Program

Links to speaker and poster abstracts:

MONDAY: P. Parrilo, M. Chan, J. Kileel, A. Cueto, poster session

TUESDAY: J. Huh, A. Seigal, M. Joswig, N. Tran, poster session

WEDNESDAY: J. Draisma, P. Zwiernik, H. Harrington, S. Sullivant, F. Sottile

THURSDAY: L. Pachter, E. Robeva, M. Michalek, J. Hauenstein, G. Blekherman

FRIDAY: A. Dickenstein, C. Vinzant, E. Miller

Grigoriy Blekherman: How to tropicalize (almost) everything and why

Tropicalization has been frequently applied in algebraic geometry to give "combinatorial shadows" of complicated objects. But one can break free from the tyranny of algebra, and tropicalize manifestly non-algebraic objects. If a set S is closed under coordinate-wise (Hadamard) multiplication then its tropicalization (in the sense of log-limits) is a closed convex cone. Valid linear inequalities on the tropicalization cone correspond to valid binomial inequalities on the original set S. I will give several examples of combinatorial sets, where tropicalization encodes interesting information (for instance, log-concavity of a sequence is given by binomial inequalities). Two phenomena emerge (1) Tropicalizations of interesting combinatorial objects are rational polyhedral cones and (2) Tropicalization does well in distinguishing between (dual cones of) nonnegative polynomials and sums of squares both in the usual and combinatorial contexts. I do not have a good explanation for the above phenomena, and would greatly benefit from audience's help.

Melody Chan: Moduli spaces in tropical geometry

I will give a hopefully accessible introduction to some work on tropical moduli spaces of curves and abelian varieties, which Bernd got me started on in graduate school. I will report on joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe, in which we find new rational cohomology classes in the moduli space A_g of abelian varieties using tropical techniques. And I will try to touch on a new point of view on this topic, namely that of differential forms on tropical moduli spaces, following the work of Francis Brown.

Angelica Cueto: Lines in the tropics

Tropical Geometry has been the subject of great amount of recent activity over the last decade. Loosely speaking, it can be described as a piecewise-linear version of algebraic geometry. It is based on tropical algebra, where the sum of two numbers is their maximum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of geometric information about their classical counterparts.

In this talk, I will give a gentle introduction to the subject and will illustrate this powerful technique through two concrete examples from classical algebraic geometry: the 28 bitangent lines to smooth plane quartics and the 27 lines on smooth cubic surfaces in projective 3-space. This is based on joint works with Hannah Markwig and Anand Deopurkar.

Alicia Dickenstein: Linear algebra, combinatorics, and defectivity

In joint work with E. Cattani, we explore four approaches to the question of defectivity for a complex projective toric variety associated with an integral configuration A. The first one arises from the explicit tropicalization of the associated dual variety given with E.M. Feichtner and B. Sturmfels. The second one is based on the notion of iterated circuits introduced by A. Esterov. The third one uses an invariant defined by R. Curran and E. Cattani in terms of a Gale dual configuration of A. The fourth one was proposed by K. Furukawa and A. Ito in terms of Cayley decompositions of A.

We obtain formulae for the dual defect in terms of iterated circuits and Gale duals. Our proofs and extended definitions are linear-algebraic in nature and hold for finite configurations over an arbitrary field of characteristic zero. We prove in this generality the equivalence of the first three approaches. We give a Gale dual interpretation of Cayley decompositions and apply it to the study of defective configurations. However, we only prove one inequality between the first three invariants and the fourth one. I will close my talk with an open question concerning the reverse inequality.

Jan Draisma: Fewestnomials in determinantal ideals

Given a variety X in K^n, one may ask what is the smallest number of terms of a nonzero polynomial in the vanishing ideal of X. In general, this is quite a subtle question, but I will discuss three classes of varieties where we have established the answer precisely:

1. very general linear spaces in K^n,

2. rank-r matrices in K^{m x n}, and

3. skew-symmetric rank-r matrices in K^{m x m}.

In each of these cases, we also have a complete characterisation of these fewestnomials.

(Joint work with Thomas Kahle and Finn Wiersig)

Heather Harrington: Computational algebra and topology for spatial structures arising in biology

Biological processes are governed by interactions at multiple scales (genomic, molecular, cellular), which are now captured by multiple modalities (multi-indexed data) at different spatial resolutions. Understanding complex biological phenomena requires mathematical approaches to elucidate dynamics, predict mechanisms and reveal

function. With the wealth of state-of-the-art data at unprecedented depth and scales, new approaches are required to extract meaningful and interpretable biological insights. This talk will present computational algebra and topology methods, relying on persistent homology, that provide insight and quantification to geometric structures arising at multiple scales in biology, such as protein structure and cancer/immune microenvironment.

Jonathan Hauenstein: Numerical computations in algebraic geometry

The foundational problem in algebraic geometry is to compute the solutions of a system of polynomial equations. Many techniques have been developed which exploit various structures present in the system such as multihomogenity and monomial sparsity to expand the limits of computations. This talk will include some stories about pushing the limits of computation to solve some problems for Bernd as well as explore some current trends in numerical computations in algebraic geometry with a particular focus on computing and analyzing the set of real solutions arising in engineering applications. Coupling with polynomial dynamical systems in which the stationary points arise as solutions to a system of polynomial equations, one is able to explore the landscape by first computing the stationary points. This will be demonstrated on a problem in mechanism synthesis and is joint work with Aravind Baskar and Mark Plecnik.

June Huh: Combinatorics and Hodge theory

I will tell two interrelated stories illustrating fruitful interactions between combinatorics and Hodge theory. The first is that of Lorentzian polynomials, based on my joint work with Petter Br\"and\'en. They link continuous convex analysis and discrete convex analysis via tropical geometry, and their structure reveals subtle information on graphs, convex bodies, projective varieties, Potts model partition functions, log-concave polynomials, and highest weight representations of general linear groups. The second is that of matroid intersection cohomology, based on my joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang. It shows a surprising parallel between the theory of Coxeter groups, convex polytopes, and matroids. After giving an overview of the similarity, I will outline proofs of two combinatorial conjectures on matroids, the nonnegativity conjecture for the Kazhdan-Lusztig coefficients, and the top-heavy conjecture for the number of flats.

Michael Joswig: Tropical Convexity

The term "tropical convexity" was coined by Develin and Sturmfels who published a landmark paper with that title in 2004. However, the topic has much older roots and is deeply connected to linear and combinatorial optimization and other areas of mathematics. The purpose of this survey talk is to sketch how that article contributed to shaping the field of tropical geometry as we know it today.

Joe Kileel: Moment tensor decompositions

In this talk, I will discuss algebraic and computational issues that arise when we decompose moment tensors into low-rank factorizations. I will describe different types of moment tensor decompositions, and mention some new ways of computing them. Applications range from statistics to structural biology.

Mateusz Michalek: Algebra for oscillators: Khovanskii and Graver bases

The study of steady states for systems of oscillators leads to very interesting polynomial equations. Computationally, one can guess the maximal number of solutions, as a function of the number of oscillators. We prove this observed bound is correct, using the theory of Khovanskii bases.

This is a joint work with Paul Breiding, Leonid Monin and Simon Telen.

Ezra Miller: Multigraded algebra: past, present, and future

Multigraded algebra has been entwined with resolutions and with toric geometry since before multigraded algebra was formalized in the late 1970s. This talk surveys multigraded commutative algebra from its historical roots to current developments and

open questions, including older motivations from combinatorial aspects of the subject as well as newer ones from modeling shape in topological data analysis and from sheaf theory in quantum noncommutative toric geometry. Both of the latter two involve non-discrete grading groups but still take cues from combinatorial considerations.

Lior Pachter: Cell dynamics from snapshot diagrams

Pablo Parrilo:

Elina Robeva: Log-concave graphical models

We study the problem of maximum likelihood estimation of a log-concave density that factorizes according to a given undirected graph G. We show that the maximum likelihood estimate (MLE) exists and is unique with probability 1 as long as the number of samples is at least as large as the smallest size of a maximal clique in a chordal cover of the graph G. Furthermore, we show that the MLE is the product of the exponentials of several tent functions, one for each maximal clique of the graph.

While the set of log-concave densities in a graphical model is infinite-dimensional, our results imply that the MLE can be found by solving a finite-dimensional convex optimization problem. We provide an implementation and a few examples.

This is joint work with Kaie Kubjas, Olga Kuznetsova, Pardis Semnani and Luca Sodomaco.

Anna Seigal: Ranks of tensors and connections to data analysis

Linear algebra is the foundation to methods for finding structure in matrix data. There are many challenges in extending this to the multi-linear setting of tensors. I will discuss the comparison of rank and symmetric rank for a tensor, including recent progress in finding tensors whose rank and symmetric rank differ (in joint work with Kexin Wang, building on work of Yaroslav Shitov). We will see connections to classical algebraic geometry, via cubic surfaces, and small open problems of a combinatorial nature.

Frank Sottile: Critical Points Of Discrete Periodic Operators

It is believed that the dispersion relation of a Schrodinger operator with a periodic potential has non-degenerate critical points. In work with Kuchment and Do, we considered this for discrete operators on a periodic graph G, for then the dispersion relation is an algebraic hypersurface. We showed how tools from computational algebraic geometry may be used to study the dispersion relation. A first step was to determine the number of critical points for a particular graph.

With Matthew Faust, we use combinatorial algebraic geometry to give an upper bound for the number of critical points, and also a criterion for when that bound is obtained. The dispersion relation has a natural compactification in a toric variety, and the criterion concerns the smoothness of the dispersion relation at the toric infinity. This toric compactification enables other questions from physics to be expressed in terms of algebraic geometry.

Seth Sullivant: On the Algebraic Geometry of Graphical Models

Graphical models are statistical models that use a graph to represent conditional independence relations between collections of random variables. Graphical models are fundamental tools in statistics and especially machine learning, because they can use a simple graphical structure to describe complex relations between large collections of random variables. On the other hand, in the case of discrete or jointly normal random variables, conditional independence relations are fundamentally algebraic constraints on the natural parameter spaces of the models. Bernd's work with Dan Geiger and Chris Meek ("On the toric algebra of graphical models") initiated the algebraic study of these models, and many of his further papers developed the algebraic study in different directions. I'll discuss what we've learned in the past 20 years.

Ngoc Tran: Exploiting combinatorial structures in data to forecast science, uncover causations, and settle economics conjectures

In many problems, one observes noisy data coming from a hidden or complex combinatorial structure. My research aims to understand and exploit such structures to arrive at an efficient and optimal solution. This talk showcases four successes, achieved with different tools, from four different fields: networks forecasting, hydrology, auction theory, and random forests. I will also try to state concrete open problems for students, postdocs, and potential collaborators.

Cynthia Vinzant: Determinantal representations and the principal minor map

The principal minor map takes an n by n matrix to the vector of its 2^n principal minors. The quest for an algebraic description of the image of this map dates back to the 19th century classical algebraic geometry and was reinvigorated by work of Holtz-Sturmfels and Lin-Sturmfels. In this talk, I will describe a connection between this map and certain classes of determinantal representations. For symmetric and Hermitian matrices, this results in a set-theoretic description of the image of the principal minor map using the orbit of a finitely-many polynomial equations and inequalities under a large group action and recovers Oeding's characterization of the principal minors of symmetric complex matrices as the variety of the orbit of Cayley's hyperdeterminant. I will also give examples to show that that no such finite characterization is possible for general matrices. This is based on joint work with Abeer Al Ahmadieh.

Piotr Zwiernik: Non-independent component analysis

A seminal result in the literature on the Independent Component Analysis states that for AY = X, if the components of the unobserved random vector X are independent and at most one is Gaussian, then the matrix A is identified up to sign and permutation from the distribution of Y (P. Comon, 1994). In this paper we study to which extent the independence assumption can be relaxed by replacing it with restrictions on the higher order cumulants of X. From the mathematical perspective this introduces a new identifiability question for symmetric tensors under zero restrictions on some of its entries. We document minimal conditions for identifiability and propose efficient estimation methods based on the new identification results. The proof strategy employed highlights new geometric and combinatorial tools that can be adopted to study identifiability via higher order restrictions in linear systems.

(joint work with Geert Mesters)

MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY

8:30 - 8:45 coffee coffee coffee coffee coffee

8:45 - 9:00 opening remarks coffee coffee coffee coffee

9:00 - 9:45 P. Parrilo J. Huh J. Draisma L. Pachter A. Dickenstein

10:00-10:45 M. Chan A. Seigal P. Zwiernik E. Robeva C. Vinzant

10:45-11:15 coffee break coffee break coffee break coffee break coffee break

11:15-12:00 J. Kileel M. Joswig H. Harrington M. Michalek E. Miller

12:00-2:15 lunch lunch lunch lunch closing remarks

2:15 - 3:00 A. Cueto N. Tran S. Sullivant J. Hauenstein

3:00 - 3:30 poster session poster session coffee break coffee break

3:30- 4:15 poster session poster session F. Sottile G. Blekherman

4:30-5:15 cake & stories more stories BERND: what is next?

Banquet 6-9 pm