Meeting on Algebraic Geometry for Applications
April 9, 2016 (Saturday)
This is a small (local) conference which goal is to bring together researchers and their students who are using algebraic geometry.
Daniel Bernstein - North Carolina State University
Brent Davis - Colorado State University
Timo de Wolff - Texas A&M
Elena Dimitrova - Clemson University
Joshua Hallam - Wake Forest University
Alex Kasman - College of Charleston
Sebastian Pokutta - Georgia Institute of Technology
Location: Matin Hall M first floor, Department of Mathematical Sciences
Time Talk 8:45 - 9:00 Registration and welcome message 9:00 - 9:45
Sebastian Pokutta - A polyhedral characterization of border basesBorder bases arise as a canonical generalization of Groebner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we adapt the classical border basis algorithm to allow for computing border bases for arbitrary order ideals, which are independent of term orderings. We also show that finding a maximum weight order ideal that supports a border basis is NP-hard, and that the convex hull of admissible order ideals has no polynomial polyhedral description. (joined work with Gábor Braun)
9:55 - 10:40
Daniel Bernstein - Toric Varieties in StatisticsIn the first half of my talk, I will explain how certain statistical models can be viewed as toric varieties, and how the algebraic theory can be leveraged to develop algorithms for hypothesis testing. In the second half, I will describe some "niceness" properties that a toric ideal can satisfy, and their implications for statistical algorithms. Then, I will present some recent results and work in progress related to these properties for hierarchical models. This is joint work with Seth Sullivant.
10:40 - 11:00 Break 11:00 - 11:45
Timo de Wolff - New Certificates for Nonnegativity via Circuit Polynomials and Geometric ProgrammingDeciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. Since this problem is NP-hard, one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century the standard certificates are sums of squares (SOS); see particularly Hilbert’s 17th problem. In this talk, we introduce polynomials supported on circuits. For this class nonnegativity is characterized by an invariant, which can be derived from the initial polynomial immediately. In consequence, we obtain an entirely new class of nonnegativity certificates, which are independent of SOS certificates. Our certificates crucially extend geometric programming approaches for the computation of lower bounds in polynomial optimization. Particularly, for polynomials with simplex Newton polytope our approach is significantly faster and often yields better than bounds than semidefinite programming, which is the standard method for polynomial optimization. These results generalize earlier works by Fidalgo, Ghasemi, Kovacec, Marshall, and Reznick. The talk is based on joint work with Sadik Iliman.
11:45 - 2:00 Lunch break 2:00 - 2:45
Elena Dimitrova - Properties and applications of vanishing ideals of points over finite fieldsThe easiest geometric object to compute over affine or projective space is a single point. It has no secrets -- in particular, its defining ideal, the set of polynomials which vanish at the point, is straightforward to describe. In contrast, vanishing ideals of multiple points is a challenge even over a finite field. Though efficient techniques for computing them, such as the BM-algorithm, exist, for most sets of points the vanishing ideal has several equally "nice" generating sets which yield multiple interpolating polynomials. In this talk, we will explore properties of vanishing ideals, particularly over finite fields. Additionally, we will see how these questions arise naturally in the design of experiments and selection of algebraic models of systems in mathematical biology.
2:55 - 3:40
Brent Davis - Quartet Cleaning using the Nearest Statistical Model of EvolutionResearch has shown that the accuracy of many quartet-based phylogenetic supertreereconstruction algorithms are sensitive to input error. Using a combination of optimization,numerical algebraic geometry (NAG), and statistical hypothesis testing we propose aquartet-based reconstruction algorithm called the nearest model method. This methodselects the best fit quartet tree given an aligned nucleotide sequence. The goal of this methodis to assist supertree algorithms by minimizing errors caused from incorrectly inferred trees.Using NAG methods, performance of the nearest model method using data generated from theJukes Cantor model of evolution is discussed. We also discuss the performance of the nearestmodel method using the interior-point method on a variety of data and models of evolution.
3:40 - 4:00 Break 4:00 - 4:45
Joshua Hallam - Combinatorial Hopf Algebras and Symmetric FunctionsHopf algebras arose in the study of algebraic topology during the 1940's. They have since been found to be extremely useful in many other disciplines, including combinatorics. In this talk we will consider a few examples of Hopf algebras that arise in combinatorics. We will see a very nice relationship between these Hopf algebras and symmetric functions. Moreover, we will see how combinatorial properties can be seen as consequences of algebraic properties of Hopf algebras. No prior knowledge of Hopf algebras or symmetric functions will be assumed.
4:55 - 5:40
Alex Kasman - Spectral Curves and the Sato Grassmannian: Algebraic Geometry for Integrable Nonlinear PDEsThe analysis of partial differential equations does not usually involve any advanced algebraic geometry. However, in the late 20th century it was discovered that a certain class of nonlinear PDEs having particle-like solutions called “solitons” also has a rich algebro-geometric structure. Vector bundles over complex projective curves and Grassmannian varieties can be used to produce and understand the solutions to these special but important differential equations. This talk will quickly review the history of this subject, present some of the key results, and also “advertise” a few of my own contributions to this surprising interface between algebraic geometry and mathematical physics.
6:30 Optional dinner (registration on site)
First Name Last Name Institution Travis Baumbaugh Clemson University Daniel Bernstein North Carolina State University Michael Burr Clemson University Benjamin Case Clemson University Brent Davis Colorado State University Timo de Wolff Texas A&M University Elena Dimitrova Clemson University Gabriel John Dusing University of Tennessee - Knoxville Michael Eldredge Clemson University Shuhong Gao Clemson University Luke Giberson Clemson University Brandon Goodell Clemson University Akshay Gupte Clemson University Josh Hallam Wake Forest University Cvetelina Hill Georgia Institute of Technology Alex Kasman College of Charleston Kisun Lee Georgia Tech Drew Lipman Clemson University Felice Manganiello Clemson University Gretchen Matthews Clemson University Sebastian Pokutta Georgia Tech Svetlana Poznanovic Clemson University Sean Sather-Wagstaff Clemson University Josephine Yu Georgia Tech
Lodging options in Clemson:
- Comfort Inn Clemson, 1305 Tiger Boulevard Clemson, SC 29631
- Days Inn Clemson, 1387 Tiger Blvd, Clemson, SC 29631
- Holiday Inn Express & Suites Clemson, 1387 Tiger Blvd, Clemson, SC 29631
- Hotel Tillman Clemson, 1303 Tiger Blvd, Clemson, SC 29631
- The James F. Martin Inn, 240 Madren Center Drive, Clemson, SC 29634
Parking:Parking enforcement will exercise relaxed enforcement in the following areas from 8:00 am – 6:00 pm on April 9: E-6, Parkway Dr. and Calhoun Dr. (E-6 is the best option being it the parking area closest to the Department.) Relaxed enforcement means that a guest parked in a valid parking space will not receive a citation for not having a permit. Guest parking in a metered parking space are expected to pay the meter as regulations are enforced seven days a week from 7:00 am – 10:00 pm.