CourseKTH.html

ALGEBRAIC GEOMETRY: COMPUTATIONS AND APPLICATIONS

This is a one-semester advanced graduate course to be held concurrently with the research program on

"Algebraic Geometry with a View towards Applications" at the Mittag-Leffler-Institute.

TEACHERS: It will be co-taught by Alicia Dickenstein from Buenos Aires and Bernd Sturmfels from UC Berkeley.
TIME: The class will meet Mondays at 10:00-12:00 in Room 3721. The first meeting will be held on January 17, 2011.
PREREQUISITES: Undergraduate Algebraic Geometry, specifically familiarity with Gröbner bases as in the text book Ideals, Varieties and Algorithms by Cox, Little and O’Shea. Experience with mathematical software (e.g. Mathematica, Macaulay2, Singular, Sage) will be useful.
OBJECTIVE: This class offers an opportunity for graduate students and postdoctoral fellows with widely different backgrounds and interests to learn from each other, and to engage with the international community of experts who are based in Stockholm during the spring semester.
FORMAT:Each Monday a new topic will be presented, and there will a homework set on that topic, to be handed in on the following Monday. After the lecture, we’ll go for lunch, and participants are encouraged to meet for informal discussions in the afternoon. Related research seminars might follow. Some of us might also go out for dinner, and we are counting on the local students to show us their favorite spots.
EXAMINATION:The course grade will based on homework and in-class participation. Collaboration on the homework is strongly encouraged, but every student is asked to turn in her or his own homework set.
SYLLABUS: Algebraic Statistics, Convex Algebraic Geometry, Elimination, Implicitization, Tropical Algorithms, Bio-chemical applications.
REFERENCES: In addition to research articles, take a look at the books D.Cox, J.Little, D.O’Shea: Using Algebraic Geometry, Springer, 1998. A.Dickenstein and I.Emiris (eds.): Solving Polynomial Equations: Foundations, Algorithms, and Applications, Springer, 2005.
B.Sturmfels: Solving Systems of Polynomial Equations, AMS, 2002.

AFTER LUNCH, at about 13:30 there will be the opportunity to discuss the topics of the course, in the same room.

COURSE MATERIAL/SCHEDULE	HOMEWORK
Slides on Convex Algebraic Geometry, from the lecture on 17/1	Homework I from the lecture on 17/1
Slides on Tropical Geometry, from the lecture on 24/1	Homework II from the lecture on 24/1
Slides on Algebraic Statistics, from the lecture on 31/1	Homework III from the lecture on 31/1
Nonnegative Polynomials and Sums of Squares, from the lecture on 7/2	Homework IV from the lecture on 7/2
Slides on "The Central Curve in Linear Programming", from the lecture on 14/2	Homework V from the lecture on 14/2
Slides from the lecture on 28/2 will be on Numerical Algebraic Geometry
Slides from the lecture on 7/3 will be on Polynomials and Biochemical reaction networks I	Homework VI from the lecture on 7/3
Slides from the lecture on 14/3 will be on Polynomials and Biochemical reaction networks II	HomeworkVII from the lecture on 14/3
The lecture on April 4 will be on Intersection Theory and will be given by D. Eklund
The lecture on April 11 will be on Koszul complexes and will be given by C. Berkesch
The lecture on April 18 will be on "Basics of Elimination Theory"	Homework from the lecture on 18/4 (due May 2)
The lecture on Monday May 2, at 10:15 will be on "Implicitization of rational hypersurfaces I" (slides)	Homework from the lecture on 2/5
The lecture on Monday May 2, at 13:30 will be on "Implicitization of rational hypersurfaces II" (slides)	Homework from the lecture on 2/5
The lecture on Monday May 9 will on Discriminants (slides)	Homework from lecture om 9/5
The lecture on May 16 will be on Resultants. Reference here. A recap from the previous lecture.