VIGRE2 Vertical Intergration of Research and Education Department of Mathematics, University of Utah

MATH 4800








Fall 2011 course

Topic: Introduction to Tropical Geometry
Instructor: Tommaso de Fernex
Catalog Number: MATH 4800-1
Credits: 3
Time: M,W 4:10 - 5:30 PM, LCB 219

Prerequisites: Good background on the basic 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 a $500 tuition benefit for up to 10 undergraduates who are US citizens, nationals, and permanent residents.

The purpose of this course is to give an introduction to algebraic curves and tropical curves. 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. For more information, please refer to

Past Courses:
Fall 2006, Math Finance
Spring 2007, Fractals
Fall 2007, Metric Spaces, The Contraction Mapping Principle, Fractals & Other Applications
Spring 2008, Knot Theory
Fall 2008, Random Walk: Modeling, Theory, and Applications
Spring 2009, Graph Theory
Fall 2009, Metamaterials
Spring 2010, Symetry&Groups
Fall 2010, To A_D_E and Beyond; Dynkin Diagram and Classification Problems