Summer Semester 2009, DM 409B, Thursday 4:30 p.m. - 6:15 p.m.

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.

Fall Semester 2007, GC 276, Friday 3:15 p.m. - 5:00 p.m.

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 n-dimensional affine k-space, 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, co-prime, 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.

Summer Semester 2007, DM 409, Tuesday 3:30 p.m. - 5:00 p.m.

Meeting on August 7th

Basics of Commutative Algebra VIII


This final meeting is devoted to the Hilbert's Nullstellensatz and the link
algebra-geometry 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 Cayley-Hamilton
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, non-rationality 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.

Spring Semester, 2007, GC 275 B, Friday 2:00 p.m.- 3:15 p.m. (or 3:30 p.m. - 5:00 p.m.)

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 algebra-geometry 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 Algebra-Geometry 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.

Fall Semester, 2006, GC 271 A, Friday 10:00 a.m. - 11:30 a.m.

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 math-major.


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
Gauss-Bonnet Theorem and applications...

Summer Session, 2006, GC 287 B, Friday 2:00p.m. - 3:30 p.m.

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 min-max 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 sub-varieties of those varieties. When the closed sub-varieties 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
sub-varieties 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 non-central 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.