TOMMASO DE FERNEX

Associate Professor - University of Utah

MATH 4800 - REU INTRODUCTION TO TROPICAL GEOMETRY - FALL 2011

Meeting: M,W 4:10-5:30 PM, LCB 219

Office Hours: By appointment

Instructor: Tommaso de Fernex

Contact: Office JWB 322, email defernex@math.utah.edu

Course Webpage: http://www.math.utah.edu/~defernex/4800-F11.html

Prerequisites: Good background on the basics concepts in Linear Algebra and Abstract Algebra (rings, fields, polynomials, matrices, etc) and some familiarity with the basic notions in Topology (open and closed sets, continuity, etc).

Tuition Benefit. The NSF VIGRE program provides tuition benefit for up to 10 undergraduates who are US citizens, nationals, and permanent residents.

Course Content: The purpose of this course is to give an introduction to algeraic curves and tropical curves.

Freeely quoting from the references below: The origins of algebraic geometry lie in the study of zero sets of systems of polynomials. These objects are algebraic varieties, and they include familiar examples such as plane curves and surfaces in three-dimensional space.

In tropical algebra, the sum of two numbers is their minimum and the product of two number is their sum. This algebraic structure is known as the tropical semiring or as the min-plus algebra. It makes perfect sense to define polynomials and rational functions over the tropical semiring. The functions they define are piecewise-linear. Also, algebraic varieties can be defined in the tropical setting. They are now subsets of R^n that are composed of convex polyhedra. Thus, tropical algebraic geometry is a piecewise-linear version of algebraic geometry.

Before starting with tropical geometry, we will spend some time introducing the basic concepts in algebraic geometry in the context of plane curves. We will then quickly move to tropical geometry. Rather than giving a rigorous, abstract treatment of the theory which would go beyond the purpose of this course, we will discuss many examples (mostly focusing on the case of plane tropical curves) and explore some of the applications to algebra, geometry, and other fields.

A rought list of topics expected to be covered in the course is given below. Topics are subject to change according to the level of the class.

  • Algebraic Geometry. Real plane conics. From the real numbers to the complex numbers, and from the affine plane to the projective plane. Conics revisited, cubics, and their group law. Topology of curves and Riemann surfaces, genus of a smooth curve. Plane curves. Resultant, Study's lemma (special case of Hilbert's Nullstellensatz), and the Bezout theorem. Kontsevich's theorem on counting rational curves.
  • Tropical Geometry. Tropical ring, examples and applications. First examples of tropical curves. Amoebas and their tentacles. Formalize approach: fields and valuations, Puisuex power series, initial forms and Grobner basis. Tropical curves and hypersurfaces. Fundamental theorem of tropical geometry. Tropical conics and higher degree plane curves. Genus of a tropical curve. Intersection and tropical Bezout theorem. Tropical proof of Kontsevich theorem.

An excellent first introduction to what tropical geometry is about can be found in the Notices of the AMS paper

Material on tropical geometry will be selected from the following references, all available for download:

Part of the material on plane algebraic curves will be selected from the books:

Coursework and Grading There will be homework assignments throughout the semester, and as the course progresses I will encourage students to find and work on a project to a topic of their choice related to the course. Some references and possible topics will be suggested in class. The last classes at the end of the semester will be devoted to presentations given by the students on their projects. Grade will be assigned based on class participation, homework, and final presentation.

Withdrawals: Until Friday, October 21st you may withdraw from the class with no approval at all. After that date you may withdraw only after petitioning the appropriate Dean's office.

ADA Statement: The Americans with Disabilities Act requires that reasonable accommodations be provided for students with physical, sensory, cognitive, systemic, learning, and psychiatric disabilities. Please contact me at the beginning of the term to discuss any such accommodation for the course.