Meeting: TH 09:10-10:30 - JTB 110
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-S15.html
Prerequisites: Enrollment in this class is by permission of the instructor only. While there are no fixed prerequisites, given the advanced topic of the course it is recommended to have a some familiarity with the basic notions in Algebra and Topology (rings, fields, polynomials, power series, open and closed sets, continuity, compact surfaces like sphere and torus, complex numbers, etc). No prior knowledge of tropical or algebraic geometry is needed: we will start from scratch.
Tuition Benefit: The RTG grant in Algebraic Geometry and Topology provides tuition benefit for up to 15 undergraduates who are US citizens, nationals, and permanent residents. Departmental support is also available for other students.
Course Content: 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
Syllabus: 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 curves) and explore some of the connnection between algebra, geometry, topology, 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.
References: Algebraic geometry:
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.
Communications: I will use the automatic system for email communications provided by the Campus Information System. Refer to General Calendar Dates for important dates.
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.
Homework:
April 21: Guest Lecture by Drew Ellingson
Title: Tropical Grab Bag
Abstract: We will explore two interesting topics in Tropical Geometry.
First, we look to tropical theta characteristics. A classical theorem states that a plane quartic curve has exactly
28 odd theta characteristics, corresponding to exactly 28 bitangents. Recent work by Yoav Len and Matt Baker has
discovered that tropical plane quartics have 7 effective theta characteristics, and conjectured that the classical 28
specialize to the tropical 7 in groups of 4.
Second, we turn to tropicalization of the trigonal locus in M_g. This tropicalization (almost always) defines a top
dimensional cell in the moduli of tropical planar curves. To study this, we introduce Hirzebruch Surfaces, Petri's
Theorem, and lots of really cool combinatorics!
Final Presentations:
April 23 | Aaron Huston | Elliptic integrals |
---|---|---|
Ethan Lake | The group structure on smooth tropical cubics | |
April 28 | Scott Neville | Combinatorial tropical determinants |
Taylor Pope | Tropical Riemann-Roch | |
Hunter Simper | Classification of tropical projective conics | |
May 5 | Connor James | Elliptic curves and cryptography |
Brady Bowen | Moduli space of elliptic curves | |
Oliver Richardson | Minkowski sum reducibility of tropical curves |