research.
 Meeting on February 10th
 This Friday, we combine the sessions on Finite Dim. Algebras and Homological Algebra, 10:00 a.m.  12:50 p.m.,
to discuss group cohomology and applications. The Algebraic Number Theory continues by Chapter 1 of BorevichShafarevich
and the Master's Research session is on embedded resolution of singularities.
 Meeting on February 3rd
 As regularly scheduled.
 Meeting on January 27th
 No meeting this time.
 Meeting on January 20th
 Organizational meeting, general discussions.
 Structure of the Seminar and topics to be covered
 This Semester's Seminar will be centered around two basic areas in Algebra and Geometry:
Algebraic Geomerty of toric varieties with a view toward String Theory and Mirror Symmetry of
varieties of general type and noncommutative algebra (finite dimensional algebras) with a view
toward their applications in geometry of compact complex homogeneous spaces and group representations.
The mentioned areas are subjects of study in the Master's Research and Finite Dimensional Algebras
Independent styudy classes respectively.
 Meeting on October 28th
 FinDimAlg: Chapter 3, Sect 2, 3; Mirror Symmetry: Intro to toric varieties.
 Meeting on October 21st
 FinDimAlg: Chapter 3, Section 2,3; Mirror Symmetry: differential forms and applications.
 Meeting on October 14th
 FinDimAlg: Chapter 3; Mirror Symmetry: connections on vector bundles and the related machinery.
 Meeting on October 7th
 Finite Dimensional Algebras (10:00 a.m. 12:00 Noon): finishing the exercises from Chapter 2 and
moving onto Section 1 of Chapter 3
Mirror Symmetry (12:15 p.m.  2:00 p.m.): Metrics, connections, curvature.
 Meeting on September 30th
 In the algebraic part of the seminar, 10:00 a.m. 11:45 a.m., we are finishing with the problems to
Chapter2 (a couple of important problems are left to be done, and then move to Chapter 3 where the theory
of non semisimple algebras begins.
In the geometric part (Mirror Symmetry), 12:00 Noon  2:00 p.m., we are discussing vector bundles on
real and complex manifolds and geometric structures on them following Chapter 1 of that book.
 Meeting on September 23rd
 In the algebraic part of the seminar, 10:00 a.m. 11:45 a.m., we are finishing with Chapter 2
and exercises thereof, and begin the theory from Chapter 3, sections 3.1 and 3.2 (DrozdKirichenko).
In the geometric part (Mirror Symmetry), 12:00 Noon  2:00 p.m., we are discussing vector bundles on
real and complex manifolds following Chapter 1 of that book.
 Meeting on September 16th
 We are covering Chapter 2 from DrozdKirichenko and are discussing examples and the exercises
in the end of the chapter. From 12:00 till 2:00p.m., the Algebraic Geometry part of the seminar,
we are discussing sections from the Hori et al book "Mirror Symmetry".
 Meeting on September 9th
 The meeting is mostly organizational. Questions on the book DrozdKirichenko "Finite Dimensional
Algebras" as well as on algebraic geometry will be discussed if need be.
 Structure of the Seminar and topics to be covered
 The work of the Seminar this semester is a continuation of what we did in Fall'10. It
is organized around the Independent Study classes on Algebraic Geometry. The format is: reading
seminar, and the participants will be preparing and presenting topics from the books we are studying
(J. Rotman "Advanced Modern Algebra", K. Hulek "Introduction to Algebraic Geometry", AtiyahMacDonald
"Introduction to Commutative Algebra", W. Fulton "Introduction to Toric Varieties", etc.). We continue
our study of algebraic varieties with a view toward toric varieties, their use in Mathematics, and
applications to Physics. The students enrolled in the Independent study classes are expected to actively
participate in the work of the Seminar, and present topics to their peers. The overall grade
in the courses will depend on that and on the independent work of the students on projects
they will be assigned during the seminar.
 Meeting on February 18th
 We continue with projective varieties, graded rings, and the projective Hilbert's Nullstellensatz.
 Meeting on February 11th
 We continue with the projective spaces and varieties.
 Meeting on February 4th
 We begin with Sections 2.1 and 2.2 of Hulek.
 Structure of the Seminar and topics to be covered
 This Semester, the Seminar will be organized around the Independent Study classes on
Algebraic Geometry. It will have the format of a reading seminar: the participants will be
preparing and presenting topics from the books we are studying (J. Rotman "Advanced Modern
Algebra", K. Hulek "Introduction to Algebraic Geometry", AtiyahMacDonald "Introduction to
Commutative Algebra", W. Fulton "Introduction to Toric Varieties", etc.). Our goal is to
learn as much as we can about toric varieties, their use in Mathematics, and applications
to Physics. The students enrolled in the Independent study classes are expected to activly
participate in the work of the Seminar, and present topics to their peers. The overall grade
in the courses will depend on that and on the independent work of the students on projects
they will be assigned during the seminar.
 Meeting on November 19th
 There will be two different sessions this time. The first is on regular and rational functions,
and the related Commutative LAgebra constructions. This will happen between 2:15 p.m. and 4:00 p.m.
The second session starts at 5:00 p.m. and is devoted to preparation for the Putnam Competition on
December 4th.
 Meeting on November 12th
 We are discussing the important construction of product of affine varieties, and then move
on to introducing rational functions on a variety. The important notion of localization will
be discussed in detail.
 Meeting on November 5th
 This meeting we are covering some topics from Commutative Algebra which have important
geometric implications: finite ring extensions, and the Noether Normalization Theorem. These
occupy Sections 1.1.3  1.1.6 in the book by Hulek.
 Meeting on October 22nd
 We are finishing the proof of the Hilbert's Nullstellensatz, discuss examples, and establish
the equivalence of the two categories discussed previously. An idea of affine schemes will be
given as well.
 Meeting on October 15th
 We are discussing the categorical viewpoint on affine varieties, and motivate the considerations
related to the Hilbert's Nullstellensatz. We prove that theorem as well.
 Meeting on October 8th
 We continue with irreducible components, (polynomial) maps between algebraic sets, interpetation
thereof in terms of the algebras of functions, categorical language (the category og affine algebraic
varieties), and more examples.
 Meeting on October 1st
 This time, we are discussing thetopology of affine algebraic sets, introducing their algebras of
regular functions, and heading toward the Hilbert's Nullstellensatz.
 Meeting on September 24th
 We are prepared for the Hilbert's Basis Theorem, and the introduction of affine algebraic sets,
Zariski topology in the affine spaces, and affine varieties. These will be the topics for the meeting.
 Meeting on September 17th
 We continue our study of polynomial rings and affine algebras.
 Meeting on September 10th
 After introducing the rings of polynomials last time, we are proving the Hilbert Basis
theorem, and are introducing affine spaces, affine algebraic sets, and the Zariski
topology on those. Rings of sunctions on affine algebraic sets, and affine algebras will be
discussed.
 Meeting on September 3rd
 Rings of polynomials of finite number of variables over a field
It will be explained how these rings are constructed, their basic properties: ideals,
generating sets of ideals, quotient of such rings, etc. The celebrated Hilbert's Basis
theorem will be proved. This is a preparation to our study of affine algebraic varieties.
 Meeting on April 9th
 This meeting will have three parts:
2:15 p.m.  3:15 p.m.  Algebraic Topology (don't forget the problem from the last time!)
3:30 p.m.  4:30 p.m. Colloquium in DM 168 ("Big Divisors on Projective Varieties")
4:45 p.m.  6:00 p.m. continuation of the Seminar.
 Meeting on April 2nd
 Basic Theorems on Homotopy; The Fundamental Group (continued).
Projective spaces and projective algebraic varieties (continued).
 Meeting on March 26th
 Basic Theorems on Homotopy; The Fundamental Group
Projective spaces and projective algebraic varieties.
 Meeting on February 26th
 Cell Complexes  exercises; coordinate rings, sheaves of functions on an affine variety
 Meeting on February 19th
 Cell Complexes; operations on spaces; basic theorems on homotopy
Rational functions on affine varieties
We continue our considerations from the last time. The emphasys in the first part, AT,
will be put on the basic theorems on homotopy. Don't forget to think of the contractibility
S^{/infty}.
 Meeting on February 12th
 Cell Complexes; operations on spaces; basic theorems on homotopy
Rational functions on affine varieties
In the topology part: discussion of suspensions, joins, wedge sums, smash products,
and related constructions.
In the Algebraic Geometry part: rational functions on affine varieties.
 Meeting on February 5th
 Cell Complexes; examples; basic theorems on homotopy
Cartesian product of affine varieties, properties. Rational
functions on affine varieties.
In the topology section, we are doing the first problems from the problem section to
Chapter 0.
 Meeting on January 29th
 Homotopy, Cell Complexes;
The Category of Affine Algebraic Varieties, Rational Functions and Maps
This is a continuation of the topics from the previous meeting.
 Meeting on January 22nd
 Homotopy, Cell Complexes;
The Category of Affine Algebraic Varieties, Rational Functions and Maps
The Algebraic Topology part of the Seminar begins at 2:00 p.m. We are discussing the
way of simplifying a topological space which does not change the algebraic groups the
general theory associates with it: the notion of homotopy. Our considerations will be
based on numerous examples (from Hatcher's book). Then, we introduce the most important
and useful topological spaces  cell complexes. These are the basis on which the exposition
in the book is built.
In the Algebraic Geometric part of the Seminar, starting at 4:15 p.m., we are discussing
the categorical point of view on affine varieties over an algebraically closed field.
The role of the Zariski topology will also be discussed (as an example of a useful
topology). Next, we introduce and discuss rational functions and maps of affine
varieties.
 Meeting on January 15th
 Basics from General Toplogy; Mappings of affine algebraic varieties
Our Spring Seminars will consist of two parts: Algebraic Topology part, and Algebraic
Geometry part. The former will discuss topics from Allen Hatcher's book, and is intended
to help the independent study class on Algebraic Topology. On the first meeting on
the subject I will cover some basic notions from general topology needed for that
class. The latter of the two parts is a continuation of our study of toric varieties
and their applications to Physics. The main speaker here is Luis Saumell.
 Meeting on November 20th
 Affine Algeraic Sets IV; BanachTarski  BanachTarski
This time, we will have a second talk, by Carlos Bajo, on the famous BanachTarski
paradox. In short, it says that a ball in the three dimensional Eucluden space
can be dissembled into pieces which, after moving them rigidly can be assembled
into two balls, each of the same volume as the first one.
 Meeting on November 6th
 Affine Algebraic Sets III
Due to the presense of new students at the Seminar, we had last time a brief
review of the theory of algebraic sets we have already done there. Next
Friday, we continue with the what we planned for the last meeting
(see below).
 Meeting on October 30th
 Affine Algebraic Sets III
We will discuss the Zariski topology on affaine algebraic sets and the related
features: principal open sets, irreducibility. The coordinate rings,
radical idelals, etc. will be introduced. The presentation will be geared
toward Hilbert's Nullstellensatz. The material is basic, and belongs in the
Topics in Algebraic Structures course.
 Meeting on October 16th
 Hilbert's Basis Theorem; Affine Algebraic Sets II
This is a continuation of the discussion started last time. Luis will explain
the definition and first properties of the affine algebraic sets: the
link between Algebra of polynomials and Geometry. The level of the
exposition is Topics in ALgebraic Structures.
 Meeting on October 9th
 Hilbert's Basis Theorem; Affine algebraic sets I
We begin the study of the topics mentioned in the first three meetings.
The first theorem to be discussed and proved next is the Hilbert's Basis
theorem. We will need it to show that any affine algebra is Noetherian. Based
on that, we will discuss and prove the first properties of the affine
algebraic sets  the building blocks of the Algebraic Geometry. The
material to cover is standard, and wihin the abilitites of the math and physics
majors students.
 Meeting on October 2nd
 Introductory remarks on toric varieties III
This is the last of the introductory talks. We will discuss some simplest
examples of toric varieties. The goal is to exhibit the link with convex cones.
 Meeting on September 25th
 Introductory remarks on toric varieties II
We discuss algebraic tori over the complex numbers, convex cones in R^n, and
the construction of affine toric varieties. We give only general ideas. Later on
in the Seminar, the concepts and constructons will be made rigorous.
 Meeting on September 18th
 Introductory remarks on toric varieties
We are giving a rather brief overview of what one needs to know from math in
order to understand the basics of the theory of toric varieties.
 Meeting on May 28th
 Plane Algebraic Curves
We are discussing the most elementary geometric and algebraic properties of
the curves of low degree (2, 3, 4) in the plane. The role of complex numbers
for understanding the geometry of the curves will be discussed among the other
things.
 Meeting on May 21st
 Introduction to Category Theory III
The topics this time: functors, natural transformations, and properties. We will
introduce some new important categories, and will discuss examples.
 Meeting on May 15th
 Introduction to Category Theory II
We are continuing the topic from the last time. Then, we discussed some well known
to the students examples. Now, we are proceeding by formally introducing the
concepts of category functor and their basic fetures.
 Meeting on May 8th
 Organizational; Introduction to Category Theory
This meeting is mostly organizational: the topics of the Seminar for the Summer
will be discussed as well as a better time for the meetings if need be.
Featured Presentation (M. Yotov):
 A Quick Introduction to Category Theory
 The first couple of lectures will introduce some basic language from Category
Theory (a Mathematical Area) and give motivating examples for the use of and
need for this language.
Prerequisites: the concepts to be introduced are so general that no specific
knowledge in Math is needed in order to grasp them. The examples would be faster
and easier to understand if the participants know a bit of set theory, a
bit of Linear Algebra, and a bit of Algebraic Structures. In case some of the
latter are not in their math background, the participants may find the
considerations on the Seminar as a valuable introduction to the mentioned areas.

 Meeting on October 19th
 Introduction to Algebraic Geometry II
James Fullwood continues with more about Zariski topology on the affine space.
He will explain the generalization of this toplogy to the case of spectra of
commutative rings.
 Meeting on October 12th
 Introduction to Algebraic Geometry I
James Fullwood will introduce the notion of affine algebraic variety, Zariski topology
on the ndimensional affine kspace, and will present some of their basic
properties. The presentation will be illustrated with a number of interesting examples.
 Meeting on October 5th
 Introduction to Commutative Algebra III
We continue our account of Unique Factorization Domains and Noetherian rings.
 Meeting on September 28th
 Introduction to Commutative Algebra II
We continue with Noetherian rings and Hilbert's Basis Theorem.>
 Meeting on September 21st
 Introduction to Commutative Algebra I
We are starting our serious work by introducing the ring of polynomials of finite
number of variables. The notions to be discussed are ideals (prime, coprime, maximal),
modules over these rings with ample amount of examples. The aim is to
get to the classical Hilbert's Basis Theorem. The material from Algebra to be covered
belogs to the must know part for every math major. It can be considered as
an introduction to the Algebraic Structures course offered by the Department of
Mathematics.
 Meeting on September 14th
 Organizational
On this first meeting, we are discussing the program of the Seminar for the Fall
semester. Roughly speaking, we are following the format of the meetings and the
choice of topics from the previous years, but the emphasis will be put on more
involved techniques from Commutative Algebra and Algebrais Geometry. The
presentations will be accessible for all with some background on Algebra and
first two parts of the Calculus sequence taught at the Department.
See you all there.
 Meeting on August 7th
 Basics of Commutative Algebra VIII
This final meeting is devoted to the Hilbert's Nullstellensatz and the link
algebrageometry related. We will discuss Zariski topology for geometric,
Noetherian, and general commutative rings. The theory will be illustrated
by numerous examples.
 Meeting on July 31st
 Basics of Commutative Algebra VII
We are discussing this time finite ring extensions. From our perspective,
the most important facts will be the Noether normalization of an affine f.g.
algebra. By using it, we are proving a weak version of Hilbert's
Nullstellensatz. The fine geometry of this normalization will be discussed
as well.
 Meeting on July 17th
 Basics of Commutative Algebra VI
The topic this time is Noethrian rings and modules. Several examples and
methods will be discussed. The main theorem (Hilbert's Basis Theorem) will be
proved.
 Meeting on July 10th
 Basics of Commutative Algebra V
Some properties of modules over a ring will be discussed in depth. Examples
of exact sequences and their use will be demonstrated. We are moving
on to some finiteness conditions for rings and modules next.
 Meeting on July 3rd
 Basics of Commutative Algebra IV
We continue with properties of modules over a commutative ring. Various examples
will be discussed.
Note the change of time for the meeting!
 Meeting on June 26th
 Basics of Commutative Algebra III
The topic of the meeting: modules and first properties. The discussion includes:
homomorphism/isomorphism theorems, generation of modules, the CayleyHamilton
theorem and its various applications, basics of Homological Algebra
(exact sequences, split exact sequences and more).
To understand better the concepts to be introduced on this meeting, one needs
more serious knowledge of Linear Algebra. See you all there.
 Meeting on June 19th
 Basics of Commutative Algebra II
We continue with the basic properties of commutative rings and their ideals.
In particular, we are discussing the Chinese Remainder Theorem, its use
to characterizing the idempotent elements of a commutative ring; the nilpotent
radical of a ring, and its geometric significance; the dual numbers
and its relation to the differential calculus in a ring.
Knowledge of basics of field extensions will be helpful in understanding some of
the examples to be considered on this meeting. See you all there.
 Meeting on June 12th
 Basics of Commutative Algebra
We are switching to Commutative Algebra (following Reid's "Undergraduate
Commutative Algebra"). The topic for this time is "Ideals in commutative rings
and their geometric interpretation". The material should be accessible
for students with rudimentary knowledge of Calculus, Algebraic Structures
and Linear Algebra. See you all there!
 Meeting on June 5th
 Conics and plane cubics IV
We are getting deeper in the geometry of the plane cubic curves. Several problems
will be discussed. The topological classification of algebraic curves
(Riemann surfaces) will be explained. See you all at the Seminar.
 Meeting on May 29th
 Conics and plane cubics III
We are moving on to study the elementary properties of plane cubic curves
This includes linear systems of cubics, rational parameterization of singular
cubics, nonrationality of the smooth cubics. As applications, we are
proving Pascal's mystic hexagon theorem (Pascal proved it when he was
16 in 1640), and are defining addition law on the smooth cubics.
All who are familiar with the material covered on the previous two meetings
will find, I hope, this one very exciting.
See you all at the Seminar.
 Meeting on May 22nd
 Conics and plane cubics II
Please note the change in the schedule: we will meet on the Tuesdays.
This time, we are discussing Chapter 1 of Undergraduate Algebraic Geometry.
We are applying the theory to solving the problems in the Exercise
sections of this chapter.
The discussion should be accessible for students having Calculus
and Linear Algebra background (Algebraic Structures' knowledge would
be a plus for understanding what will be covered at the Seminar).
 Meeting on May 14th
 Conics and plane cubics
This Summer's Seminar is devoted to introduction to Algebraic Geometry
and Commutative Algebra following the books by Miles Reid "Undergraduate
Algebraic Geometry" and "Undergraduate Commutative Algebra". The first
meeting will be a discussion of the plane conics and cubics. Apart from
the theory, a lot of examples will be discussed and computations performed.
The topics are accessible for all students with minimal Algebraic Structures
background and knowledge from the course of Calculus.
 Meeting on February 23rd
 Plane Algebraic Curves III
This Friday, at 1:30 p.m., we are continuing with the topic Plane Algebraic Curves.
This time, we are explaining the geometric interpretation of the prime ideals of
the polynomial ring k[X, Y] as points and curves in the plane. This will establish
the (complete) interrelation algebrageometry on the level of plane objects. We will
begin with the much better known case of ideals in k[X] and their geometric interpretation.
 Meeting on February 9th
 Plane Algebraic Curves II
The main topics for this second talk are:
Projective algebraic plane curves;
Intersection of such curves;
The Bezout theorem for algebraic curves.
The talk should be accessible for all students with basic backgroundon Algebra
and Calulus.
 Meeting on February 2nd
 Plane Algebraic Curves I
After the general introduction to the interplay AlgebraGeometry in Mathematics
on our first meeting, we are beginning a series of two or three talks on how
this works in the case of plane algebraic curves.
Topics to be covered this Friday:
Affine algebraic curves; Study's lemma (this is the Hibert's Nullstellensatz
for curves); Decomposition of a curve into components,irreducibility and
connectedness; Minimal polynomial of a curve and degree of a curve; Resultants
and Discriminants (needed for the proof of Study's lemma.
The material of these talks are accessible for studenmts with basic background
in Algebra and Calculus.
 Meeting on January 26th
 Opening meeting
On this meeting, we will be discussing the topics to be covered in Spring 2007.
A new feature for this semester will be the multiple invited speakers, faculty
members, who will give introductory talks to the areas of Mathematics they are
interested in. The goal of these talks will be to advertize those areas to the
students, and invite them to do research with experts available at the Department.
Faithful to the primary goal of the Seminar, we are also covering important
Commutative Algebraic and Algebraic Geometric topics (mostly at introductory level).
Apart from the talks by faculty members, we are expecting active participation,
in form of talks, by students as well.
 Meeting on December 1st
 Invariants, Symmetry, Parity and More...
We have a double session, with the Math Club, of problem solving this Friday starting
at 10:00 am in GC 271 A. We'll talk about problems involving different type of invariants,
symmetry, parity and more.
The Putnam contest is this Saturday, Dec. 2. It has two sessions: 10:00am  1:00pm and
3:00pm  6:00 pm. We will write again to tell you the room where we are going to meet for
the contest. For those of you registered, try to be fresh and well rested that day.
 Meeting on November 24th
 Elementary modular arithmetic techniques in problem solving
This time we are showing various problems in which "modular arithmetic" techniques
help find quick and elegant solutions. Some of the rpoblems are taken from different
mathematical competitions. The methods are elementary and should be accessible for all
participants.
 Meeting on November 3rd
 Recurrence relations and Generating Functions
We continue presenting different topics from Mathematics which are both interesting
by themselves and useful for those who want to attend the Putnam Competition this
coming December. This time we are discussing some methods from combinatorial analysis
and apply them to solving problems from different math competitions and olympiads. The
presentation will be accessible for broad audience of students.
 Meeting on October 27th
 Polynomials
This time, we are going over some well known properties of polynomials with complex
coefficients. Various problems (from different math competitions) will be solved as
applications of the theory. The topic, and the problems, will be inetersting also to
all who want to take part in the coming Putnam Competition.
 Meeting on October 20th
 Numbers: algebraic and transcendental
Some time ago, when we were discussing the famous ancient geometric problems,
we used also the fact that the number \pi was not constructible. Actually, this number
is not algebraic (over the rational numbers). The same is the case with the Napier
number e as well. In this talk we are discussing in some more depth the notion of
transcendental numbers. For the presentation, we will be using only knowledge from
Calculus I and II and some rudiments from Algebra. This is why we will not be able
to prove that the abovementioned numbers are transcendental. But we will prove they
are irrational, and will give explicit examples of transcendental ones.
 Meeting on October 13th
 An algebraic proof that A_n is simple for n>4
The previous three talks at the Seminar were devoted to the theory of algebraic
equations of degree 2, 3, 4, and 5. We mentioned that the quintic can't be solved
in radicals, and stated that this fact was related to the special property of the
alternating group A_5 of not having enough normal subgroups (i.e. it is a simple group).
This week, Prof. L.Ghezzi will explain (with proof) that all A_n, n>4, are simple.
Actually, the simplicity of A_n is the reason why the general equation of degree n>4
is not solvable in radicals.
The proof will be algebraic, all the concepts involved defined, and the presentation
accessible for every curious mathmajor.
 Meeting on October 6th
 First steps into Galois Theory III
We are finishing the triptych on Galois Theory, solving equations in radicals, and relations
to "elementary" geometry. We will learn how to solve the quartic (via a resolvent), and will
see why the "general" quintic isn't solvable. The link to the geometry of the icosahedron
will be explained as well.
 Meeting on September 29th
 First steps into Galois Theory II
Keeping the presentation simple/accessible, we are getting deeper into the theory
of field extensions, and are explaining algebraically the formulae for solving
the algebraic equations of degree up to 4. The ultimate goal is to get to
the equation of degree five (the quintic). The wonderful geometry related will be explained as well.
 Meeting on September 22nd
 First steps into Galois Theory
We are discussing, on well known and simple examples, the ideas from Galois Theory.
As an application, we are developing the theory of constructible numbers over the field of rational numbers.
 Meeting on September 15th
 Organizational meeting
We are discussing the topics for the Fall Semester at the Seminar.
Some suggestions are:
Galois Correspondence and Geometric Constructibility
Regular polyhedra and the geometry of the degree five polynomial
Mobius Transformations and applications
Symplectic geometry and Mechanics
Hadamard matrices and applications
Covariant Derivatives on hypersurfaces
GaussBonnet Theorem and applications...
 Meeting on August 11th
 "Plane cubic curves III"
In this final part of the presentation, we are discussing the question of existence and structure of
the set of rational points on an elliptic curve.
With this talk we are completing the Summer Session of the Seminar. Meeting on August 4th
 "Plane cubic curves II"
In the second part of the presentation, we are discussing the associativity law of the addition
operation on a smooth cubic curve as well as the number theoretical aspects of the elliptic
curves defined over the rational numbers. Meeting on July 28th
 "Plane cubic curves"
This topic is a continuation of what we did about the degree two plane curves (conic sections)
earlier this Summer. The arithmetic and geometry of the degree three plane curves (elliptic curves)
are much more rich and interesting than the conic sections' ones. The plan for the talks is
to give an algebraic classification of all cubic curves,
to discuss the possible "canonical forms" of these curves,
to introduce some interesting geometric objects on them (inflexion points etc.),
to discuss the group structure of the points on them,
to discuss the complex analysis approach to these curves.
As a special topic, we will briefly discuss the relation of the elliptic curves to the theory of numbers
and the last Fermat's Theorem (I will use the book by Silverman for our course on Number Theory
as a main source here). Meeting on July 21st
 Robert Salom (FIU) will finish his presentation on Riemann surfaces of certain analytic functions
 Meeting on July 14th
 Robert Salom (FIU) will speak on Riemann surfaces of analytic functions
In this talk we will see, on an intuitive level, how the Riemann surfaces of certain
analytic functions are constructed. Meeting on July 7th
 Postponed for July 14th
 Meeting on June 30th
 Cancelled.
Next Friday, July 7th, Robert Salom (FIU) will be talking about Riemann surfaces
of analytic functions. An abstract of the talk will follow soon. Meeting on June 23rd
 "Pascal's and Brianchon's theorems about conic sections".
We will show how to prove these celebrated theorems by using elementary Euclidean geometry,
and then by using basic machinery from Algebraic Geometry (curves of degree two and three,
degenerated curves, pencils of curves, and other related notions in the plane). The second
approach will allow us to prove these theorems in their natural general setup.
Most part of the exposition will be accessible to sophomore students. Meeting on June 16th
 "Some metric properties of the conic sections".
We are going to discuss some classical (metric) properties of these curves. These include
the (famous) optical as well as some related minmax properties of those curves.
The exposition will be elementary (accessible to any mathematically curious undergraduate
student). Meeting on June 9th
 Jorge Castillo (FIU) will explain the construction of rings of fractions in Commutative Algebra.
This algebraic construction has a very important geometric aspect: whenapplied to
affine algebras (these are the rings of functions of an affine algebraic variety), it
provides rings of regular functions on open subsets or rings of rational functions on
closed subvarieties of those varieties. When the closed subvarieties are irreducible,
the rings of fractions we get are local (they have only one maximal ideal). So either
we get functions locally (on open subsets), or we get local rings (provided the
subvarieties are irreducible). This is the reason why the general construction of
rings of fractions is (often) called "localization".
Several examples will be discussed.
 Meeting on June 2nd
 Due to the poor weather conditions, the previous meeting was postponed for today.
 Meeting on May 26th
 "Conic sections from geometric and algebraic point of view".
We will discuss in more detail the noncentral curves, degenerate degree two curves
(this will complete the classification of all degree two curves in the plane), the
use of complex numbers in this theory, and (if the time permits) the degree two curves
in the projective plane.
 Meeting on May 19th
 "Conic sections from elementary geometric and algebraic point of view".
We will start by expressing the conic sections (elementary geometrically) as conic
sections. Then we will express these plane curves by algebraic equations. After that,
using Linear Algebra, we will begin the classification of all degree two curves in the
real affine plane.