Daniel Brosch -- The flag algebra of rooted binary trees

While trees are usually considered sparse objects, they turn dense when considering only the leaves to be the "vertices" of a tree. In this setting one defines an (induced) subtree, given by a subset of leaves, to be the smallest tree containing these leaves, where inner vertices of degree 2 are contracted. We can then naturally extend Razborov's flag algebras to this setting, allowing the application of sums-of-squares and moment techniques to extremal questions about trees. We introduce and apply these methods to investigate the inducibility of trees and density profiles of trees.

Diego Cifuentes -- On semidefinite relaxations for noisy estimation problems

We consider estimation problems, in which the goal is to recover a point in an algebraic variety given some noisy estimate. Some special cases include matrix (or tensor) PCA, rotation synchronization, or camera triangulation. We also consider completion problems where, in addition to the noise, some observations are missing. Some special cases include matrix (or tensor) completion, or phase retrieval. We will show that these NP-hard problems can often be solved exactly by means of semidefinite programming under non-adversarial noise.

Harm Derksen -- Invariant Theory and Field Theory for Computer Vision

Polynomial and Rational functions that are invariant under spatial symmetries are very useful in computer vision. In my talk I will focus on the problem of computing invariants of 3D objects such as points, lines, conics from a single 2D projection. In many situations, one can prove that no non-constant invariants can be computed from a single 2D image. I will also discuss joint work with John Christian (Georgia Tech) and Ryan Watkins (NASA) on using

invariants of craters in images of the moon's surface for space navigation.

Papri Dey -- Spectral Factorization and Polynomials with Lorentzian signature

This talk has two parts. In the first part, I shall talk about a polynomial time algorithm to compute spectral factorization of a nonnegative matrix polynomial. This provides an approximation theory to certify positivity of a nonnegative matrix polynomial via computing the Jordan Normal Form of a complex matrix. In the second part, I shall introduce the notion of polynomials with Lorentzian signature which unifies the notions of Lorentzian polynomials and hyperbolic polynomials and satisfies the Hodge-Riemann (HR) relations of degree <=1. The first part of the talk is joint work with Nick Ryder, Ravi Kannan, and Nikhil Srivastava.

Heide Gluesing-Luerssen -- Decompositions of q-Matroids using Cyclic Flats

A q-matroid is the q-analogue of a (classical) matroid. Its ground space is a finite-dimensional vector space over a finite field, and the rank function is defined on the lattice of subspaces. The required properties of the rank functions (boundedness, monotonicity, and submodularity) are the natural generalization of classical matroids. After an introduction to the basic notions of q-matroids and their relevance for coding theory, we turn to the cryptomorphisms known for matroids and give a brief overview of their q-analogues. This will bring us to cyclic flats and our first result which states that the cyclic flats along with their rank values fully determine the q-matroid. Thereafter, we will consider direct sums of q-matroids and show that the collection of cyclic flats of a direct sum consists exactly of all direct sums of the cyclic flats of the two summands. This will allow us to define and characterize irreducible q-matroids and show that every q-matroid can be decomposed into irreducible ones. The decomposition is unique in a natural way.

Amy Huang -- An old problem in linear algebra relevant to quantum information theory and complexity theory, and what modern algebraic geometry can tell us about it.

A linear subspace of the space of b x c matrices is of bounded rank r if no matrix in the space has rank greater than r. Such spaces have been studied for a long time, but little is known about them. I'll explain classical and modern results about them, and why people in complexity theory and quantum information theory care about them. 

Svetlana Poznanović  -- Using Polytopes to Improve RNA Branching Predictions

Minimum free energy prediction of RNA secondary structures is based on the Nearest Neighbor Thermodynamics Model. While such predictions are typically good, the accuracy can vary widely even for short sequences, and the branching thermodynamics are an important factor in this variance. Recently, the simplest model for multiloop energetics - a linear function of the number of branches and unpaired nucleotides - was found to be the best. We develop a branch-and-bound algorithm that finds the set of optimal parameters with the highest average accuracy for a given set of sequences. The search uses the branching polytopes for RNA sequences. Our analysis shows that previous ad hoc parameters are nearly optimal for tRNA and 5S rRNA sequences on both training and testing sets.