List of Titles and Abstracts

Daniele Agostini: Catalecticant ideals for secant varieties

The secants of an algebraic variety have a straightforward geometric definition, but their algebraic aspects, such as their equations are in general unknown. There are some natural equations, called catalecticant, that are conjectured to generate the ideal of the secants if the embedding is sufficiently positive. We confirm this conjecture in some cases, by working with the Hilbert scheme of points.

This is joint work in progress with Jinhyung Park.

Ana Cannas da Silva: Toric Real Loci via Moment Polytopes

Any symplectic toric manifold admits anti-symplectic involutions anti-commuting with the toric action. Toric real loci are the lagrangian submanifolds obtained as fixed point sets of such involutions. We use a particular polyhedral description — called a kaleidoscope — for a toric real locus, to understand, in an elementary geometric way, its orientability and Euler characteristic. The kaleidoscope definition relies on the restriction of the moment map to a toric real locus and provides a user-friendly and computationally simple description of that toric real locus.

This is joint work with João Camarneiro

Jan Draisma: Subranks of bilinear maps

The (border) subrank of a bilinear map f : U × V → W is the maximal number of independent scalar multiplications  that can be carried out (respectively, approximately carried out) using f. It is dual to the ordinary rank of f, which is the minimal number of scalar multiplications needed to evaluate f. While ordinary rank has been the subject of much research in applied algebraic geometry, subrank has become a prominent research theme only over the last years. Accordingly, many basic questions are still open. I will discuss some of the recent literature on *generic* (border) subrank over an algebraically closed field, as well as the subrank of bilinear maps over the real numbers. The first topic involves a strengthening of the Hilbert-Mumford criterion. And a sample result from the second topic says that if U = V = W = C^n and f is the component-wise multiplication of complex scalars, regarded as a real bilinear map, then the real subrank of f is n: no more than n pairs of real scalars can be multiplied with a device that multiplies n pairs of complex scalars. 

Based on joint work with Biaggi-Chang-Rupniewski and Biaggi-Eggleston.

Claudia Fevola: KP Solitons: Tropical Curves meet Grassmannians

The KP equation, a partial differential equation describing nonlinear wave motion, has solutions linked to algebraic curves. Solitons, a special class of solutions, arise from rational nodal curves. Kodama and Williams explored real regular solitons and their connection to totally positive Grassmannians. Building on Abenda and Grinevich’s work, I will discuss the interplay between real regular solitons, dual graphs of singular curves, matroids, and cells in the totally positive Grassmannian.

This is based on forthcoming work with Simonetta Abenda, Türkü Özülüm Çelik, and Yelena Mandelshtam.

Hanieh Keneshlou: Hilbert scheme of points of low degree and cactus varieties

Hilbert schemes are fundamental objects in algebraic geometry parameterizing families of ideals in a polynomial ring. Since being introduced by Grothendieck, they have paved the way for the construction of most moduli spaces and have found connections to various research lines, yet their geometry is to be understood. In this talk, we address the study of components of the Hilbert scheme of points of low degrees, and we further discuss the irreducibility of certain Grassmann cactus varieties tightly closed to Hilbert schemes and polynomial ranks.

This is based on a joint work with Maciej Ga lazka and Klemen Sivic.

Diane Maclagan: Tropical toric vector bundles

I will describe a new definition, joint with Bivas Khan, for a tropical toric vector bundle on a tropical toric variety. This builds on the tropicalizations of toric vector bundles, and can be used to define tropicalizations of vector bundles on subvarieties of toric varieties.  I will discuss when these bundles do and do not behave as in the classical setting.

Leonid Monin: Chow quotients and rational smoothness

Many interesting algebraic varieties can be realized as Chow quotients of a projective variety equipped with an action of an algebraic torus. Notable examples include the spaces of complete quadrics and complete collineations, the moduli space of marked stable rational curves, wonderful models of hyperplane arrangements, and many more. In this talk, we will focus on Chow quotients of projective varieties under the action of a one-dimensional torus. In particular, we will describe certain conditions under which the Chow quotient is rationally smooth. As an application of our criterion, we construct rationally smooth models for Richardson varieties. Finally, if time permits, I will explain how these results connect to recent developments in Kazhdan–Lusztig–Stanley theory.

Based on joint work in progress with Mateusz Michalek and Botong Wang.

Rahul Pandharipande: On points on lines and lines on planes

I will discuss aspects of the virtual geometry of the moduli space of n lines on the projective plane. The various points of view on the geometry (KSBA, log GW theory, Chow quotients, matroidal decompositions, and so on ...) all contribute to the study. Beyond the construction of the virtual class, I will present an outline of a theory of descendent integration.

The talk represents joint work in progress with Dan Abramovich and Dhruv Ranganathan.

Marta Panizzut: Positive del Pezzo geometry

Positive geometry is a recent branch of mathematical physics which presents exciting connections with real, complex and tropical algebraic geometry. In this talk, we introduce the topic by developing the positive geometry of del Pezzo surfaces and their moduli spaces. We will analyze their connected components, their likelihood equations and their  scattering amplitudes.

The talk is based on joint work with Early, Geiger, Sturmfels and Yun. 

Javier Sendra Arranz: Hilbert Schemes of points of fold-like curves and its combinatorics

Hilbert schemes of points are important objects in algebraic geometry. In general, the geometry of Hilbert schemes of points is highly challenging. However, for a smooth curve, the Hilbert scheme of points coincides with the symmetric product of the curve. This simplicity disappears when the smoothness condition is dropped, and the geometry becomes significantly more complex. In this talk, we present new results on the geometry of the Hilbert scheme of points on curves with rational n-fold singularities. We determine the irreducible components, analyze their singularities and describe the punctual Hilbert scheme. Furthermore, we explore the rich combinatorial connections between these geometric objects and hypersimplicial complexes.

Joint work with Ángel David Ríos Ortiz.

Johannes Schmitt: Using computer algebra to study intersection theory of moduli spaces

The moduli spaces Mbar_g,n of stable curves are fundamental objects in algebraic geometry, parameterizing algebraic curves of genus g with n marked points. While their full cohomology can be extremely complicated, these spaces contain a rich subring of tautological classes that admits algorithmic descriptions. In this talk, I will demonstrate how the SageMath package admcycles enables systematic exploration of the intersection theory of these moduli spaces through computer algebra. I will showcase how computational experiments with admcycles have led to new mathematical discoveries across several directions: finding explicit formulas for leaky double Hurwitz numbers through pattern recognition, discovering counterexamples to long-standing conjectures (including non-formulas for lambda_g and the failure of the Gorenstein conjecture in compact type), and exploring deep connections between seemingly unrelated cycle classes such as strata of differentials and Witten's r-spin class. The talk will include live demonstrations of the package's capabilities, from basic calculations with psi, kappa and lambda-classes to more sophisticated computations involving double ramification cycles, hyperelliptic loci, and strata of differentials. Throughout, I will emphasize how computer-assisted exploration can guide mathematical intuition, both in formulating new conjectures and in constructing counterexamples to existing ones.

Rainer Sinn: Spurious local minima and syzygies

I will explain the Burer-Monteiro approach to sum-of-squares relaxations in polynomial optimization, a nonlinear relaxation of a convex optimization problem motivated by complexity considerations. The main question in the talk is about (the existence of) local minima (versus the global minimum) of this optimization problem. To study the question, we use algebraic techniques in a geometric setup, mainly syzygies in homogeneous coordinate rings of projective varieties of minimal degree. 

This is joint work with Greg Blekherman, Mauricio Velasco, and Shixuan Zhang.

Lucas Slot: Computational complexity of the moment-SOS hierarchy for polynomial optimization

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O'Donnell (2017) and later Raghavendra & Weitz (2017) show, there exist examples where the SOS-representations used in the hierarchy have exponential bit-complexity. In this talk, we discuss several recent works that formulate sufficient conditions on polynomial optimization problems that guarantee efficient computation of the moment-SOS relaxations is possible.

Based on joint works with  Sander Gribling, Sven Polak, and Marilena Palomba, Luis Felipe Vargas, Monaldo Mastrolilli.

Bernd Sturmfels: Maximal Mumford Curves from Planar Graphs

We discuss a recent article with Mario Kummer and Raluca Vlad in real non-archimedean geometry.  A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves whose graph is planar.

Kexin (Ada) Wang: Computing Arrangements of Hypersurfaces

In this talk, I will present a Julia package, HypersurfaceRegions.jl, for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in R^n. I will demonstrate the use of the package through various examples. The package is based on the method from the paper "Smooth Connectivity in Real Algebraic Varieties" by Cummings et al, which involves computing critical points of rational equation. I will discuss some theory about the number of critical points expected.

Lena Weis: Shellings of tropical hypersurfaces

The shellability of the boundary complex of an unbounded polyhedron is investigated. For this concept to make sense, it is necessary to pass to a suitable compactification, e.g., by one point. This can be exploited to prove that any tropical hypersurface is shellable. Under the hood there is a subtle interplay between the duality of polyhedral complexes and shellability. Translated into discrete Morse theory, that interplay entails that the tight span of an arbitrary regular subdivision is collapsible, not not shellable in general.

This talk is based on joint work with George Balla and Michael Joswig.