Aldo Conca (University of Genoa): Resolution of Ideals Associated to Subspace Arrangements
Abstract: The ideal of definition of a linear subspace in a projective space has a very simple structure and hence a very simple free resolution, i.e. a Koszul complex. What can we say for the ideal that defines a finite collection of linear subspaces, a subspace arrangement, in a projective space? Here we can take the intersection of the ideals defining the individual subspaces or their product. For the intersection, the structure of the resolution remains largely mysterious. For the product instead the resolution can be described and it turns out that it is supported on a polymatroid associated with the subspace arrangement. Joint work with Manolis Tsakiris (Chinese Academy of Sciences).
María Angélica Cueto (Ohio University): Lines in the tropics
Abstract: 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.
Wolfram Decker (Technical University of Kaiserslautern): OSCAR – A new Computer Algebra System
Abstract: The OSCAR project develops a comprehensive Open Source Computer Algebra Research system for computations in algebra, geometry, and number theory, with particular emphasis on supporting complex computations which require a high level of integration of tools from different mathematical areas. The project builds on and extends the four cornerstone systems GAP, Singular, polymake, and Antic, as well as further libraries and packages, which are combined using the Julia language. Though still under development, OSCAR is already usable. In this talk, I will give examples of what is possible right now, and outline future lines of development.
Hamza Fawzi (University of Cambridge): Lifts of convex sets
Abstract: A central question in optimization is to maximize, or minimize, a linear function on a given convex set. Such a problem may be easy or hard depending on the geometry of the convex set. In this talk, we will consider the following question: given a convex set, is it possible to express it as the projection of a simpler convex set in a higher-dimensional space? We will focus on the class of polyhedral lifts and spectrahedral lifts where the simplicity of the lift can be naturally quantified, and show how tools from algebra and geometry can be used to obtain obstructions on the existence of simple lifts. We will illustrate these tools by proving separations between various classes of conic optimization paradigms, namely second-order cone optimization, semidefinite optimization, and polynomial optimization.
Jesús Fernández-Sánchez (Polytechnic University of Catalonia): Algebraic and semi algebraic conditions for phylogenetic reconstruction
Abstract: The main goal of phylogenetics is the inference of the evolutionary relationships (phylogenies) between species. These phylogenies are usually represented by means of a tree graph, where leaves represent the current species and the interior nodes the possibly extincted ancestors, and the starting point for inferring such a tree is usually an alignment of nucleotide sequences. However, the impact of phylogenetics is usually beyond the theoretical knowledge of the evolutionary history, and may play a relevant role to determine the origin of pathogens and for the traceability of cancer cells genes among other applications.
The usual approach to model the substitution of nucleotides in an evolutionary process is by means of a Markov process on the tree. The joint probabilities of nucleotides can be computed then as polynomials in terms of the substitution probabilities. In this talk, we will discuss the advantages of taking these probabilities as the parameters of the models (in contrast with the continuous-time models usually considered by biologists) and we will show how this gives way to using tools and results from algebraic geometry and commutative algebra that assist in the design of new methods of phylogenetic reconstruction. We will also see that taking into account the semi algebraic information available in the data can improve the performance of these methods and lead to the design of new ones. Finally, we will show some new results obtained by these methods on real data.
M. Casanellas, J. Fernandez-Sanchez, M. Garrote-Lopez (2021). SAQ: Semi-Algebraic Quartet Reconstruction. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 18(6), 2855–2861
M. Casanellas, J. Fernández-Sánchez, J. Roca-Lacostena (2022), The embedding problem for Markov matrices, to appear in Publicacions Matemàtiques
J. Fernández-Sánchez, M. Casanellas (2016). Invariant Versus Classical Quartet Inference When Evolution is Heterogeneous Across Sites and Lineages. Systematic Biology, 65(2), 280–291
Joachim Jelisiejew (University of Warsaw): Minimal border rank tensors and oMEGA
Abstract: Minimal border rank tensors appear prominently as input for the laser method in complexity theory: finding new such tensors may yield new lower bounds for the famous exponent omega.
These tensors have a rich and complicated geometry and are amenable to a range of methods, from computational, through algebraic, to geometric. In the talk I will review these, focusing on recent developments and open questions. Time permitting, I will also mention how proving some conjectures on the geometric side would allow us to upgrade the laser method further.
Parts of the talk are joint work with Mateusz Michałek and with Joseph Landsberg and Arpan Pal.
Tanja Lange (Eindhoven University of Technology): Elliptic curves for future-proof cryptography
Abstract: Elliptic-curve cryptography is now used in many systems from web browsers, to chat apps, to passports. However, the security of these systems rests on the discrete-logarithm problem, a problem for which we currently know only exponential-time attacks but which can be broken in polynomial time running Shor’s algorithm on a sufficiently large quantum computer.
While such computers do not yet exist today it is high time to change our systems because attackers can record today’s messages and decrypt them once they have a large quantum computer. Post-quantum cryptography is the area of cryptography which studies systems under the assumption that the attacker has a quantum computer.
This talk will explain one branch of post-quantum cryptography, namely isogeny-based cryptography. Isogenies are maps between elliptic curves and they can be used for curves over finite fields to build systems for key exchange, encryption, and signatures.
Claudiu Raicu (University of Notre Dame): Cohomology of line bundles on flag varieties
Abstract: A fundamental problem at the confluence of algebraic geometry, combinatorics and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. I will give an overview of the subject and discuss some recent developments, illustrating some of the difficulties that occur when passing from characteristic zero to positive characteristic.
Marie-Françoise Roy (Institut de Recherche Mathématique de Rennes): Effective methods for Hilbert 17 th problem, real Nullstellensatz and Positivstellensatz
Abstract: Artin’s solution to Hilbert 17 th problem was one of the great achievements of modern algebra, nearly one century ago and real Nullstellensatz and Positivstellensatz were proved several decades later by similar methods. Primitive recursive degree bounds for these problems rely on proof theoretical methods combined with algebraic constructions. Computer algebra, more specifically subresultants, has been the extra ingredient needed to obtain elementary degree bounds. I shall describe the results known at this point and suggest possible future improvements. This talk is based on work of Henri Lombardi, joint work of myself with Henri Lombardi and Daniel Perrucci and a joint research project with Daniel Perrucci.