Welcome to the Weekly A3G Seminars of our group in Applied, Algorithmic and Combinatorial Algebraic Geometry.

Seminar room: B.02.18 (Maths building). 

Organizers:  Fatemeh Mohammadi, Emiliano Liwski and Sebastian Seemann

Registration:

Please email Fatemeh if you are interested in attending the meeting or have any questions.

Spring Semester 2025

• June 16, 2025: 10:00  Daisie Rock (KU Leuven)

When can we stop assuming "discrete"?: I will cover some of my research in converting discrete concepts into not-discrete concepts. I will point out the places where we still use discreteness in some way or if we can remove the assumption all together. The topics include quiver representations, cluster categories, associahedra, triangulations of polygons, and some lattice theory.

• June 5, 2025: 11:00 Ilaria Colazzo (University of Leeds)

Classifying Set-Theoretic Solutions to the Pentagon Equation via Matched Pairs of Groups: In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, revealing an unexpected connection with matched pairs   of groups. We will begin by introducing the necessary definitions and concepts. The focus will then shift to the core components: the  irretractable solutions. We will explore how these solutions are related to matched pairs of groups. Finally, I will explain how each  irretractable solution can be extended to recover all bijective solutions.

• June 5, 2025: 10:00 Leandro Vendramin (Vrije Universiteit Brussel)

What is a skew brace? Skew braces are ring-theoretical algebraic structures to study non-degenerate set-theoretic solutions of the Yang–Baxter equation. A typical example of a skew brace is a Jacobson radical ring. In this talk, we will discuss the basic properties of skew braces and how these structures are related to the celebrated Yang–Baxter equation. We will also discuss connections with groups and non-commutative rings.

• May 19, 2025:  11:00 Kevin Piterman (Vrije Universiteit Brussel)

Posets of decompositions into subobjects: In mathematics, we usually try to decompose a given object into smaller pieces (subobjects) such that there is a "nice" way to assemble them again and recover the structure of the original object. This is the case, for instance, when we decompose a module into indecomposable summands, a topological space into its connected components, or even a group into free factors or direct product factors. Moreover, some algebraic and even cohomological properties of the automorphism group of our original object may be recovered (in some way) from the automorphism group of these smaller subobjects. In many cases, to achieve these results, one needs to understand the topology and combinatorics of certain posets of subobjects and decompositions.

In this talk, we discuss a general framework that encompasses some of the constructions used to approach these problems in a unified way. Roughly speaking, we fix an object, a certain product of subobjects (such as direct sum for modules, or free product for groups), and take the posets of all those subobjects that are complemented with respect to this product. From this construction, we can define posets of (partial) decompositions and establish general combinatorial and topological properties without resorting to the specific choice of our initial object and product. Then, we will specialise these constructions to particular examples such as matroids, vector spaces with non-degenerate forms, or free groups, where there are still many open questions.

• May 19, 2025:  10:00 Emine Yıldırım (University of Leeds)

Frieze patterns in algebra: Friezes are in the intersection of representation theory of algebras, cluster algebras and surface combinatorics. This leads to interesting questions that can be solved combining techniques from all these areas. In our joint work with K. Baur, L. Bittman, E. Gunawan, and G. Todorov, we investigate friezes coming from affine type D by looking at them from different perspectives and further we prove a certain property, called "growth coefficient", for these friezes.

• May 5, 2025: 16:00 (online) Sarah Lumpp (TU Munich) as part of the Algebraic Statistics Meetings joint with Aalto and Helsinki University

Introduction to graphical continuous Lyapunov models: Stationary distributions of multivariate diffusion processes have recently been proposed as probabilistic models of causal systems in statistics and machine learning. By assuming each observation to arise as a one-time cross-sectional snapshot of a temporal process in equilibrium, they allow to model dependence structures that may include feedback loops. Specifically, the graphical continuous Lyapunov model consists of steady-state distributions of multivariate Ornstein-Uhlenbeck processes where sparsity assumptions on the drift matrices are represented with a directed graph. These distributions are Gaussian with covariance matrices that are parametrized as solutions of the continuous Lyapunov equation. In this talk, I will give an introduction to the model with a focus on the conditional independence structure as well as identifiability of the drift parameters in specific cases.

• May 5, 2025: 11:00 Lisa Vandebrouk (KU Leuven)

A generating set for the matroid ideal and a decomposition of the circuit variety: The matroid ideal is the ideal of the Zariski closure of the realization space of matroids. In this talk, we will find a set of generators for the matroid ideal, up to radical, for matroids of rank 3 including cactus matroids and the Pappus and Pascal matroid. Moreover, we will find an irredundant, irreducible decomposition of the circuit varieties of various matroids of rank 3.

• April 28, 2025: 11:00 Sebastian Debus (Chemnitz University)

The Wonderful Geometry of the Vandermonde Map: The Vandermonde map, consisting of the first d power sum polynomials, plays a fundamental role in algebraic geometry and polynomial optimization, particularly in the study of nonnegative symmetric polynomials and trace inequalities of real symmetric matrices. This talk explores the rich geometry underlying the image of the probability simplex under the Vandermonde map. This set is called a Vandermonde cell. We investigate the boundary structure of Vandermonde cells, their limits, and their surprising links to cyclic polytopes.

This is joint work with Jose Acevedo, Greg Blekherman, and Cordian Riener.

• March 31, 2025: 10:00 (online): Nobukazu Kowaki (Osaka University)

Tropical hyper plane arrangements and combinatorial mutations of matching field polytopes of Grassmannians: A sequence of combinatorial mutations of matching field polytopes preserves the property of giving rise to a toric degeneration of Grassmannians. In this talk, we find a way to check that two matching field polytopes are combinatorial mutation equivalence using tropical hyperplane arrangements

• March 24, 2025: 11:00, B.02.18: Matthias Orth (KU Leuven)

Free resolutions: computation and structural analysis: Free resolutions are an important tool in commutative algebra for studying the structure of modules over polynomial rings and their quotient rings. In this talk I will first explain a method for computing free resolutions via Gröbner bases and a generalized Schreyer's Theorem. We discuss a directed graph associated with this method of computation. As the method is based on Gröbner bases, there is a natural connection with monomial ideals. I will argue why it is important to study the resolutions of monomial ideals that satisfy combinatorial stability conditions, and why it can be useful to perform the analysis of free resolutions in the right coordinate system. If time permits, I will present some applications that have appeared in recent and ongoing collaborations on the topics Lefschetz properties, Rees ideals of monomial curve parametrizations, and topological data analysis.

• March 17, 2025: 13:00, B.05.100: Pardis Semnani (University of British Columbia)

Prime decomposition of a class of determinantal ideals with applications to the intersection axiom in conditional independence statements: We study a class of determinantal ideals associated with the conditional independence statements in the presence of hidden variables. We present a decomposition for such ideals, and show that this decomposition is prime assuming certain constraints on the size of the state space of the observed variables. We also conjecture a generalization of this prime decomposition that allows for more relaxed size constraints. As an application of this prime decomposition, a generalization of the intersection axiom is achieved. This talk is based on ongoing work with Yulia Alexandr, Kristen Dawson, Hannah Friedman, Fatemeh Mohammadi, and Teresa Yu.

• March 17, 2025: 11:00, B.02.18: Filip Jonsson Kling (Stockholm University)

The strong Lefschetz property via Gröbner bases and lattice paths: In this talk, we will sketch part of a new proof that an algebra defined by the ideal generated by the squares of all variables of a polynomial ring do have the strong Lefschetz property. This will be done by finding a nicely described Gröbner bases for a family of related ideals involving elementary symmetric polynomials. In particular, we will show how the initial terms of the Gröbner basis elements enumerate a family of lattice paths and how this enumeration will aid us in establishing the strong Lefschetz property. If time permits, I will say some words on how Catalan numbers show up and what this has to do with relations of powers of general linear forms.

This is based on joint work with Samuel Lundqvist, Fatemeh Mohammadi, Matthias Orth and Eduardo Sáenz-de-Cabezón.

• TBA 2025: B.02.18: Daisie Rock (KU Leuven)

Applications of quiver representations and some generalisations: I will present a short introduction to representations of quivers (directed graphs) and some of their applications followed by my work, which generalises these ideas. The representations of a quiver are equivalent to the representations of the corresponding path algebra. Many finite-dimensional algebras are quotients of path algebras, making the study of path algebras and their representations interesting in their own right. There are also applications to topological data analysis (directly), and high energy physics (not so directly). I will point at these applications as well. Finally, I will present some of the things I have worked on, which generalise the applications above to a context that drops the finite/discrete assumptions that typically come with quivers.

• March 06, 2025: 13:30, B.02.14: (Special session) 

Erdenbayar Bayarmagnai:  Algebraic tools for computing loop invariants:

Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, generating invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and the assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case, where the polynomials can have arbitrary degrees. 

Using tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants within a given vector subspace, for a branching loop with nondeterministic conditional statements. These algorithms combine linear algebra subroutines with computations on polynomial ideals. They differ depending on whether the initial values of the loop variables are specified or treated as parameters. Additionally, we present a much more efficient algorithm for generating polynomial invariants of a specific form, applicable to all initial values. This algorithm avoids expensive ideal computations.

     Matthias Orth: Quasi-stable ideals, free resolutions, and Hilbert schemes:

To start the talk, I will define the notions of quasi-stability and Pommaret bases for monomial ideals. After a generic linear change of coordinates, any homogeneous polynomial ideal is in quasi-stable position - this roughly means that one has a Pommaret basis also for the polynomial ideal. I will discuss the syzygies and the free resolution induced by such a basis, their application to the study of homological invariants, and mention some generalizations of this construction.

In a second part of the talk, I will introduce the notion of marked bases - which can be seen as term-ordering-free Pommaret bases. In the zero-dimensional case, marked bases are closely related to border bases. We will see how, by computing certain syzygies of a marked basis, one can parameterize a marked scheme and obtain an open cover of a Hilbert scheme via such marked schemes.

Sebastian Seemann: Vandermonde cells, positive geometries, boundaries: 

Vandermonde cells are semi-algebraic sets obtained as the image of a polynomial map. We study these cells from the perspective of positive geometry, which warrants an analyis of their boundary structure and defining equations. In particular, all planar Vandermonde cells are positive geometries and we provide a recursive formula for their canonical forms. In higher dimensions, the computations become significantly more difficult and we provide neccessary conditions the boundaries have to satisfy for the Vandermonde cells to be a postive geometry.  

Emiliano Liwski: An algorithm to identify the set of minimal matroids in dependency posets:

We present an algorithm for identifying the minimal matroids associated with a given point-line configuration M. These matroids correspond to the smallest extensions of M in the dependency order. Applying this algorithm, we obtain the irreducible decomposition of the circuit varieties arising from point-line configurations.

• March 05, 2025: 14:30, B.02.18: Teresa Krick (Universidad de Buenos Aires)

An Effective Positivstellensatz over the Rational Numbers for Finite Semialgebraic Sets: This talk belongs to the field of computer algebra, and has its roots in Hilbert's 17th problem and its further developments. It is motivated by questions in semidefinite programming. I will present recent joint results with Lorenzo Baldi and Bernard Mourrain, on the exact representation of multivariate integer polynomials that are nonnegative on finite semialgebraic sets, described by means of a zero-dimensional ideal, as sums of squares of rational polynomials. We provide existential results for the strictly positive case and a sufficient condition for the nonnegative case, and degree and height (i.e. bit-size) bounds.

• February 10, 2025: 10:00, B.02.18: Raphael Bennett-Tennenhaus (Bielefeld University) 

Semilinear clannish algebras: Clannish algebras are certain quotients of path algebras introduced by Crawley-Boevey. Each relation is either a path, or a quadratic polynomial in a loop that factors with distinct roots over the ground field. I will discuss a larger class of rings. On the one hand, the more general notion of a semilinear clannish algebra retains the property of having a tame module category.  On the other hand, the definition specifies to interesting examples of rings. By permitting irreducible quadratics, we recover representations arising in work of Geuenich and Labardini-Fragoso. By allowing arrows to skew scalars by automorphisms, we recover representations arising in work of Kottwitz and Rapoport. This talk is based on joint work with Crawley-Boevey (2204.12138) and joint work with Labardini-Fragoso (2303.05326).

• February 5, 2025: 13:00, Online: Joris Koefler (MPI, Leipzig)

Taking the Amplituhedron to the Limit: The amplituhedron is a semialgebraic set given as the image of the non-negative Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters, namely the number of particles n, to infinity. We study this limit amplituhedron for m=2 and any k, relating to the number of negative helcity particles. We determine its algebraic boundary in terms of Chow hypersurfaces. This hypersurface in the Grassmannian is stratified by singularities in terms of higher order secants of the rational normal curve. In conclusion, we show that the limit amplituhedron is a positive geometry with a residual arrangement that is empty.

Fall Semester 2024

• October 15, 2024:  Emiliano Liwski

An algorithm to identify the set of minimal matroids in dependency posets: We introduce an algorithm to identify the minimal matroids associated with a given point-line configuration M. These minimal matroids represent the smallest extensions of M in the dependency order. Utilizing this algorithm, we determine the irreducible decomposition of the circuit varieties associated with point-line configurations.

• October 22, 2024: Sebastian Seemann  

How to stab a Polytope: We discuss the semi-algebraic set of k-dimensional linear subspaces in the Grassmannian that intersect a Polytope. We discuss a decomposition into stabbing chambers and relations to sign vectors associated to the (n-k-1)-dimensional faces of the polytope. In particular, we will discuss how these sign vector characterize regions in the Schubert-divisor complement of the Grassmannian. 

• October 28, 2024: Erdenebayar Bayarmagnai

Loop invariants in algebraic geometry: Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, the generation of invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case where the polynomials exhibit arbitrary degrees.

Applying tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants for a while loop, up to a specified degree. These algorithms differ based on whether the initial values of the loop variables are given or treated as parameters. Furthermore, we introduce various methods to address cases where the algebraic problem exceeds the computational capabilities of our methods. In such instances, we identify alternative approaches to generate specific polynomial invariants.

• November 12, 2024: Rémi Prébet 

Connectivity in Real Algebraic Sets: Algorithms and Applications: Computational real algebraic geometry, positioned at the interface of mathematics and computer science, addresses algorithmic problems on real solution sets to systems of polynomial constraints with real coefficients. 

I will start by showing how fundamental algorithmic subroutines can be combined to solve challenging problems in robotics, such as deciding cuspidality of mechanisms, with potential for further applications in differential equations depending on the audience's interest. Next, I will present recent algorithmic improvements on the most challenging subroutines: answering connectivity queries in real algebraic sets. This introduces the notion of roadmaps, which reduce the problem to studying real algebraic curves, leading us to focus on their connectivity properties, from an algorithmic point of view. Throughout the presentation, I will showcase original implementations leveraging the latest software developments. 

     This talk gathers joint works with D.Chablat, N.Islam,  A.Poteaux, M.Safey El Din, D.Salunkhe, É.Schost and P.Wenger.

• November 25, 2024: Emiliano Liwski

An algorithm to identify the set of minimal matroids in dependency posets II: Building on my previous talk, we explore potential extensions of the algorithm, expanding from rank three to higher rank matroids. These minimal matroids correspond to the smallest extensions of a matroid in the dependency poset.

• December 2, 2024: Fatemeh Mohammadi

Matroids in Applied Algebraic Geometry and Algebraic Statistics: Matroids are combinatorial structures that generalize various notions of independence. Motivated by problems of realizability and causality in statistics, we study the varieties associated with matroids, which can be reducible and exhibit arbitrary singularities. We focus on several families of matroids that arise in statistics. I will discuss these matroids and their connections to finding minimal matroids for hypergraphs in general. As an application, we will explore the intersection axiom for conditional independence (CI) models in the presence of hidden variables. 

• December 16, 2024: Sebastian Seemann

Positive geometries and Vandermonde cells: In this talk I will review the definitions and standard examples of positive geometries. After discussing polytopes and some generalisations we continue by defining Vandermonde cells and fitting them into the framework of positive geometry. For this, the standard definition of positive geometries needs to be relaxed and in order to determine a canonical form we need implicit equations of the boundary, which we deduce from a theorem going back to 1959.