Lionel Lang (Wednesday, 13:40–14:40)
Galois group of systems of polynomial equations

In this talk, I want to report on recent progress in computing the Galois group of systems of polynomial equations with prescribed supports. Already in the case of two variables, we do not yet have a complete answer. By gathering evidence from various univariate setups, I will attempt to explain what that answer might be. The methods used will be topological and derived from toric and tropical geometry. This talk is intended for a broad audience. (Joint work with A. Esterov.)

Claudia Yun (Wednesday, 15:00–16:00)
The topology of the moduli spaces of tropical unramified p-covers

Tropical geometry is a technique that allows us to study algebro-geometric objects through combinatorics. Let p be a prime number. The moduli space M_{g, Z/p} parametrizes isomorphism classes of étale (Z/p)-Galois covers of smooth curves of genus g. It is itself an étale covering of M_g, the moduli space of smooth curves of genus g.

On the other hand, an abstract tropical curve is a decorated metric graph. We study an associated moduli space of unramified tropical (Z/p)-covers of curves of genus g. Its rational homology is conjecturally identified with the top weight cohomology of M_{g, Z/p}. When g > 2, we identify a nested sequence of contractible loci in the moduli space of tropical covers and show that it is simply connected. In the case g = 2, we also determine the homotopy type of the tropical space, showing that it is contractible for p = 2 and 3 and is a wedge of spheres for larger p. This is joint work with Yassine El Maazouz, Paul Alexander Helminck, Felix Röhrle, and Pedro Souza.

Marta Panizzut (Wednesday, 16:10–17:10)
Positive geometry of del Pezzo surfaces

Positive geometry is a recent branch of mathematical physics which present 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.

Nancy Abdallah (Thursday, 10:00–11:00)
Nets in P2 and Alexander Duality

A net in the projective plane is a configuration of lines A and points X satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac–Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid M and rank r, we associate a monomial ideal (a monomial variant of the Orlik–Solomon ideal) to the set of flats of M of rank at most r. In the context of line arrangements in P2, applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.

Jakob Palmkvist (Thursday, 11:20–12:20)
Non-associative structures in extended geometry

I will present a generalisation of vector fields on a vector space, where the vector space is generalised to a highest-weight module of a semisimple finite-dimensional Lie algebra satisfying a certain condition. The generalised vector field is an element in a non-associative superalgebra defined by the module and the Lie algebra. Also the Lie derivative of a vector field with respect to another is generalised and expressed in a simple way in terms of this superalgebra. It reproduces the generalised Lie derivative in the general framework of extended geometry, developed in collaboration with Martin Cederwall (arXiv:1711.07694 External link, opens in new window.). In special cases it reduces to the generalised Lie derivative in double and exceptional field theory, which unify diffeomorphisms with gauge transformations in supergravity theories.

Stefan Wagner (Thursday, 13:40–14:40)
Geometric structures, C*-algebras, and applications to T-duality

In this presentation, we provide an introductory overview of the interplay between geometric structures and C*-algebras, focusing on their applications to T-duality in mathematical physics – a duality arising in string theory and noncommutative geometry. We begin by examining the classical geometric framework of fiber bundles, emphasizing principal bundles and gerbes. The discussion then extends to the role of C*-algebras in modern geometry, specifically to the noncommutative geometry of principal bundles. Finally, we suggest potential directions for future research inspired by recent developments within the mathematical framework of T-duality, including the exploration of non-associative principal bundles.

Alan Sola (Thursday, 15:00–16:00)
Stable polynomials and ideals of admissible numerators

Given a polynomial p with no zeros in the polydisk, or equivalently the poly-upper half plane, we study the problem of determining the ideal of polynomials q having the property that the rational function q/p is bounded near a boundary zero of p. We give a complete characterization of this ideal in several important special cases, and we construct several illuminating examples to complement our results.

This reports on joint work with K. Bickel, G. Knese, and J. E. Pascoe.

Dag Nilsson (Friday, 10:00–11:00)
Model equations for water waves with an emphasis on full-dispersion models

Model equations are a common way to describe phenomena in fluid mechanics for certain physical settings. Examples of model equations are the KdV equation, NLS equation, Whitham equation, KP equations. In this talk I will start by giving giving a brief introduction to model equations, describing in particular what is meant by full-dispersion model equations. After this I will go into more detail about a specific model equation, the full dispersion Kadomtsev–Petviashvili (FDKP) equation

u_t + m(D)u_x + 2uu_x = 0,

where D = –i(∂_x, ∂_y), β is a parameter describing the strength of the surface tension and

m(D) = (1 + β|D|^2)^(1/2)[tanh(|D|)/|D|]^(1/2)[1 + 2D_2^2/D_1^2]^(1/2).

This is a model equation describing three-dimensional long waves of small amplitude. The FDKP equation is a fully dispersive version of the classical KP equation, similar to how the Whitham equation is a fully dispersive version of the KdV equation. The solitary waves they found can be approximated by rescalings of KP-solitary waves.

I will consider the weak surface tension regime (β < 1/3) and describe how to prove existence of solitary wave solutions for the FDKP-equation. The proof is variational and relies upon a series of reductive steps which transform the FDKP-functional to a perturbed scaling of the Davey–Stewartson functional, for which solitary waves are found.

This talk is based on a joint work with Mats Ehrnström (NTNU) and Mark Groves (Saarland University).

Per Enflo (Friday, 11:20–12:20)
Construction of Invariant Subspaces for Operators on Hilbert Space

I will present a method to construct invariant subspaces – non-cyclic vectors – for a general operator on Hilbert space. It represents a new direction of a method of "extremal vectors", first presented in Ansari–Enflo [1]. One looks for an analytic function l(T) of T, of minimal norm, which moves a vector y near to a given vector x. The construction produces for most operators T a non-cyclic vector, by gradual approximation by almost non-cyclic vectors. But for certain weighted shifts, almost non-cyclic vectors will not always converge to a non-cyclic vector. The construction recognizes this, and when the construction does not work, it will show, that T has some shift-like properties. And for those T, one uses the information obtained to produce non-cyclic vectors.

The method also leads to problems and conjectures in analysis, which may be of interest in themselves.

Reference:
[1] S. Ansari, P. Enflo, "Extremal vectors and invariant subspaces", Transactions of Am. Math. Soc. Vol. 350 no. 2, 1998, pp. 539–558.