Nonlinear Algebra - Algebraic Geometry

Algebraic geometry is a branch of mathematics that deals with the study of solution sets of systems of polynomial equations, which are known as algebraic varieties.
 
In recent years, the field of algebraic geometry has expanded its reach beyond theoretical mathematics and has found applications in other areas of science and mathematics.
 
Both symbolic and numerical computations are necessary to apply these methods in practice. Scientists in the Nonlinear Algebra Group work at the intersection of algebraic geometry and its applications in physics, polynomial optimization, partial differential equations, algebraic statistics, tensors, game theory, and algebraic vision.  
 
This interdisciplinary approach has not only provided solutions to complex problems in different fields but has also led to new developments in classical and applied algebraic geometry, which are further inspired by the insights and intuitions emerging from natural sciences.

by Claudia Fevola

The Kadomtsev–Petviashvili (KP) equation is a nonlinear partial differential equation describing the evolution of water waves. Krichever [1] made a substantial contribution to the interplay between integrable systems and algebraic geometry by providing an algebro-geometric procedure to construct KP solutions from a complex algebraic curve. The family of solutions built by Krichever is expressed by means of the Riemann theta function, which is a complex analytic function consisting in an infinite sum of exponentials supported on the integer lattice $\mathbb{Z}^g$. Furthermore, this result led to an alternative formulation of a classical problem in algebraic geometry, namely the Schottky problem.

We study KP solutions arising from algebraic curves with at worst nodal singularities. This is done by analyzing the limiting behavior of solutions associated to the Riemann theta function when the smooth curve is defined over a non-archimedean field. In the limit, the curve becomes singular and the Riemann theta function reduces to a finite sum where the support is a certain subset of the integer lattice $\mathbb{Z}^g$. Such a degenerate theta function gives rise to a subset of KP solutions called solitons.

We introduce the Hirota variety, which parametrizes solitons, and give a combinatorial procedure to determine its defining ideal. Using computer algebra software programs, we implement an algorithm to compute solutions arising from hyperelliptic curves. The importance of introducing the Hirota variety and understanding its geometry lies in the fact that it provides a different formulation and approach to the Schottky problem. Check [2, 3] if you want to know more about nodal curves and solitons.

by Sebastian Falkensteiner

Algebraic curves and varieties are an old topic of geometric and algebraic investigation. They have found applications, for instance, in architectural designs, in cryptographic algorithms, and are the central objects in computer-aided geometric design. Algebraic varieties can be represented in different ways, such as implicitly by defining polynomials, global parametrically by rational or algebraic functions, or locally parametrically by power series expansions. All these representations have their individual advantages; an implicit representation lets us decide easily whether a given point actually lies on the variety, a parametric representation allows us to generate points, and with the help of a power series expansion, we can trace a curve through a singularity.

Rational and local parametrizations of algebraic curves are well-studied [4]. Rational parametrizations exist if and only if the (geometric) genus is zero. Local parametrizations always exist [5]. Necessary field extensions are known, and rational algorithms are provided in both cases. For rational parametrizations of algebraic surfaces, the situation is similar [6]. More general types of global parametrizations, namely radical parametrizations, are considered in [7, 8].

Local parametrizations of surfaces and higher-dimensional varieties are just slightly studied [9]. A modern presentation including a fast version of the algorithm for computing such local parametrization is under investigation.

Algebraic varieties, including parameters, arise in various applications such as level curves, algebraic aspects in machine learning, non-linear differential equations, etc. For some choices of the parameters, the behavior of the variety itself or that of parametrizations changes. Parametric algebraic curves are studied in [10] and give interesting relations to Hilbert's irreducibility theorem and Tsenh's theorem, two classical results on polynomials involving parameters.

by Javier Sendra-Arranz

In the frame of normal form games, it is of interest to study different notions of equilibria. The most classical one is the set of totally mixed Nash equilibria. From an algebraic point of view, this set can be seen as the intersection of a Segre variety with the open simplex in a projective space. This notion of equilibria describes the situation where the players behave independently. On the contrary, the notion of dependency equilibria deals with the situation where all the players behave collectively. In [11], this notion of equilibria is studied using algebraic geometry through the study of the Spohn variety. In particular, the intersection of the Spohn variety and the open simplex equals the dependency equilibria.

We study the different notions of equilibria arising from the dependencies among the players from an algebro-geometric point of view. For instance, the situation where only two players behave collectively is studied in [12] by introducing the Nash Conditional Independence curve.

by Javier Sendra-Arranz

Hurwitz theory is the study of branched coverings of algebraic curves (Riemann surfaces). A classical question is to count the number of degree $d$ simple branched covering of the Riemann sphere from a genus $g$ curve up to isomorphism. This number is known as the Hurwitz number $H_{g,d}$. In [13] the notion of plane Hurwitz number was introduced. The plane Hurwitz number $\mathfrak{h}_d$ is the number of degree $d$ simple branched coverings of the Riemann sphere from genus $\binom{d-1}{2}$ plane curves that are linear projections. These integers are only known for $\mathfrak{h}_3 = 40 $ and $\mathfrak{h}_4 = 120$.

We study related questions concerning plane Hurwitz numbers as the real counting of such coverings, similar counting problems on other surfaces as $\mathbb{P}^1\times\mathbb{P}^1$ or the blow-up of the plane at a point, or the study of coverings coming from projections of canonical curves. Check [14] if you want to know more about plane Hurwitz numbers.

by Samantha Fairchild

The uniformization theorem guarantees an equivalence of Riemann surfaces and algebraic curves. In practice, however, recovering an algebraic curve requires the use of transcendental functions, that is, functions which cannot be represented as polynomials. We focus on building tools and algorithms which allow us to detect the algebraic structure of a Riemann surface given with analytic presentation. Topics include recovering algebraic curves from point clouds, numerically approximating a basis of holomorphic differentials via Poincare Theta series, and numerically approximating a curve from a Riemann matrix via Riemann Theta functions.

by Pierpaola Santarsiero

Over the last 60 years, multilinear algebra made its way into the applied sciences. Tensors have become increasingly central in many applications because they provide a natural and compact way to store information. A standard way to extract information from tensors is to consider structured decompositions with respect to some notion of rank, which typically has a natural geometric interpretation.

Algebraic geometry provides the natural tools to study these structured decompositions because the indecomposable tensors forming a given decomposition are parametrized by projective algebraic varieties: for example, the Segre variety parametrizes all indecomposable tensors with respect to the standard rank, while Veronese varieties collect all symmetric indecomposable tensors. Higher rank tensors are then studied by means of secant varieties. These are very classical objects, and their interest dates back to the beginning of the 20th century when the Italian school started a systematic study of dimensions of such varieties with the works of Palatini, Scorza, and Terracini. Due to its connection with tensors, the interest in secant varieties overcame time and remains strong up to the present day.

Magnifying lens on the identifiability problem

A tensor of a given rank is identifiable if it admits a unique rank decomposition up to reorder and up to scalar multiplication. One of the main advantages of working with tensors instead of matrices is that tensors "very often" admit a unique rank decomposition. Under this perspective, after translating applied problems of different fields in the language of tensors, the uniqueness of the tensor rank decomposition represents a unique way of interpreting the initial data of the corresponding application. Working in the applied fields, one may also be interested in the identifiability of specific tensors. Indeed, when translating an applied problem in the language of tensors, one may be forced to deal with a very specific tensor with a precise structure for reasons related to the nature of the applied problem itself. Knowing how to deal with zero-dimensional schemes is a very useful way to tackle these problems. Check [15, 16] if you want to know more about the identifiability of tensors.

by Sachi Hashimoto

Let $X$ be a nice (smooth, projective, geometrically integral) curve of genus $g > 1$ defined over $\mathbb{Q}$. The problem of describing $X(\mathbb{Q})$, the set of rational points of $X$, has fascinated mathematicians for centuries. Faltings's theorem says that this set is finite. However, there is no general algorithm to determine $X(\mathbb{Q})$. Nonetheless, there are surprising connections between the geometry of the curve and its arithmetic structure. A theme of my research is studying and developing algebro-geometric algorithms to determine the set of rational points on certain classes of curves.

For example, in recent work with Duque-Rosero and Spelier [17], we compare the quadratic Chabauty method of Balakrishnan, Besser, Dogra, and Müller and its geometric analog due to Edixhoven and Lido. The quadratic Chabauty method is an explicit method to determine $X(\mathbb{Q})$ by studying $p$-adic heights on a certain non-abelian analog of the Jacobian of $X$, called a Selmer variety. Geometric quadratic Chabauty is an analog of this method that works directly in the Jacobian and torsors over the Jacobian. We shed light on the connections between the algebro-geometric and cohomological approaches to finding rational points.

by Leonie Kayser

A fundamental problem in computational algebraic geometry is to solve systems of homogeneous polynomial equations with finitely many solutions. The complexity of this problem depends strongly on the Hilbert function of the ideal generated by these equations. In many applications, the ideal is non-saturated. This means it is strictly contained in the vanishing ideal, whose Hilbert function is well understood. This happens, for instance, in tensor decomposition, where we are handed only the generators in a fixed (small) degree. We study the Hilbert function in such cases and deduce new complexity bounds for solving systems of equations.