There is an increasing interest in the use of algebraic tools in Life Sciences. Techniques from Algebraic Geometry and Computational Algebra have proved themselves useful in areas such as Biochemistry or Epidemiology. Understanding structural properties of biological processes, performing model selection or infering parameters involved in those models can sometimes be done by the geometric interpretation of experimental data, or through the study of parametric families of polynomial equations modeling the process.

The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann [Adv. in Appl. Math. (2001)], or BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa as points in a CAT(0) cube complex. However, it is sometimes necessary to analyze collections of trees on nonidentical taxa sets (i.e., with different numbers of leaves), and in this context it is not evident how to apply BHV space. Ren et al. [preprint, 2017] approached this problem by describing a combinatorial algorithm extending tree topologies to orthants in higher-dimensional tree spaces, so that one can quickly compute which topologies contain a given tree as partial data. In joint work with Megan Owen, we refine and adapt their algorithm to work for metric trees to give a full characterization of the subspace of extensions of a subtree. For a collection of subtrees, we define a piecewise-linear space of possible supertrees, and use expanding neighborhoods of the extension spaces to quantify the degree of compatibility.

Algebraic tools have been incorporated in phylogenetic reconstruction methods during the last decade. In order to improve these methods, it is important to take into account the semi-algebraic constraints imposed by the stochasticity of the parameters of the evolutionary models considered. We present a a new method of phylogenetic reconstruction that incorporates both algebraic and semi-algebraic constraints. We will provide results on simulated and real data to illustrate the success of the method.

The interaction of neurons in response to a stimulus determines different brain functions. In the geometric model, the neural data is a cloud of points (manifold) in a higher-dimensional space where a point represents the response of thousands of neurons to a single stimulus. The advance in the technology recording the co-activity of thousands of neurons launches the challenge of reliable estimation of the neural dynamics in higher-dimensional space; indeed, the number of samples (points) is smaller than the number of possible neural interactions (spacial dimensions, features). Therefore, one may ask whether the current methods in geometry and machine learning analysis are appropriate for neural research.

For a preliminary intuition of the features in a higher-dimensional neural space, we pose three questions aiming to understand the qualitative shape, linearity, and spread of the neural manifold. Then we suggest an appropriate method of analysis. In this talk, I will describe how shape and linearity impact our understanding of the data, particularly when sampled from known algebraic varieties and a real dataset.

Discrete Bayesian networks, also known as discrete DAG models, are ubiquitous to represent relations between multivariante data. However, encoded in their definition is the inability to represent context-specific conditional independence (CI) relations, that is CI relations that hold only  for a subset of the conditioning variables. I this talk I will introduce a class of discrete statistical models to represent context-specific conditional independence relations for discrete data and illustrate their usage for causal discovery in a coronary heart disease data set. The goal is to introduce this model as a novel option to represent diverse context-specific settings that appear in the sciences.

The purpose of COST Action CA18232 Mathematical models for interacting dynamics on networks (www.mat-dyn-net.eu) is to bring together leading groups in Europe working on a range of issues connected with modeling and analyzing mathematical models for dynamical systems on networks. We will shortly present the semigroup approach to such problems and make a short overview of other main topics of the Action. We will also discuss the possibilities female and young researchers have within the framework of the Action.

Each partial differential equation can be written as an abstract initial value problem on an appropriate Banach space. Its solution is then given by an operator semigroup, the approximation of which leads to a numerical method. In the present talk we review how the space discretisation of a partial differential equation can be related to the approximation of the operator appearing in the corresponding abstract initial value problem, and how the time discretisation can be done by approximating the semigroup operators. Using the idea introduced we will derive numerical methods for solving various types of problems.

The nonlinear Schrödinger equation with growing potential has been extensively studied by both mathematicians and physicists from the fundamental well-posedness of Cauchy problem to the existence and stability of standing waves.
In this talk, we deal with NLS equation on discrete graphs assuming the linear potential is discrete. Making use of the generalized Nehari manifold approach, we prove the existence and multiplicity of standing waves for both self-focusing and defocusing cases. Our approach is variational and based on the critical point theory applied to the energy functional restricted to the generalized Nehari manifold. This is a joint work with Sedef Karakılıc (Dokuz Eylul University).

Controllability of coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Roughly speaking, the main challenge is controlling a system with less inputs than equations. In this talk this is successfully done for a system of Korteweg-de Vries equations posed on an oriented tree shaped network. The couplings and the controls appear only on boundary conditions.

More precisely, we consider a tree-shaped network of N+1 edges, connected at one vertex.
On each edge we pose a nonlinear Korteweg-de Vries (KdV) equation. On the first edge we put no control and on the other edges we consider Neumann boundary controls.
The main goal is to study the local exact controllability of the nonlinear KdV equation on the tree shaped network of N+1 edges with N controls. The main result of this talk gives a positive answer if the time of control is large enough and the lengths of the edges are small enough.

To show this result, we prove first the exact controllability result of the KdV equation linearized around 0. Our proof is based on an observability inequality for the linear backward adjoint system obtained by a multiplier approach.
We then get the local exact controllability result of the nonlinear KdV equation applying a fixed point argument.

This is joint work with Eduardo Cerpa (Pontificia Universidad Católica de Chile) and Emmanuelle Crépeau (Univ. Grenoble Alpes, France).

Some statistical models can be parameterized by the positive part of a toric variety. On first glance, this setup might seem rather artificial. Yet, those models appear naturally, which I demonstrate with the example of a gambler. In order to be able to compute Bayesian integrals for such models, we endow toric varieties with the structure of a probability space.
This presentation is based on joint work with M. Borinsky, B. Sturmfels, and S. Telen. Exploiting the combinatorial nature of toric varieties, we provide algorithms for sampling from (tropical) densities on toric varieties.

Studying convex cones inside the cone of positive semidefinite (PSD) polynomials is an important field of
research in real algebraic geometry and polynomial optimization. In this talk, we combine two such well
established cones, which are sums of squares (SOS) and sums of nonnegative circuit polynomials (SONC)
and consider PSD polynomials, that decompose into a SOS and a SONC part. We call the resulting set
the SOSONC cone. For this newly established cone, we present separation results showing that for all
nontrivial cases from Hilbert’s 1888 Theorem, the SOSONC cone lies properly in between the union of
the SOS and SONC cones as well as the PSD cone. Moreover, we address the question how membership
to the SOSONC cone can be decided in an efficient way.

The SOS cone of all real forms of degree 2d representable as finite sums of squares (SOS)
of half degree d real forms is included in the PSD cone of all positive semidefinite real forms (PSD) of even degree 2d for a fixed number of n+1 variables. Hilbert (1888) states that these cones in fact coincide if and only if n+1=2, d=1 or (n+1,2d)=(3,4). In this talk, we construct an explicit filtration of intermediate cones between the SOS and PSD cone by extending local positive semidefinite real quadratic forms along projective varieties generated by s (s≥0) real quadratic forms over the Veronese variety. Indeed, the Veronese variety is a projective variety finitely induced by real quadratic forms. We analyze this filtration for proper inclusions by using a result of Blekherman et al. (2016) on projective varieties of minimal degree, Hilbert’s 1888 Theorem and techniques based on Robinson (1969) and Choi-Lam (1977) exemplary in the quaternary quartics case.

A real polynomial which can be written as a sum of squares of polynomials is clearly non-negative. On the other hand, not every non-negative polynomial can be written as a sum of squares. Hilbert examined and classified situations where both notions agree already in 1888. In this talk we focus on the situation of polynomials invariant by a finite group and show how results from invariant theory and representation theory allow us to study this situation in more detail. In particular we will overview how Hilbert’s classification changes in the case of finite reflection groups.
(based on joint work with S. Debus, C. Goel, and S. Kuhlmann)

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and many more areas.
In this talk I will introduce the notion of “uniform determinantal representation”, and derive a lower bound on the size of the matrix, showing a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows.
I will also relate uniform determinantal representations to vector spaces of singular matrices, in particular compression spaces.
The paper I will report on is a joint work with van Doornmalen, Draisma, Hochstenbach, and Plestenjak.

In 1965, Motzkin and Strauß translated the NP-hard problem of finding the stability or the clique number of graphs into the problem of finding the optimal value of certain even quartic forms on the unit sphere. Using the technique of pure states on ideals from real algebraic geometry, we prove certain sums of squares certificates for these optimal values in the vein of Reznick’s Positivstellensatz. This has computational consequences for approaches via semidefinite programming to the mentioned NP-hard problem. This is joint work with Monique Laurent and Luis Felipe Vargas.

Joint work with Sarah Nakato and Roswitha Rissner.

For a long time, the study of non-uniqueness of factorization of
elements into irreducibles was focused on rings of integers in
number fields, where this phenomenon was first discovered.
There, nevertheless, non-uniqueness is comparatively tame,
because uniqueness of factorization into prime ideals is
always lurking in the background.

In stark contrast, non-uniqueness of factorization in rings of
integer-valued polynomials is often quite wild. The ring of
integer-valued polynomials on a domain D, denoted Int(D), is the
subring of K[x] (where K is the quotient field of D) consisting
of such polynomials as map elements of D to elements of D.

In many cases we can show that every finite set of positive
integers not containing 1 occurs as the set of lengths of
factorizations of an element (where the length of a factorization
is the number of irreducible factors). Another measure of wildness
is that there is no transfer homomorphism into a Krull monoid.

In some cases we can characterize the non-absolutely irreducible
elements. Those are irreducible elements some of whose powers
factor into irreducibles in non-unique ways. In number fields,
the existence of such elements characterizes the rings of integers
with non-unique factorization. It is not known whether this holds
in general.

A globalization of the concept of valuation domain is obtained through the definition of Pr\”ufer domain (H. Pr\”ufer, 1932), i.e. a domain whose localizations at prime ideals are valuation domains.
Pr\”ufer $v$-multiplcation domains further generalizes the concept of Pr\”ufer domain using the tool of $t$-operation on ideals. This generalization allows to involve in this class of domains the polynomial rings $D[X]$, which are never Pr\”ufer (unless $D$ is a field).

If $D$ is a domain with quotient field $K$, then ${\rm Int}(D) := \{f(X) \in K[X]\mid f(D) \subseteq D\}$ is the ring of integer-valued polynomial rings over $D$. I studied P$v$MD property for these rings and , in collabration with A:C. Finocchiaro, I applied some topological coniderations to the case ${\rm Int}(D)$ – P$v$MD.

Fundamental aspects of the classical theory of factorization can be widely generalized by combining the languages of monoids and preorders. In this talk, we show a general theorem on the existence of certain factorizations, referred to as ≼-factorizations, for the ≼-non-units of a (multiplicative) monoid H endowed with a preorder ≼, where an element u of H is a ≼-unit if u ≼ 1_H ≼ u and a ≼-non-unit otherwise. The “building blocks” of these factorizations are the ≼-irreducibles of H (namely, the ≼-non-units a of H that cannot be written as a product of two ≼-non-units each of which is strictly ≼-smaller than a); and it is interesting to look for sufficient conditions for the ≼-factorizations of a ≼-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). During the presentation we also address this kind of questions, by inquiring into the interaction between minimal ≼-factorizations (i.e., a refinement of ≼-factorizations used to counter the “blow-up phenomena” that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the “local behaviour” of the pair (H, ≼). We present a variety of arithmetic results, enriched with examples and remarks.

This is a joint work with S. Tringali.

We study for which domains of Krull dimension one the class of direct sums of finitely generated torsion-free modules is closed under direct summands. We show that in the local case this happens if and only if any indecomposable fiinitely generated torsion free module has a local endomorphism ring. We also deal with the non-local case in the particular situation of h-local domains.

This talk is based on joint work with Roman Alvarez and Pavel Prihoda.

Recent advances in nonlinear problems in the calculus of variations have had important applications in the physical and biological sciences, engineering and so on. Topics such as lower semicontinuity and relaxation of nonlinear energy functionals, notions of convexity, generalised Orlicz spaces, asymptotic analysis and free boundary problems have proved instrumental to the understanding, prediction and treatment of instability phenomena, double phase behaviours in materials science, image segmentation problems, thin structures, fracture mechanics, shape optimisation and phase transitions, among others.

In this introductory talk to the minisymposium we will mention some of the ideas and methods that can be used to treat nonlinear problems in the calculus of variations, with the aim of providing some background for the subsequent talks in this session.

Vector-valued generalized Orlicz spaces can be divided into anisotropic, quasi-isotropic and isotropic. In isotropic spaces, the Young function depends only on the length of the vector, i.e. Φ(v) = φ(|v|). In the quasi-isotropic case Φ(v) ≈ φ(v|) so the dependence is via the length of the vector up to a constant. In the anisotropic case, there is no such restriction, and the Young function depends directly on the vector. Jihoon Ok and I obtained maximal local regularity results of weak solutions or minimizers of

when A or F are general quasi-isotropic Young functions. In other words, we studied the problem without recourse to special function structure and without assuming Uhlenbeck structure. We established local $C^{1,α}$-regularity for some α ∈ (0, 1) and $C^α$-regularity for any α ∈ (0, 1) of weak solutions and local minimizers. Previously known, essentially optimal, regularity results are included as special cases. In addition, I briefly comment on recent advances on the anisotropic case.

Preprints are available at https://www.problemsolving.fi/pp/.

When studying the minimization of supremal functionals of the form

several concepts of convexity come into play. We revisit these concepts in order to clarify the relations between them and to settle the ground to address minimization
under the lack of lower semi-continuity.

This is a joint work with E. Zappale.

We consider a Plateau problem in codimension 1 in the non-parametric setting. A Dirichlet boundary datum is given only on a part of the boundary of
a convex domain in $\mathbb{R}^2$. Where the Dirichlet datum is not prescribed, we allow the solution to have a free contact with the plane domain. We show existence of a
solution, and prove some regularity for the corresponding area-minimizing surface. Finally we compare the solutions we find with classical solutions provided by Meeks
and Yau, and show that they are equivalent at least in the case that the Dirichlet boundary datum is assigned in at most 2 connected components of the boundary of the domain. This result was obtained in collaboration with Giovanni Bellettini & Riccardo Scala.

In this talk we present an atomistic to continuum model for a graphene sheet undergoing bending, within the small displacements approximation framework. Under the assumption that the atomic interactions are governed by a harmonic approximation of the 2nd-generation Brenner REBO (reactive empirical bond-order) potential, we determine the variational limit of the energy functionals. If some specific contributions in the atomic interaction are neglected, the variational limit is non-local.

We then analyze the results and by making a connection with the classical theory of plates we will be lead to introduce a new material property: the bending Poisson coefficient. Finally, we consider some extreme cases of our model and this will bring us to microscopically piece-wise rigid plates. The talk is based on joint works with C. Davini, A. Favata, and A. Micheletti.