Mathematics 277: Toric Varieties in Mathematics, Physics, and Computation. (Spring 2003)
Instructor:
Christian Haase
Description:
Algebraic geometry is the study of algebraic varieties:
geometric objects that are defined by polynomial
equations. Toric varieties form a subclass of varieties that
can be defined from combinatorial data: a polyhedral fan. They
thus provide a link between algebraic geometry and polyhedral
combinatorics.
This relation has been fruitful in both directions. We will see
instances of polytope theorems whose proof uses algebraic
geometry as well as variety theorems whose proof uses
polytopes.
But toric varieties have spread further. To name some
applications, there is
- Batyrev's interpretation of mirror symmetry in terms of
polytope duality,
- test set algorithms for integer programming as Gröbner
basis computation of toric ideals,
- solving systems of polynomial equations;
- contingency table sampling in statistics.
Format:
During the semester (part A) there will be reading assignments
and one or two lectures per week by a participant or a faculty
member on a basic subject, so that by the end of the term, we
are familiar with the fundamentals.
The Gergen Lectures by Bernd Sturmfels (March 1-3) are part of
the course.
We will then spend a weekend in April somewhere nice (part B)
where each participant will give a talk on a more specialized
subject. The choice of subjects will depend on interests and
background of the participants.
Possible topics are
- Polytopes
- the g-theorem for simple polytopes
- secondary and fiber polytopes
- Algebraic geometry
- semi-stable reduction over curves
- weak factorization
- abelian/toric McKay correspondence
- Commutative algebra
- sparse resultants
- normal and Koszul semi-group rings
- Physics
- stringy Hodge numbers and mirror symmetry
- monomial <-> divisor mirror map
- Computation
- solving systems of polynomial equations
- integer programming
- global optimization and sums of squares
- toric varieties and statistics
- linear PDE with constant coefficients
But the choice of subjects will depend on the interests of the
participants.
Schedule:
THERE WILL BE AN ORGANIZATIONAL MEETING .
End of January: project title.
End of February: table of contents, prerequisites, references.
End of March: draft of the talk/term paper.
Prerequisites
Curiosity, commitment, some algebra - linear algebra(!), (polynomial-)
rings, ideals. A previous encounter with (complex) projective space is
advantageous.
We WILL go over the necessary basics in polytope theory and
algebraic geometry relatively quickly (few proofs) in
Part A.
The target audience is graduate students and advanced
undergrads from mathematics, physics, statistics, and computer
science.
References
The main text during part A will be
- William Fulton Introduction to Toric Varieties
PUP (1993).
Other standard references are
- Vladimir Danilov The geometry of toric varieties
Uspekhi Mat. Nauk 33 (1978).
- Güunter Ewald Combinatorial Convexity and Algebraic Geometry
Springer GTM 168 (1996)
- George Kempf, Finn Knudsen, David Mumford, and
Bernard Saint-Donat Toroidal Embeddings I
Springer LNM 339 (1973)
- Tadao Oda Convex Bodies and Algebraic Geometry
Springer (1988)
More literature will be listed at the course website.
The references for part B will mostly be research articles.
Course Website
More information will soon be available at
Blackboard
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Last modified: 13 October 2003