Salma Kuhlmann’s research while affiliated with University of Konstanz and other places

We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.

This workshop was dedicated to the newest developments in real algebraic geometry and their interaction with convex optimization and operator theory. A particular effort was invested in exploring the interrelations with the Koopman operator methods in dynamical systems and their applications. The presence of researchers from different scientific communities enabled an interesting dialogue leading to new exciting and promising synergies.

We formulate a class of nonlinear {evolution} partial differential equations (PDEs) as linear optimization problems on moments of positive measures supported on infinite-dimensional vector spaces. Using sums of squares (SOS) representations of polynomials in these spaces, we can prove convergence of a hierarchy of finite-dimensional semidefinite relaxations solving approximately these infinite-dimensional optimization problems. As an illustration, we report on numerical experiments for solving the heat equation subject to a nonlinear perturbation.

In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplications nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert's 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of both the SOS and the SONC cone. This follows by exploiting a new necessary condition for membership in the SONC cone.

We develop a first-order theory of ordered transexponential fields in the language $\{+,\cdot,0,1,<,e,T\}$, where e and T stand for unary function symbols. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural valuation. We establish necessary and sufficient conditions on the value group of an ordered exponential field (K,e) to admit a transexponential function T compatible with e. Moreover, we give a full characterisation of all countable ordered transexponential fields in terms of their valuation theoretic invariants.

The cone $\mathcal{P}_{n+1,2d}$ ($n,d\in\mathbb{N}$) of all positive semidefinite (PSD) real forms in n+1 variables of degree 2d contains the subcone $\Sigma_{n+1,2d}$ of those that are representable as finite sums of squares (SOS) of real forms of half degree d. In 1888, Hilbert proved that these cones coincide exactly in the Hilbert cases (n+1,2d) with n+1=2 or 2d=2 or (n+1,2d)=(3,4). To establish the strict inclusion $\Sigma_{n+1,2d}\subsetneq\mathcal{P}_{n+1,2d}$ in any non-Hilbert case, one can show that verifying the assertion in the basic non-Hilbert cases (4,4) and (3,6) suffices. In this paper, we construct a filtration of intermediate cones between $\Sigma_{n+1,2d}$ and $\mathcal{P}_{n+1,2d}$. This filtration is induced via the Gram matrix approach (by Choi, Lam and Reznick) on a filtration of irreducible projective varieties $V_{k-n}\subsetneq \ldots \subsetneq V_n \subsetneq \ldots \subsetneq V_0$ containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n+1 variables of degree d. By showing that $V_0,\ldots,V_n$ are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with $\Sigma_{n+1,2d}$. Likewise, for the special case when n=2, $V_{n+1}$ is also a variety of minimal degree and the corresponding intermediate cone also coincides with $\Sigma_{n+1,2d}$. We moreover prove that, in the non-Hilbert cases of (n+1)-ary quartics for $n\geq 3$ and (n+1)-ary sextics for $n\geq 2$, all the remaining cone inclusions are strict.

Hahn groups endowed with the canonical valuation play a fundamental role in the classification of valued abelian groups. In this paper we study the group of valuation (respectively order) preserving automorphisms of a Hahn group G. Under the assumption that G satisfies some lifting property, we prove a structure theorem decomposing the automorphism group into a semidirect product of two notable subgroups. We characterise a class of Hahn groups satisfying the aforementioned lifting property. For some special cases we provide a matrix description of the automorphism group.