Abstract: Amoebas originated in algebraic geometry and have since proven to be subtle and challenging objects of study. At the same time, they have made surprising appearances across several seemingly unrelated areas of mathematics. In this talk, I will walk you through the basic theory of amoebas and briefly highlight their connections to other fields. Who knows — perhaps you will discover amoebas in your own research as well. I will also tell the story of how we used tools from differential geometry to compute the area of an amoeba.

Abstract: Computing the number of (complex) realizations of minimally rigid graphs has obtained some interest in the recent years. I will survey a number of results (from several different authors) about computations in the plane and on the sphere, exhibiting a range of techniques from classical tools in computer algebra to algebraic/tropical geometry and matroids.

Abstract: The Laplace operator is a fundamental tool for studying the geometry of manifolds. Motivated by electric networks, Laplacians on graphs are defined using edge weights that play the role of conductance. When the weights are constant, the graph Laplacian reduces to the combinatorial Laplacian, which captures rich combinatorial information about the graph.
For a graph embedded on a surface, it is natural to consider a geometric Laplacian whose edge weights reflect the underlying metric. On Euclidean surfaces, the 1-skeleton of a geodesic triangulation satisfies the well-known cotangent formula, which expresses these weights in terms of the Euclidean geometry. In this talk, we present the hyperbolic analogue of this construction and discuss its applications to discrete harmonic maps into hyperbolic surfaces, which realize the weighted graph as the 1-skeleton of an equivariant convex polyhedral surface.
This is joint work with Ivan Izmestiev.

Abstract: Identifiability shows up everywhere, from rigidity questions in discrete geometry to tensor completion problems and secant varieties in algebraic geometry. At its core lies a simple question: when do a collection of algebraic relations determine the underlying points uniquely?
I will present a new framework that tackles this question for systems where algebraic constraints interact with combinatorial structure. The key idea is to understand how the combinatorics of a graph or hypergraph control local and global identifiability. Starting from the classical setting of graph rigidity, where relations come from Euclidean distances, we extend the theory to hypergraphs equipped with general algebraic measurements. This perspective allows us to obtain several necessary or sufficient conditions for identifiability, using tools from both rigidity theory and the geometry of secant varieties.
The results come from several joint works: (a) with James Cruickshank, Anthony Nixon, and Shin-ichi Tanigawa; (b) with Sean Dewar, Georg Grasegger, Kaie Kubjas, and Anthony Nixon; and (c) with Louis Theran and Jessica Sidman.

Abstract: The geometry of the image of the nonnegative orthant under the power-sum polynomials maps is called the Vandermonde cell. We analyze the geometry of this object in a finite number of variables and concentrate on the limit as the number of variables approaches infinity. We explain how the geometry of the limit plays a crucial role in undecidability results in nonnegativity of symmetric polynomials, deciding validity of trace inequalities in linear algebra, and extremal combinatorics (recently observed by Grigoriy Blekherman, Annie Raymond, and Fan Wei. We also show how differences in this geometry amount for the fact that undecidability does not hold for the normalized power sum map.

Abstract: This talk will present progress in a project investigating simplicial surfaces, addressing combinatorial and geometric aspects, using computational approaches. One challenging problem is to construct realizations of simplicial surfaces (embeddings), given an incidence structure and prescribed edge lengths. We report on three recent results:

• classification of symmetric embeddings of an icosahedron composed of equilateral triangles in three-dimensional Euclidean space, where, e.g., a flexible icosahedron was found,
• constructing embedded icosahedra composed of several congruence types of triangles by inscription into an octahedron,
• existence of perfect toroidal chains of isosceles tetrahedra.

Abstract: Systems of polynomial constraints with real coefficients encode naturally geometric properties in the realm of Euclidean geometry and hence they arise naturally in areas such as robotics, biology and rigidity theory. The expectations when solving such systems of constraints depends very much on the application. Topical algorithmic questions encompass real root finding, grabbing sample points in each connected component of the solution set, counting these connected components and quantifier elimination over the reals.
In this talk, we will review recent progress on these algorithmic problems, with a focus on computer algebra algorithms that provide exact answers. Complexity issues and software will be discussed.

Abstract: A framework $(G,p)$ is rigid if any $(G,q)$ sufficiently close to $(G,p)$ that has the same edge lengths is congruent to $(G,p)$. Since the frameworks $(G,q)$ with the same edge lengths as $(G,p)$ are a semialgebraic set, rigidity can be tested by computing a description of this set and then looking at the dimension of the component containing p. However, for all but tiny examples, this approach, while accurate, is infeasible. In practice, sufficient conditions for rigidity, such as infinitesimal rigidity and prestress stability are used instead, since they can be checked with linear algebra or semidefinite programming, respectively. The question of devising, efficient, higher order rigidity tests has been considered in the structural engineering and robotics literature, with various definitions of higher order rigidity proposed. All of these are known to have theoretical and practical limitations. Meanwhile, in the physics community, energy based approaches to rigidity are commonly used. But this literature tends to study a specific energy in any given application or setting, and whether results transfer to other energies is unclear.
I will describe an energy-based approach that assigns to each rigid framework a numerical “rigidity order” that quantifies how rigid it is. The rigidity order is universal over all “stiff bar” energies, a class that includes all the energies commonly encountered in applications, including spring energies and the Lennard-Jones potential. It generalises the notions of first and second order rigidity, and it can be characterised in terms of “higher order flexes”. In certain specific cases, the rigidity order can be computed efficiently. There is also a variant of rigidity order for “prestressed energies” that generalises prestress stability.
This is joint work with S J Gortler and M Holmes-Cerfon.