* Please note this course is now online *

Wednesdays from 2:00 PM till 4:00 PM
Zoom meeting ID: 997 1289 2235

In the course, I will discussed relations between algebra and geometry which are useful in both directions. Newton polyhedron is a geometric generalization of the degree of a polynomial. New-ton polyhedra connect the theory of convex polyhedra with algebraic geometry. Toric varieties provide a tool for developing such connection. Dehn{Zommerville duality for simple polyhedra leads to the computation of the cohomology ring of smooth toric vari-eties. Riemann-Roch theorem for toric varieties provides a valuable information about the number of integral points in convex polyhedra and unexpected multidimensional generalization of the classical Euler-Maclaurin summation formula. Newton polyhedra allow to compute many discreet invariants of generic complete intersections. Topological version of Parshin reciprocity laws and Grothendieck residues help to study zero dimensional complete intersection.

Newton-Okounkov bodies connect the theory of convex bodies (not necessary polyhedra) with algebraic geometry. It provides a simple proof of the classical Alexandrov{Fenchel inequality (generalizing the isoperimetric inequality) and suggests analogues of these inequality in algebraic geometry. Tropical geometry and the theory of Grobner bases relate piecewise linear geometry and geometry of lattice with algebraic geometry. These relations allow to describe the ring of conditions for complex torus and for other horospherical homogeneous spaces.  All needed facts from algebraic geometry and will be discussed in details during the course.

Instructor: Prof. Khovanskii

Course number: MAT1191HS

Textbooks:

    G.Kempf, F.Knudsen, D.Mamford, B.Saint-Donat. \Toroidal Embeddings", Springer Lecture Notes 339, 1973.
    K.Kaveh, A.Khovanskii. \Newton{Okounkov bodies, semigroups of integral points, graded algebras and intersection theory", Annals of Mathematics, V. 176, No 2, 925{978, 2012. Some extra papers will be handout during the course.

Grading: 1 final presentation or written report.