April 13 - 14, 2007
The study of Schubert varieties has grown out questions in enumerate
geometry from the 19th century. This field has flourished over the
past fifty years and now has applications in algebraic geometry,
representation theory, combinatorics, physics, computer graphics, and
economics. We will define Schubert varieties in the context of
Grassmannians and flag varieties. These varieties have many
interesting properties determined by combinatorial data like
partitions, permutations, posets, and graphs. We present five fun
facts on Schubert varieties and some open problems in the field.
This is a spectacular subject, both by its many applications
and its intricate proof. We describe in the talk the main ideas
(after Zariski, Hironaka, Abhyankar and various other people)
how to use blowups for the resolution of singular varieties,
and how to build up an induction procedure to choose at each
step of the resoluton process the center of the next blowup and
to show that the singularities have improved after blowup. The
key constructions and phenomena are illustrated by
visualizations of surface singularities.
As the talk will give mostly the intuitive ideas, we would like
to refer the interested reader for details to three notes we
have written on the subject. There, also quite complete lists
of other references can be found.
The three references to my talk are on my web-site
www.hh.hauser.cc
This talk gives a brief introduction to the theory of toric
varieties. In the first half of the talk I give seven different
definitions of a toric variety and compare them briefly. In the second
half I describe some of the uses of toric varieties. The first of these
is as a testing ground for algebraic geometric conjectures, as many
geometric questions about toric varieties can be turned into combinatorial
questions. A second use is as a good ambient variety in which to embed
and study other varieties, using the fact that smooth toric varieties are
natural generalizations of projective space.
This talk is intended as a brief introduction to certain aspects of the
theory of algebraic curves and surfaces.
To a projective algebraic variety one associates its arithmetic genus,
using the constant term of the Hilbert polynomial. Since a complex
projective algebraic curve can be viewed as a compact Riemann surface, it
also has a topological genus, equal to the number of holes in the Riemann
surface. Hirzebruch's Riemann-Roch theorem says that the topological genus
is equal to the arithmetic genus. I sketch a proof of this fundamental
result, using algebra, geometry, and topology. For algebraic surfaces
there is a corresponding result, Noether's formula, expressing the
arithmetic genus in terms of topological invariants, which can be proved
in a similar way. The last part of the talk discusses the theory of curves
on surfaces, in particular the links to classical enumerative geometry and
to modern string theory in theoretical physics.
While algebraic geometry is traditionally done over the complex
field, most problems from real life are modelled on real numbers and
ask for real, not for complex, solutions. Thus real algebraic
geometry --- the study of algebraic varieties defined over the real
numbers, and of their real points — is important. While most
standard techniques from general algebraic geometry remain important
in the real setting, there are some key concepts that are fundamental
to real algebraic geometry and have no counterpart in complex
algebraic geometry.
In its first part, this talk gives an informal introduction to a few
such key concepts, like real root counting, orderings of fields, or
semi-algebraic sets and Tarski-Seidenberg elimination. We also sketch
typical applications. In the second part, relations between
positivity of polynomials and sums of squares are discussed, as one
example for a currently active and expanding direction. Such
questions have already been among the historic roots of the field.
New techniques and ideas have much advanced the understanding in
recent years. Besides, these ideas are now successfully applied to
polynomial optimization.
Intersection theory is a big subject that has played an important role in
algebraic geometry, and any attempt at a comprehensive introduction in 90
minutes would surely fail. With this in mind, I have decided instead to
attempt to convey some of the beauty and flavor of intersection theory by
way of discussing a few concrete classical examples of intersection theory
on surfaces.
The material in the talk is covered in almost any text in algebraic
geometry. There are many approaches to finding 27 lines on a cubic
surface. If one wishes to see a more rigorous version of the presentation
given in this talk, then please see Hartshorne's treatment in Chapter V.4
of the book cited in the refereces. Chapter V of Hartshorne's book is a
very nice introduction to intersection theory on surfaces. For a more
general orientation in the subject with good historical context, one might
wish to read Fulton's "Introduction to intersection theory in algebraic
geometry." Fulton's other book cited in the references is the standard in
the subject, and the other texts listed offer additional points of view.