The articles in this volume study various cohomological aspects of algebraic varieties:- characteristic classes of singular varieties;- geometry of flag varieties;- cohomological computations for homogeneous spaces;- K-theory of algebraic varieties;- quantum cohomology and Gromov-Witten theory.The main purpose is to give comprehensive introductions to the above topics through a series of "friendly" texts starting from a very elementary level and ending with the discussion of current research. In the articles, the reader will find classical results and methods as well as new ones. Numerous examples will help to understand the mysteries of the cohomological theories presented. The book will be a useful guide to research in the above-mentioned areas. It is adressed to researchers and graduate students in algebraic geometry, algebraic topology, and singularity theory, as well as to mathematicians interested in homogeneous varieties and symmetric functions. Most of the material exposed in the volume has not appeared in books before.Contributors:Paolo AluffiMichel BrionAnders Skovsted BuchHaibao DuanAli Ulas Ozgur KisiselPiotr PragaczJ?rg Sch?rmannMarek SzyjewskiHarry Tamvakis
Author(s): Editor: Piotr Pragacz
Series: Trends in Mathematics
Edition: 1
Year: 2005
Language: English
Pages: 325
Contents......Page 6
Preface......Page 10
Notes on the Life and Work of Alexander Grothendieck......Page 12
References......Page 28
1. Lecture I......Page 30
2. Lecture II......Page 37
3. Lecture III......Page 42
4. Lecture IV......Page 48
5. Lecture V......Page 53
References......Page 60
Introduction......Page 62
1. Grassmannians and flag varieties......Page 63
2. Singularities of Schubert varieties......Page 80
3. The diagonal of a flag variety......Page 90
4. Positivity in the Grothendieck group of the flag variety......Page 100
References......Page 111
1. Introduction......Page 115
2. K-theory of Grassmannians......Page 116
3. The bialgebra of stable Grothendieck polynomials......Page 120
4. Geometric specializations of stable Grothendieck polynomials......Page 122
5. Degeneracy loci......Page 123
6. Grothendieck polynomials......Page 126
7. Alternating signs of the coefficients c[sub(w,µ)]......Page 128
References......Page 130
1. Computing homology: a classical method......Page 132
2. Elements of Morse theory......Page 137
3. Morse functions via Euclidean geometry......Page 144
4. Morse functions of Bott-Samelson type......Page 151
References......Page 159
1. Completely integrable systems......Page 161
2. Random matrices and enumeration of graphs......Page 170
3. Witten’s conjecture and Kontsevich’s solution......Page 178
4. Some of the further developments......Page 184
References......Page 185
1. Introduction......Page 188
2. Characteristic map and BGG-operators......Page 189
3. Structure constants for Schubert classes......Page 191
4. A combinatorial proof of the Pieri formula......Page 194
References......Page 198
1. History......Page 200
2. Calculus of constructible functions......Page 205
3. Stratified Morse theory for constructible functions and Lagrangian cycles......Page 210
4. Characteristic classes of Lagrangian cycles......Page 214
5. Verdier-Riemann-Roch theorem and Milnor classes......Page 218
6. Appendix: Two letters of J. Schürmann......Page 222
References......Page 224
1. Introduction......Page 227
2. Grothendieck groups......Page 229
3. K-theory of fields......Page 236
4. Quillen Q-construction......Page 240
5. K'[sub(•)] of noetherian schemes......Page 250
6. K[sub(•)] of certain varieties......Page 258
7. Applications......Page 276
References......Page 293
1. Lecture One......Page 295
2. Lecture Two......Page 300
3. Lecture Three......Page 304
4. Lecture Four......Page 310
5. Lecture Five......Page 314
References......Page 320
