Conference

Abstracts

Andreas Bernig (Goethe-Universität Frankfurt): Integral geometry beyond convexity 

We give a survey of the algebraic structures on convex valuations, yielding to a large set of integral-geometric formulas, in particular to the kinematic formulas. We also sketch the more general theory of valuations on manifolds and its applications to kinematic formulas on isotropic Riemannian manifolds. We then motivate the use of more general classes of sets, like sets of positive reach and subanalytic sets. The latter case is joint work with Vadim Lebovici and will be explained in more detail in his talk. 

Greg Blekherman (Georgia Tech): Eigenvalues of Locally Positive Semidefinite Matrices, Grassmanian Optimization and Equiangular Tight Frames

A symmetric matrix is k-Locally Positive Semidefinite if all of its k by k submatrices are positive semidefinite. I will explain that understanding eigenvalues of these matrices has connections to natural optimization problems on the Grassmanian, existence of real equiangular tight frames, and simple-looking geometric optimization problems. In some cases, I will present solutions, but many problems will be left for the audience. Joint work with Jose Acevedo, Sebastian Debus, Seokbin Lee and Cordian Riener.

Peter Bürgisser (TU Berlin): Probabilistic intersection rings of homogeneous spaces 

Suppose X_1, ….,X_s are submanifolds of a compact homogeneous space M, in general position, with finite intersection. We may think of M as a real or complex projective space, or a Grassmann manifold. The signed count of intersection points can be described in terms of the real cohomology algebra of M: intersection corresponds to multiplication. Our goal is to understand the expected (unsigned) count of intersection points. Integral geometry provides methods to compute this when the X_i are moved at random with respect to the Haar measure. We motivate and outline the functorial construction of a commutative Banach algebra, whose multiplication mirrors the intersection of randomly moved submanifolds. We call it the probabilistic intersection algebra of M. The elements of M are classes of zonoids (certain convex bodies) in the exterior algebra (we call them Grassmann zonoids) and their multiplication is induced by the wedge product. This establishes a connection to the theory of connect bodies. The probabilistic intersection algebra contains the cohomology algebra as a direct summand. There is an intimate relationship with Alesker’s multiplication of valuations of convex bodies. So far, we understand the probabilistic intersection algebra  of real and complex projective space. We took first steps towards analysing the probabilistic intersection algebra of real Grassmann manifolds, which is infinite dimensional. There is a close connection to harmonic analysis. (Joint work with Paul Breiding, Antonio Lerario and Leo Mathis.) 

Georges Comte (Université Savoie Mont Blanc): Some nonarchimedean integral geometry: the motivic Vitushkin variations

I will explain how, in a joint work with I. Halupczok, using motivic integration theory and some specific regular stratification, we introduce two new objects in the framework of definable nonarchimedean geometry. The first one is a convenient partial preorder on the set of constructible motivic functions, and the second one is an invariant, denoted V0, which is a nonarchimedean substitute for the number of connected components. Based on V0, we obtain the existence of higher nonarchimedean substitutes of real measure geometric invariants Vi, called the Vitushkin variations. We then establish the nonarchimedean counterpart of a real inequality involving the metric entropy and our invariants Vi. 

Emil Horobeţ (Sapientia Hungarian University): The problem of real vs complex in distance optimization 

In this talk, we study several aspects of metric algebraic geometry where we have to make a choice (either of the metric or the variety, etc.), whether we work over the reals or over the complex. These choices lead to several problems like complexifications, isotropicity, realifications and non-differentiability. We present these problems and provide the general framework of discriminants to tackle them. 

Daniel Hug (Karsruhe Institute of Technology): Integral meets stochastic geometry

The stationary Boolean model Z is a fundamental concept and benchmark model in stochastic geometry that describes random spatial structures formed by the union of random shapes placed at random positions in space, driven by a Poisson process. A major problem consists in retrieving information about the underlying particle process X from knowledge about (observations of) the union set Z. In dimensions 2 and 3 it was shown (Weil 2001) that the intensity of the particle process X is determined by certain local quantities such as the densities of mixed volumes of Z. In the talk, we explain how decomposition results for mixed volumes and mixed measures of translative integral geometry can be combined with Alesker's representation theorem for translation invariant valuations to settle the problem in general dimensions (Hug, Weil 2019). Boolean models have been explored also in hyperbolic space. The derivation of asymptotic formulas for mean values and covariances of intrinsic volumes motivates the derivation of new integral geometric formulas for which no Euclidean analogue is known (Hug, Last, Schulte '25) and which will be discussed in the last part of the talk. 

Nidhi Kaihnsa (University of Copenhagen): Distance Optimisation in Polyhedral Norms

Given a set, X, of points in space, R^n, and a metric, a Voronoi diagram partitions R^n  into regions by identifying all the points closer to a fixed point in the set with respect to the given metric. The region corresponding to that point is called its Voronoi cell. Set of points in the space whose distance to X is optimised by (at least) two different points in X define its medial axis. In this talk I will present results on Voronoi cells of varieties with respect to the polyhedral norms. In particular, I will discuss the stratification of varieties based on the Voronoi cones and the description and computations of the medial axis. Time permitting, I will also talk about stratification of Grassmannian via polyhedral norms. This is based on the joint works with Duarte, Lindberg, Torres, and Weinstein.

Kaie Kubjas (Aalto University): Geometry of low nonnegative rank matrix completion

We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most r, where r is some natural number. For nonnegative matrices of rank one and two, nonnegative rank is equal to rank. This is our main tool for studying nonnegative rank-1 and -2 completions.  We show that a partial matrix with nonnegative entries has a nonnegative rank-1 completion if and only if it has a rank-1 completion. This is not true in general when r ≥ 2. For 3×3 matrices, we characterize all the patterns of observed entries when having a rank-2 completion is equivalent to having a nonnegative rank-2 completion. For nonnegative matrices of rank at least three, the nonnegative rank can differ from the usual rank, and therefore, studying the completion problem becomes more challenging. We introduce a geometric characterization for nonnegative rank-r completion employing families of nested polytopes which generalizes the geometric characterization for nonnegative rank, and then use this characterization to obtain first results about nonnegative rank-3 completion. This talk is based on joint work with Lilja Metsälampi.

Salma Kuhlmann (Universität Konstanz): Closure of cones in semi-normed real algebras and integral representations of characters; with a view towards algorithms in polynomial optimisation

Let A be a real unital commutative semi-normed algebra.  We let C be a cone in A and K a subset (closed in the product topology) of the character space X(A) of A. This talk consists of three parts. Firstly we analyse algebraically the following topological equality: "The closure of C in A = the positivity cone of K in A". Secondly we interpret analytically this equality, thereby parametrising the topological dual of A via (integration along) Radon measures (with support K) on X(A). Thirdly we give specific applications to solve three concrete moment problems (which play a prominent role in e.g. algorithms for polynomial optimisation).

Vadim Lebovici (Sorbonne University): Additive kinematic formulas for subanalytic sets

The fundamental theorem of algebraic integral geometry (FTAIG) proven by Bernig and Fu states that the additive kinematic formulas for convex bodies are dual to an operation of convolution on the space of valuations. I will explain how we use this result with Andreas Bernig to prove additive kinematic formulas for subanalytic subsets of the Euclidean space and of the 3-sphere. The Minkowski sum of convex bodies has to be generalized by a notion of convolution introduced by Schapira in the late 80s using Euler characteristic computations. The key is then a comparison of this notion of convolution with the convolution on valuations, providing a positive answer to a question left opened by Alesker. 

Julia Lindberg (Georgia Tech): The maximum likelihood degree of sparse polynomial systems

In this talk, we consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the model. We give a convex geometric formula for the maximum likelihood degree and discuss algorithmic implications of this. If time permits, I will discuss analogous results for algebraic degrees.

Jana Maříková (University of Vienna): Measures in non-standard o-minimal structures 

O-minimal structures aim to capture the notion of being geometrically tame, with the archetypal example being the semialgebraic sets.  For instance, definable functions are differentiable on a large subset of their domain and the derivatives are again definable in the same structure.  The situation regarding integration, or just the existence of well-behaved measures is less clear. While the theory of an o-minimal structure that originates on the reals (in a sense that will be made precise), admits a nice measure on its definable sets, the existence of such a measure is unknown in general.  We survey what is known on this topic, give some partial answers, and touch upon the use of measures in model theory in general. This talk can be viewed as a very loose continuation of Tamara Servi's minicourse in the previous week, though knowledge of its contents will not be assumed. 

Gaiane Panina (St. Petersburg State University): A new proof of the Milnor – Wood theorem, and beyond

The Milnor--Wood inequality states that if a (topological) oriented circle bundle over an orientable surface of genus g has a smooth transverse foliation, then the Euler class of the bundle satisfies |E| <= 2g-2. We give a new proof of the inequality based on a (previously proven by the authors) local formula which computes E from the singularities of a quasisection and present some more applications of this approach. (Based on a joint work with Timur Shamazov and Maksim Turevskii)

Raman Sanyal (Goethe-Universität Frankfurt): Zonotopal Wachspress coordinates

Every polytope is the canonical projection of a simplex. Fibers over points correspond to representations in terms of vertices, and sections of the projection yield generalized barycentric coordinates. Wachspress coordinates are particular sections with many interesting algebraic and geometric properties. Zonotopes are Minkowski sums of line segments or, equivalently, projections of cubes. In this talk, I will discuss zonotopal Wachspress coordinates, which give natural representations of points in zonotopes as sums. Zonotopal Wachspress coordinates enjoy similar nice properties that reflect the geometry of the hyperplane arrangement associated with the collection of segments. This is work in progress with Tom Baumbach and Martin Winter.

Luca Sodomaco (MPI-MIS Leipzig): Distance degrees and distance polynomials

The (Euclidean) distance degree of an algebraic variety X is among the most studied topics in Metric Algebraic Geometry. It equals the number of complex critical points of the squared distance function from a generic point u (a polynomial quadratic objective function) restricted to the locus of nonsingular points of X. The univariate polynomial (in an unknown t) whose roots are the squared distances between u and a critical point x is the (Euclidean) distance polynomial of X. The coefficients of this polynomial are themselves polynomials in the coordinates of u. Evaluating the distance polynomial of X at some value t=ε gives the equation of the ε-offset hypersurface of X. In this talk, we will overview the most relevant properties of the coefficients of the distance polynomial of X. Time permitting, we will discuss an ongoing project with Emil Horobeț on the polynomial map that sends a point u to the tuple of the coefficients of the distance polynomial of X evaluated at u.