Mireille Boutine -- Can a drone hear the shape of a room?

Suppose that some microphones are placed on a drone inside a room with planar walls/floors/ceilings. A loudspeaker emits a sound impulse and the microphones receive several delayed responses corresponding to the sound bouncing back from each planar surface. These are the first-order echoes. In this talks, we will discuss the problem of reconstructing the shape of the room from the first-order echoes. The time delay for each echo determines the distance from the microphone to a mirror image of the source reflected across a wall. Since we do not know which echo corresponds to which wall, the distances are unlabeled. The problem is to figure out under which circumstances, and how, one can find out the correct distance-wall assignments and reconstruct the wall positions. Our algorithm uses a simple echo sorting criterion to recover the wall assignments for the echoes. We prove that, if the position and orientation of the drone is generic, then the wall assignment obtained through our echo sorting criterion must be the right one and thus the reconstruction obtained through our algorithm is correct. Our proof uses methods from computational commutative algebra. This is joint work with Gregor Kemper.

Tianran Chen -- Root counting, tropical intersection, homotopy, and networks of oscillators

This talk will explore a few concrete aspects of what numerical algebraic geometry can potentially contribute to network science especially the phenomenon of spontaneous synchronizations in networks of coupled oscillators. Starting with the root counting problem, we will discuss two generalizations of the BKK bound theory and their applications to the Kuramoto equations and the Nooburg's neural network model. We will also explore the roles of tropical intersection theory and homotopy continuation methods in these problems. This talk will cover joint works with R. Davis and D. Mehta.

Kathlén Kohn -- The adjoint of a polytope

This talk brings together many areas: discrete geometry, statistics, algebraic geometry, and geometric modeling. First, we recall the definition of the adjoint of a polytope given by Warren in 1996 in the context of geometric modeling. He defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics. It is the numerator of a generating function over all moments of the uniform probability distribution on a simplicial polytope. Thirdly, we prove the conjecture that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. Finally, we see that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3- or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.

Lek-Heng Lim -- Tropical geometry and deep learning

We show that ReLU-activated neural networks are exactly tropical rational maps. This observation may be used to study the expressivity of deep neural networks via the geometry of tropical hypersurfaces. The tropical perspective also applies to ConvNet, maxout networks, DAG networks, neural networks involving max-pooling or downsampling. By monitoring how topology changes across layers, we provide empirical evidence to demonstrate that, in the context of learning, a tropical rational map may be viewed as a device for "topological simplication" -- as a point cloud data set passes through the layers of a well-trained neural network, the Betti numbers in all dimensions are decreased. With enough layers, the point cloud data representing a class of objects in a classification problem would ultimately be reduced to one with Betti_0 = 1 and Betti_k = 0 for all k > 1. This is joint work with Greg Naitzat and Liwen Zhang.

Pablo Parrilo -- Switched Linear Systems and Infinite Products of Matrices

Many situations of interest can be modeled as "switched linear systems", which are collections of linear difference equations, with some logical rule for switching between subsystems. Mathematically, this boils down to understanding infinite products of matrices, all of which are elements of a given finite set. Analyzing these systems is a difficult question that appears in a number of applications, including the analysis of optimization algorithms, information theory, and wavelets. Depending on whether the switching is deterministic or stochastic, different notions can be used to quantify the resulting convergence rate, like the *joint spectral radius*, or the *Lyapunov exponent*. In this talk, we provide a gentle introduction to this class of problems, their applications, and several results regarding their exact and approximate computation.

Ngoc Tran -- Tropical geometry and economics

Tropical geometry has emerged as the variational approach to network flow problems, with a growing number of applications in numerous fields. In this talk, I will review the background to some open problems in economics that seem particularly suitable to tropical and combinatorial approaches. No prior knowledge on economics is assumed.

Cynthia Vinzant -- Completely log-concave polynomials and high-dimensional expanders

Completely log-concavity is a functional property of real multivariate polynomials that translates to strong and useful conditions on its coefficients. In particular, from such a polynomial, one define a random walk on its monomials that mixes quickly and for which the associated graph has good expansion properties. I will introduce the class of completely log-concave polynomials in elementary terms, discuss the beautiful real and combinatorial geometry underlying these polynomials, and describe applications to random walks on simplicial complexes. One nice consequence of this theory is a proof of the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.