Georgia Tech. April 11, 2015 (Saturday)

The goal of the meeting is for people at nearby universities to get to know each other.  We will have eight 30 minute talks, with ample time remaining for discussions and exchanging ideas.

Registration is now closed.

List of Speakers
Michael Burr (Clemson)
Dustin Cartwright (UT Knoxville)
Noah Giansiracusa (University of Georgia)
Robert Krone (Georgia Tech)
Luke Oeding (Auburn)
Rainer Sinn (Georgia Tech)
Mark van Hoeij  (Florida State University)
Cynthia Vinzant  (NCSU)

Schedule
9-10 registration, coffee
    10-10:30 Noah Giansiracusa
    10:30-11 Cynthia Vinzant
11-11:30 break
    11:30-12 Mark van Hoeij
    12-12:30 Dustin Cartwright
12:30-2:30 lunch
    2:30-3 Michael Burr
    3-3:30 Robert Krone
3:30-4 break
    4-4:30 Rainer Sinn
    4:30-5 Luke Oeding
Dinner

Directions
The School of Mathematics is located at 
Skiles Classroom Building, 686 Cherry St, Atlanta GA 30332
The talks will take place in room Skiles 005 on the ground floor.  
Please enter from the southwest corner of the building.

Parking
Street parking is available on weekends on Ferst Drive, Tech Parkway, and in the streets south of campus (e.g. near Luckie St and Merritts Ave).

You can also park in any visitor parking lot:
Visitor Area 1 (North Ave) 33°46'16.1"N 84°23'41.7"W
Visitor Area 2 (Student Center, Ferst Drive) map
Please take a ticket when you enter and and keep it.  We can give you a validation.  If the gate is up, then you can park for free.


Abstracts
  • Michael Burr. Continuous Amortization: Intrinsic Complexity for Subdivision-based Algorithms. 
Many algebraic geometry applications require the approximation of the solutions of a system of polynomials.  A common search method for solving such systems is to iteratively subdivide an input region.  Continuous amortization was recently introduced as a technique to compute the complexity of these subdivision-based algorithms.  Continuous amortization has proven to be a powerful technique, producing state-of-the-art and adaptive complexity bounds based on the intrinsic complexity of the input instance.  In this talk, I will present recent advances in both the theory and application of the continuous amortization technique. 
  • Dustin Cartwright. A quantitative version of Mnev's theorem

Mnev's theorem says that realization spaces of matroids can be as complicated as arbitrary systems of polynomial equations, in terms of singularities and in terms of finding rational solutions. I will give some background on this theorem and then talk about a version of Mnev's theorem over the integers, for which the size of the matroid can be bounded in an explicit way.

  • Noah Giansiracusa. A persistent homology approach to fingerprint classification and comparison

I'll give a quick introduction to topological data analysis (persistent homology and barcodes) and then present work-in-progress exploring the applicability of these methods to biometrics, specifically fingerprint classification and comparison. This is based on an undergraduate group research project I'm leading at UGA this semester. Although the mathematics comes from algebraic topology rather than algebraic geometry, I'll mention work of Carlsson et al. where invariant theory is used to interface topological data analysis with functional machine learning, and this brings in some tools and ideas from algebraic geometry.

  • Robert Krone. Equivariant Gröbner bases and syzygies
Given a polynomial ring with an action of the infinite symmetric group on its set of variables, an equivariant Gröbner basis of an ideal is a set of polynomials whose orbits under the action form a Gröbner basis.  Equivariant Gröbner bases offer a concise way to describe ideals in high dimension with symmetry and (under some conditions) can be effectively computed.  We look to adapt signature based Gröbner basis algorithms to the equivariant setup, and in doing so confront the notion of equivariant syzygies.  We prove that an ideal that is finitely generated up to symmetry has finitely generated equivariant syzygies when considered in the correct module.  This is work in progress with Jan Draisma and Anton Leykin.
  • Luke Oeding. Homotopy techniques for tensor decomposition and perfect identifiability

Given a tensor (or hyper-matrix), we would like to express it in the simplest possible way as the sum of the smallest number of decomposable (or rank-1) tensors. While there are many algorithms that attempt to accomplish this task, it is known to be a very difficult problem. Moreover, such a decomposition may not be unique. When a generic tensor of a given format has a unique decomposition, we say that tensors of that format are "generically identifiable." 
We propose a new method to find tensor decompositions via homotopy continuation. This technique allows us to find all decompositions of a given tensor (at least for relatively small tensors). Our experiments yielded a surprise - we found two new tensor formats, (3,4,5) and (2,2,2,3), where the generic tensor has a unique decomposition. Using techniques from algebraic geometry, we prove that these cases are indeed "generically identifiable". 
This is joint work with J. Hauenstein, G. Ottaviani and A. Sommese.
  • Rainer Sinn. Generic Spectrahedral Shadows

Spectrahedral shadows are projections of linear sections of the cone of positive semidefinite matrices. We characterize the polynomials that vanish on the boundaries of these convex sets when both the section and the projection are generic.
  • Mark van Hoeij. Computing an Integral Basis for an Algebraic Function Field.

 This talk will present several practical applications of integral bases, and present an efficient algorithm to compute an integral basis for the case of an algebraic function field with small prime characteristic.
  • Cynthia Vinzant. An algebraic view on phase retrieval

    The problem of phase retrieval is to reconstruct a signal from certain magnitude measurements. This problem is closely related to low-rank matrix completion and has many imaging-related applications: microscopy, optics, and diffraction imaging, among others. In purely mathematical terms, phase retrieval means recovering a complex vector from the modulus of its inner product with certain measurement vectors. One can ask how many measurements are necessary for this recovery to be possible. I’ll discuss recent progress made on this problem by translating it into real algebraic language and using tools from computational algebraic geometry.


We thank the support from School of Mathematics at Georgia Tech.