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.