Seminars and Colloquia by Series

Hodge theory of mapping class group dynamics

Series
Algebra Seminar
Time
Tuesday, April 26, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Daniel LittUniversity of Georgia

This is joint work with Aaron Landesman. There are a number of difficult open questions around representations of free and surface groups, which it turns out are accessible to methods from Hodge theory and arithmetic geometry. For example, I'll discuss applications of these methods to the following concrete theorem about surface groups, whose proof relies on non-abelian Hodge theory and the Langlands program:

Theorem. Let $\rho: \pi_1(\Sigma_{g,n})\to GL_r(\mathbb{C})$ be a representation of the fundamental group of a compact orientable surface of genus $g$ with $n$ punctures, with $r<\sqrt{g+1}$. If the conjugacy class of $\rho$ has finite orbit under the mapping class group of $\Sigma_{g,n}$, then $\rho$ has finite image.

This answers a question of Peter Whang. I'll also discuss closely related applications to the Putman-Wieland conjecture on homological representations of mapping class groups. 

TBA

Series
Algebra Seminar
Time
Tuesday, April 19, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael BurrClemson University

Baker-Lorscheid (Hyperfield) Multiplicities in Two Variables

Series
Algebra Seminar
Time
Tuesday, April 12, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Trevor GunnGeorgia Tech

For polynomials in 1 variable, Matt Baker and Oliver Lorschied were able to connect results about roots of polynomials over valued
fields (Newton polygons) and over real fields (Descartes's rule) by looking at factorization of polynomials over the tropical and signed
hyperfields respectively. In this talk, I will describe some ongoing work with Andreas Gross about extending these ideas to two or more
variables. Our main tool is the use of resultants to transform questions about 0-dimensional systems of equations to factoring a single
homogeneous polynomial.

Stratified polyhedral homotopy: Picking up witness sets on our way to isolated solutions!

Series
Algebra Seminar
Time
Tuesday, April 5, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tianran ChenAuburn University at Montgomery

Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical method. Two of the fundamental tools in this field are the polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets put forth by Sommese and Wampler as numerical representations for non-isolated solution components. In this talk, we will describe a stratified polyhedral homotopy method that will bridge the gap between these two largely independent area. Such a homotopy method will discover numerical representations of non-isolated solution components as by-products from the process of computing isolated solutions. We will also outline the pipeline of numerical algorithms necessary to implement this homotopy method on modern massively parallel computing architecture.

Image formation ideals

Series
Algebra Seminar
Time
Tuesday, March 29, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tim DuffUniversity of Washington

Projective space, rational maps, and other notions from algebraic geometry appear naturally in the study of image formation and various camera models in computer vision. Considerable attention has been paid to multiview ideals, which collect all polynomial constraints on images that must be satisfied by a given camera arrangement. We extend past work on multiview ideals to settings where the camera arrangement is unknown. We characterize various "image formation ideals", which are interesting objects in their own right. Some nice previous results about multiview ideals also fall out from our framework. We give a new proof of a result by Aholt, Sturmfels, and Thomas that the multiview ideal has a universal Groebner basis consisting of k-focals (also known as k-linearities in the vision literature) for k in {2,3,4}. (Preliminary report based on ongoing joint projects with Sameer Agarwal, Max Lieblich, Jessie Loucks Tavitas, and Rekha Thomas.)

Symmetric generating functions and permanents of totally nonnegative matrices

Series
Algebra Seminar
Time
Thursday, March 17, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark SkanderaLehigh University

For each element $z$ of the symmetric group algebra we define a symmetric generating function

$Y(z) = \sum_\lambda \epsilon^\lambda(z) m_\lambda$, where $\epsilon^\lambda$ is the induced sign

character indexed by $\lambda$. Expanding $Y(z)$ in other symmetric function bases, we obtain

other trace evaluations as coefficients. We show that we show that all symmetric functions in

$\span_Z \{m_\lambda \}$ are $Y(z)$ for some $z$ in $Q[S_n]$. Using this fact and chromatic symmetric functions, we give new interpretations of permanents of totally nonnegative matrices.

For the full paper, see https://arxiv.org/abs/2010.00458v2.

Computing the nearest structured rank deficient matrix

Series
Algebra Seminar
Time
Tuesday, March 15, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Diego CifuentesGeorgia Tech

Given an affine space of matrices L and a matrix Θ ∈ L, consider the problem of computing the closest rank deficient matrix to Θ on L with respect to the Frobenius norm. This is a nonconvex problem with several applications in control theory, computer algebra, and computer vision. We introduce a novel semidefinite programming (SDP) relaxation, and prove that it always gives the global minimizer of the nonconvex problem in the low noise regime, i.e., when Θ is close to be rank deficient. Our SDP is the first convex relaxation for this problem with provable guarantees. We evaluate the performance of our SDP relaxation in examples from system identification, approximate GCD, triangulation, and camera resectioning. Our relaxation reliably obtains the global minimizer under non-adversarial noise, and its noise tolerance is significantly better than state of the art methods.

Degree bounds for sums of squares of rational functions

Series
Algebra Seminar
Time
Tuesday, March 8, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Grgoriy BlekhermanyGeorgia Tech

Hilbert’s 17th problem asked whether every nonnegative polynomial is a sum of squares of rational functions. This problem was solved affirmatively by Artin in the 1920’s, but very little is known about degree bounds (on the degrees of numerators and denominators) in such a representation. Artin’s original proof does not yield any upper bounds, and making such techniques quantitative results in bounds that are likely to be far from optimal, and very far away from currently known lower bounds. Before stating the 17th problem Hilbert was able to prove that any globally nonnegative polynomial in two variables is a sum of squares of rational functions, and the degree bounds in his proof have been best known for that two variable case since 1893. Taking inspiration from Hilbert’s proof we study degree bounds for nonnegative polynomials on surfaces. We are able to improve Hilbert’s bounds and also give degree bounds for some non-rational surfaces. I will present the history of the problem and outline our approach. Joint work with Rainer Sinn, Greg Smith and Mauricio Velasco.

Inflation of poorly conditioned zeros of systems of analytic functions

Series
Algebra Seminar
Time
Tuesday, January 25, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anton LeykinGeorgia Tech

Given a system of analytic functions and an approximate zero, we introduce inflation to transform this system into one with a regular quadratic zero. This leads to a method for isolating a cluster of zeros of the given system.

(This is joint work with Michael Burr.)

Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums

Series
Algebra Seminar
Time
Tuesday, January 11, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yue RenDurham University

We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank-k maxout unit is a function computing the maximum of k linear functions. For networks with a single layer of maxout units, the linear regions correspond to the regions of an arrangement of tropical hypersurfaces and to the (upper) vertices of a Minkowski sum of polytopes. This is joint work with Guido Montufar and Leon Zhang.

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