Mon 22.8. | Tue 23.8. | Wed 24.8. | Thu 25.8. | Fri 26.8. | |

08:45 | Opening | ||||

09:00 | EMS Lecture 1: C. Voisin | EMS Lecture 2: C. Voisin | Mini Symposia | EMS Lecture 3: C. Voisin | Mini Symposia |

10:00 | Coffee break | Coffee break | Coffee break | Coffee break | Coffee break |

10:30 | I. Perugia | K. Mietinen | Gender lecture: M. Hannula | ||

11:30 | K. Bringmann | M. Bruna | Lunch | ||

12:30 | Exhibition opening | Lunch | Lunch | Lunch | |

12:45 | Lunch | ||||

13:00 | N. Holden | ||||

14:00 | Mini Symposia | Mathematics and art workshop: K. Peltonen | Free afternoon / Excursion | General assembly | |

15:30 | Coffee break | Coffee break | Coffee break | ||

16:00 | Poster session | General assembly | |||

17:30 | Round of introductions | Country coordinators meeting | |||

18:00 | Dinner |

**08:45-09:00 **Opening

**Room**: Lecture hall E

**09:00-10:00 **EMS Lecture: Claire Voisin (CNRS)

**Room**: Lecture Hall E

Hyper-Kähler manifolds

Hyper-Kähler manifolds form a special class of compact Kähler manifolds with trivial canonical bundle. They are higher-dimensional generalizations of K3 surfaces, and a number of deformation classes of hyper-Kähler manifolds can be constructed starting from either a K3 or an abelian surface. In the first two lectures, I will introduce them and describe some of their general properties, from the viewpoints of Riemannian geometry, topology, and algebraic geometry. The last lecture will focus on dimension 4. I will describe a simple topological characterization of hyper-Kähler manifolds of Hilb^2(K3) deformation type and sketch the proof (this is joint work with Debarre, Huybrechts, and Macrì).

**10:00-10:30 **Coffee break

**10:30-11:30 **Ilaria Perugia (University of Vienna, Austria)

**Room**: Lecture Hall E

Nonstandard finite element methods for wave problems

Finite elements are a powerful, flexible, and robust class of methods for the numerical approximation of solutions to partial differential equations. In their standard version, they are based on piecewise polynomial functions on a partition of the domain of interest. Continuity requirements conform to the regularity of the exact solutions. By breaking the constraints of the classic finite element paradigm, new methods have been developed in order to better reproduce physical properties of the exact solutions, enhance stability, and improve accuracy vs. computational cost.

In this talk, I will focus on discontinuous, operator-adapted finite element methods for wave propagation problems. They are based on incorporating a priori knowledge about the problem into the approximating spaces, which are spanned, element by element, by functions belonging to the kernel of the differential operator. The numerical approximation of wave problems is notoriously difficult for standard finite element methods. In fact, due to the oscillatory nature of the solutions, stability requires many degrees of freedom per wavelength, in the time-harmonic case, and small time-steps according to the spatial mesh size, in the time-dependent case. Giving up to continuity requirements gives freedom in the choice of the local approximation spaces. Using operator-adapted local approximation spaces breaks the strong requirements to get stability and provides higher accuracy with respect to standard methods.

**11:30-12:30 **Kathrin Bringmann (University of Cologne, Germany)

**Room**: Lecture Hall E

Modularity of combinatorial generating functions

In my talk I will describe various generating functions arising from combinatorics which have specific symetries.

**12:30-12:45 **Exhibition opening

**Room**: Beta Gallery

The exhibition “Women of Mathematics throughout Europe: A Gallery of Portraits” is shown in the Beta Space Gallery, Otakaari 1 X during 17.08-28.08. The exhibition features 21 women mathematicians, including a new portrait for Claire Voisin, who is the EMS lecturer at the EWM General Meeting. For more information, see https://womeninmath.net/ and https://womeninmath.net/venue/espoo-finland/.

**12:45-14:00 **Lunch

**14:00-15****:30 **Minisymposia (Detailed schedule below)

**15:30-16:00 **Coffee break

**16:00-17****:30 **Minisymposia (Detailed schedule below)

**17:30-19:00 **Round of introductions

**Room**: Lecture Hall E

**09:00-10:00 **EMS Lecture: Claire Voisin (CNRS)

**Room**: Lecture Hall E

Hyper-Kähler manifolds

Hyper-Kähler manifolds form a special class of compact Kähler manifolds with trivial canonical bundle. They are higher-dimensional generalizations of K3 surfaces, and a number of deformation classes of hyper-Kähler manifolds can be constructed starting from either a K3 or an abelian surface. In the first two lectures, I will introduce them and describe some of their general properties, from the viewpoints of Riemannian geometry, topology, and algebraic geometry. The last lecture will focus on dimension 4. I will describe a simple topological characterization of hyper-Kähler manifolds of Hilb^2(K3) deformation type and sketch the proof (this is joint work with Debarre, Huybrechts, and Macrì).

**10:00-10:30 **Coffee break

**10:30-11:30 **Kaisa Miettinen (University of Jyvaskyla, Finland)

**Room**: Lecture Hall E

Why are Interactive Multiobjective Optimization Methods Useful in Decision Making?

Decisions are typically characterized by conflicting perspectives. This means that to be able to make good decisions, we must optimize several conflicting objective functions simultaneously. For this, we need multiobjective optimization methods. Because of the conflict, there are many so-called Pareto optimal solutions with different trade-offs among the objectives and we need additional preference information from a domain expert, a so-called decision maker, to find the final, most preferred solution. We can call it the best compromise.

We characterize different classes of multiobjective optimization methods and concentrate on interactive ones, where the decision maker takes actively part in the solution process.

This means that the decision maker can learn about the interdependencies among the objectives, gain insight in the trade-offs and understand what kind of preferences are achievable. Based on the learning, the decision maker can adjust the preferences and gain confidence in the final solution.

We discuss various practical challenges of decision making problems, give some examples of interactive multiobjective optimization methods and describe how they address the challenges. We also share some experiences in solving real problems in various fields of life and briefly introduce the open-source software framework DESDEO containing implementations of many interactive methods.

**10:30-11:30 **Maria Bruna (University of Cambridge, UK)

**Room**: Lecture Hall E

Continuum models of strongly interacting Brownian particles

I will discuss many-particle systems with strong interactions. These models are motivated by the study of many-particle systems in biology or industrial applications, where it is crucial to account for the finite size of particles. I will explain how these interactions can be included in the models and different methods to derive continuum PDE descriptions. In the second part of the talk, I will show how these methods can be used to model active matter or self-propelled particles such as bacteria or ants.

**12:30-14:00 **Lunch

**14:00-15:30 **Mathematics and art workshop: Kirsi Peltonen (Aalto University, Finland)

**Room**: Lecture Hall C

EWM Origami Workshop (Art and Math Workshop)

Mathematics and origami: Come and join folding simple origami models and discover their mathematical power. Participants can try various models according to their interests. Simple polyhedra from business cards, Miura-ori tessellations and beyond. Models can be used to approach concepts like self-similarity and Gaussian curvature. No previous experience in origami is expected.

**15:30-16:00 **Coffee break

**16:00-17****:30 **Poster Session

**Room**: Lecture Hall E

**17:30-19:00 **Country coordinators meeting

**Room**: Lecture Hall E

**09:00-10:00 **Minisymposia (Detailed schedule below)

**10:00-10:30 **Coffee break

**10:30-12:30 **Minisymposia (Detailed schedule below)

**12:30-14:00 **Lunch

**14:00-… **Free afternoon

**09:00-10:00 **EMS Lecture: Claire Voisin (CNRS)

**Room**: Lecture Hall E

Hyper-Kähler manifolds

Hyper-Kähler manifolds form a special class of compact Kähler manifolds with trivial canonical bundle. They are higher-dimensional generalizations of K3 surfaces, and a number of deformation classes of hyper-Kähler manifolds can be constructed starting from either a K3 or an abelian surface. In the first two lectures, I will introduce them and describe some of their general properties, from the viewpoints of Riemannian geometry, topology, and algebraic geometry. The last lecture will focus on dimension 4. I will describe a simple topological characterization of hyper-Kähler manifolds of Hilb^2(K3) deformation type and sketch the proof (this is joint work with Debarre, Huybrechts, and Macrì).

**10:00-10:30 **Coffee break

**10:30-11:30 **Gender Lecture: Markku Hannula (University of Turku, Finland)

**Room**: Lecture Hall E

**11:30-13:00 **Lunch

**13:00-14:00 **Nina Holden (ETH Zurich and Courant Institute)

**Room**: Lecture Hall E

Random curves and surfaces

How can you sample a surface uniformly at random? A natural approach is to consider a uniformly sampled planar map, which is a model for a discrete surface studied in many branches of both math and physics. When the size of the surface goes to infinity it converges to the continuum random surface known as a Liouville quantum gravity surface, which was originally introduced in the physics literature. We will give an introduction to these objects and present a powerful technique to study them known as conformal welding, where the random fractal curves known as Schramm-Loewner evolutions appear.

**14:00-15:30 **General assembly

**Room**: Lecture Hall E

**15:30-16:00 **Coffee break

**16:00-17****:30 **General assembly

**Room**: Lecture Hall E

**18:00-.. **Dinner

Dinner takes place at Ravintola Factory Otaniemi, Otakaari 5, 02150 Espoo

**09:00-10:00 **Minisymposia (Detailed schedule below)

**10:00-10:30 **Coffee break

**10:30-12:30 **Minisymposia (Detailed schedule below)

**12:30-14:00 **Lunch

## Minisymposia on Monday

**14:00-14:25** Miruna-Stefana Sorea (SISSA, Trieste, Italy and RCMA ULBS, Romania)

Algebra and Geometry behind Data - introductory talk

This is the introductory talk for the EWM GM 2022 mini-symposium called “Algebra and Geometry behind Data”. The goal of the talk is twofold. First, we present some algebraic and geometric tools and techniques that are useful in unveiling hidden structures in data analysis. Second, we show that questions inspired from data science play an important role in discovering new interesting results in algebra and geometry.

To this end, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational MLE and we give a criterion for these models to be log-linear. Moreover, we introduce a new family of lattice polytopes with rational linear precision. This is based on a joint work with Isobel Davies (Univ. Magdeburg), Eliana Duarte (Univ. do Porto) and Irem Portakal (TU Munich).

**14:30-14:55** Kathlén Kohn (KTH Royal Institute of Technology, Sweden)

The Geometry of Linear Convolutional Networks

We discuss linear convolutional neural networks (LCNs) and their critical points. We observe that the function space (i.e., the set of functions represented by LCNs) can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network’s architecture on the geometry of the function space. For instance, for LCNs with one-dimensional convolutions having stride one and arbitrary filter sizes, we provide a full description of the boundary of the function space.

We further study the optimization of an objective function over such LCNs: We characterize the relations between critical points in function space and in parameter space and show that there do exist spurious critical points. We compute an upper bound on the number of critical points in function space using Euclidean distance degrees and describe dynamical invariants for gradient descent.

This talk is based on joint work with Thomas Merkh, Guido Montúfar, and Matthew Trager.

**15:00-15:25** Chiara Meroni (Max Planck Institute for Mathematics in the Sciences, Germany)

Convex Geometry meets Quantum Physics

Alice and Bob run an experiment. They receive a photon from a source, that can be measured in two ways. From the measurement, they obtain one outcome, out of two possibilities. The correlation is the probability of equal outcomes minus that of different outcomes. The set of all correlations, called correlation body, turns out to be convex. This is a fundamental object studied in Quantum Physics, since some of the correlations cannot be modelled within classical probability. We will explore the geometry of the correlation body and draw connections to quantum key distribution.

**16:00-16:25** Maria Dostert (KTH Royal Institute of Technology, Sweden)

Learning polytopes with fixed facet directions

Geometric tomography is concerned with reconstructing shapes from geometric data such as volumes of sections and support function evaluations, a task that arises naturally in a variety of application areas, for example, robotics, computerized tomography and magnetic resonance imaging. In this talk we consider the task of reconstructing polytopes with fixed facet directions from finitely many (possibly noisy) support function evaluations. For fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction. We also discuss limitations of our results if the restriction on the normal fan is removed. This is joint work with Katharina Jochemko.

**16:30-16:55** Elima Shehu (Osnabrück University and MPI MiS, Germany)

Line Multiview Varieties

The mathematical abstraction of a pinhole camera is a projective linear map given by a 3×4 matrix. Suppose that we have a line in three-dimensional projective space and take m images of this line under using m different pinhole cameras. This produces an arrangement of m lines in two-space, called a line correspondence. I will present line correspondences for pinhole cameras from the point of view of algebraic geometry. We define the line multiview variety as the (complex) Zariski closure of the set of all line correspondences with m fixed cameras. We prove that in the case of generic camera matrices it is characterized by a natural determinantal variety and we provide a complete description for any camera arrangement. We investigate basic properties of this variety such as dimension, smoothness and multidegree. This is joint work with Paul Breiding, Felix Rydell, and Angelica Torres.

**17:00-17:25** Angélica Marcela Torres Bustos (KTH Royal Institute of Technology, Sweden)

Robustness in chemical reaction networks with low-dimensional stoichiometric space

Autoregulatory processes, such as homeostasis, have been observed experimentally in different biochemical systems. Their main characteristic is the presence of an agent or species that remains unchanged under perturbations of the system. When the biochemical systems are modelled with Chemical Reaction Networks this property is called Absolute Concentration Robustness (ACR), and under mass-action kinetics it is possible to detect its presence by studying a family of parametric algebraic varieties. In this talk I will present a full classification of the networks with low-dimensional stoichiometric space that have ACR for every set of parameters. This classification shows that for small networks ACR arises only in networks that satisfy the Feinberg condition for ACR. This is joint work with Anne Shiu and Nicolette Meshkat.

**Organizers**:

- Anna-Laura Sattelberger
- Miruna-Stefana Sorea

**14:00-14:55** Delaram Kahrobaei (USA)

Applied Group Theory in the Quantum and Artificial Intelligence Era and cybersecurity

In this talk I present an overview of the current state-of-the-art in post-quantum group-based cryptography. I describe several families of groups that have been proposed as platforms, with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. I then describe some applications of combinatorial algebra in fully homomorphic encryption, and in particular homomorphic machine learning. In the end I will discuss several open problems in this direction. See [1, 2] .

[1] D. Kahrobaei, R. Flores, M. Noce, Group-based Cryptography in the Quantum Era, The Notices of the American Mathematical Society, https://arxiv.org/abs/2202.05917, accepted, 1–15 (2022)

[2] D. Kahrobaei, R. Flores, M. Noce, M. Habeeb, Book: Applications of Group Theory in Cryptography, the Mathematical Surveys and Monographs series of the American Mathematical Society. 1–200, Under consideration (2022)

**15:00-15:25 **Jone Uria-Albizuri (University of the Basque Country, UPV-EHU, Spain)

Core-portrait description for a family of self-similar groups

An automorphism of a d-adic rooted tree T can be described by decorating the tree vertices with permutations from the symmetric group Sd.

However, we can also decorate the leaves by tree automorphisms, providing in this way a finite description of an automorphism.

On the other hand, if the action of a group G on T is what is known as contracting, there is a finite subset of the group G, called the nucleus, for which every element in G admits such a finite description with all the leaves

decorated by elements in the nucleus. The minimal such description for an element is called its core-portrait and these are some kind of normal forms for self-similar groups.

We provide the first explicit description of the core-portraits of a group.

Namely, we give such a despription for every non-symmetric Grigorchuk-Gupta-Sidki group.

**16:00-16:25** Marzia Mazzotta (Italy)

The structure group and the permutation group of a set-theoretic solution of the Yang-Baxter equation

In the ’90s Drinfel’d raised the issue of determining all the set-theoretic solutions of the Yang-Baxter equation, a basic equation that was first introduced

in the field of statistical mechanics. Briefly, a solution is a pair (X, r) where X is a set and r : X × X → X × X a map satisfying the braid relation

(r × idX) (idX × r) (r × idX) = (idX × r) (r × idX) (idX × r).

In this talk, we survey some results on two groups introduced in the milestone by Etingof, Schedler, and Soloviev in 1999, namely the structure group

G(X, r) and the permutation group G(X, r) of a solution (X, r). These two algebraic structures are crucial and bring group-theoretic tools into the

study of solutions to the point that they have become the object of study of many mathematicians over the years. In particular, we will discuss that

some properties of the solutions reflect in these algebraic structures and vice versa.

**16:30-16:55** Paola Stefanelli (University of Salento, Italy)

Groups, semi-braces, and set-theoretic solutions of the Yang-Baxter equation.

The Yang-Baxter equation is a fundamental equation of the statistical mechanics that arose from Yang’s work in 1967 and, independently, from Baxter’s one in 1972. In 1992,

Drinfel’d posed the question of finding all set-theoretic solutions of the Yang-Baxter equation, namely, maps r : X × X → X × X, with X a set, satisfying the identity

(r × idX) (idX ×r) (r × idX) = (idX ×r) (r × idX))(idX ×r).

Until now, a complete description of these solutions is unknown, even if several papers are on this topic. In this context, significant group-theoretical interpretations of solutions

can be found in the seminal papers by Gateva-Ivanova and Van den Bergh (1998), Etingof, Schedler, and Soloviev (1999), and Lu, Yan, and Zhu (2000).

In this talk, we will illustrate how certain classes of groups turn out to be useful for determining set-theoretic solutions. In particular, we will show that this is the case of the

multiplicative group of a brace and its generalizations, algebraic structures that include radical rings. Finally, we will focus on current methods for constructing semi-braces.

**17:00-17:25** Anitha Thillaisundaram (Lund University, Sweden)

Hausdorff spectrum of p-adic analytic pro-p groups

The concept of Hausdorff dimension first arose in the context of fractals. With time it was also applied to other areas of mathematics, and this was extended

in the 90s to profinite groups. A special class of profinite groups includes the p-adic analytic groups, which have a rich geometric and analytic structure, and

play a key role in the theory of groups. It is an open question still whether p-adic analytic groups can be characterised by the set of Hausdorff dimensions of their

subgroups, i.e. the so-called Hausdorff spectrum. In this talk, we survey what has been done towards answering this question, and we include recent results

concerning the Hausdorff spectrum when computed with respect to the lower p-series. This is joint work with Iker de las Heras and Benjamin Klopsch.

**Organizer**: Marialaura Noce

**14:00-14:25** Anna Sakovich (Uppsala University, Sweden)

Mathematical General Relativity: an overview

We will provide a brief overview of the field of Mathematical General Relativity, starting from the groundbreaking work of Yvonne Choquet-Bruhat of 1952 where the Einstein equations of General Relativity were formulated as a well-posed Cauchy problem. Our main focus will be on one of the major avenues of the current research, namely the construction and the study of initial data for the Cauchy problem, but we will also touch upon the evolutionary aspects such as stability and formation of singularities.

**14:30-14:55** Virginia Agostiniani (Department of Physics, University of Trento, Italy)

A potential-theoretic approach to geometric inequalities in General Relativity

We give a brief overview of how, for different classes of ambient manifolds, the derivation of suitable monotone quantities associated with a specific potential is a valuable tool for proving relevant geometric inequalities. A specific emphasis will be given to the Positive Mass Theorem and to the Riemannian Penrose Inequality.

**15:00-15:25** Melanie Graf (University of Tübingen, Germany)

Coordinates are messy

In General Relativity, an “isolated system at a given instant of time” is modeled as an asymptotically Euclidean initial data set (M,g,K). Such asymptotically Euclidean initial data sets (M,g,K) are characterized by the existence of asymptotic coordinates in which the Riemannian metric g and second fundamental form K decay to the Euclidean metric delta and to 0 suitably fast, respectively. Using harmonic coordinates Bartnik showed that (under suitable integrability conditions on their matter densities) the (ADM-)energy, (ADM-)linear momentum and (ADM-)mass of an asymptotically Euclidean initial data set are well-defined. To study the (ADM-)angular momentum and (BORT-)center of mass, however, one usually assumes the existence of Regge-Teitelboim coordinates on the initial data set (M,g,K) in question, i.e. the existence of asymptotically Euclidean coordinates satisfying additional decay assumptions on the odd part of g and the even part of K. We will show that, under certain circumstances, harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge-Teitelboim coordinates. This allows us to easily give examples of (vacuum) asymptotically Euclidean initial data sets which do not possess any Regge-Teitelboim coordinates. This is joint work with Carla Cederbaum and Jan Metzger.

**16:00-16:25** Zoe Wyatt (King’s College London, UK)

Stabilising relativistic fluids on slowly expanding cosmological spacetimes

On a background Minkowski spacetime, the relativistic Euler equations are known, for a relatively general equation of state, to admit unstable homogeneous solutions with finite-time shock formation. By contrast, such shock formation can be suppressed on background cosmological spacetimes whose spatial slices expand at an accelerated rate. The critical case of linear, ie zero-accelerated, spatial expansion, is not as well understood. In this talk, I will outline some recent works concerning the Einstein–relativistic Euler equations for geometries expanding at a linear rate.

**16:30-16:55** Alena Pravdova (Institute of Mathematics of the Czech Academy of Sciences, Czech Republic)

Black holes and other solutions to quadratic gravity

In general, field equations of quadratic gravity are too complicated to attempt to find exact solutions. However, an additional assumption that the Ricci scalar is constant, which has both a mathematical and physical motivation, leads to some simplifications of the field equations. In particular, all corrections to the Einstein equations are then proportional to the Bach tensor. Since the Bach tensor is conformally well-behaved, it enables us to construct vacuum solutions to quadratic gravity using conformal transformations. We present various examples of such solutions, including the non-Schwarzschild spherically symmetric black hole obtained by a conformal transformation of a Kundt spacetime.

**Organizer**: Carla Cederbaum

**14:00-14:25** Jana Björn (Linköping University, Sweden)

Growth estimates for p-harmonic Green functions on metric spaces and weighted R^n

As shown by Serrin in 1964, the growth at an isolated singularity

of solutions to the elliptic equation div A(x,grad u)=0 in R^n

(including p-harmonic functions with p>1) is exactly determined

by the dimension n and the parameter p associated with the equation.

In this talk I will discuss growth and integrability

properties for p-harmonic Green functions and their gradients

on weighted R^n, with a p-admissible weight, as well as

on complete metric spaces equipped

with a doubling measure supporting a p-Poincare inequality.

In these situations, the dimension n is replaced by the local growth of

the underlying measure near the isolated singularity,

and the obtained growth and integrability exponents are sharp.

**14:30-14:55** Saara Sarsa (University of Helsinki, Finland)

Second order Sobolev-regularity for p-harmonic functions

We consider p-harmonic functions, that is, weak solutions of the PDE div(|Du|^{p-2}Du)=0 for 1<p<\infty. It is well known that if u is a p-harmonic function, then the nonlinear transformation of the gradient |Du|^{\frac{p-2}{2}}Du has locally square integrable weak derivative. In this talk we discuss a more general second order Sobolev-regularity result for p-harmonic functions. We prove that if u is p-harmonic then |Du|^{\beta}Du has locally square integrable weak derivative whenever \beta is large enough, more precisely, when \beta>-1+\frac{(n-2)(p-1)}{2(n-1)}. Furthermore, in case \beta=0 we discuss a global version of this regularity result. This talk is partly based on joint work with Akseli Haarala.

**15:00-15:25** Anna Zatorska-Goldstein (University of Warsaw, Poland)

Potential estimates and local behavior of solutions to nonlinear elliptic equations and systems

We consider measure-data elliptic problems involving a second-order operator in a divergence form with Orlicz growth and measurable coefficients. We provide pointwise estimates for solutions expressed in terms of a nonlinear potential of a measure datum. We also investigate regularity consequences, in particular, we show a sharp criterion for data that is equivalent to H\”older continuity of the solutions. The talk is based on joint works: (scalar) with I. Chlebicka and F. Giannetti [arXiv:2006.02172] and (vectorial) with I. Chlebicka and Y. Youn, [arXiv:2102.09313], [arXiv:2106.11639].

**16:00-16:25** Leah Anna Schätzler (Paris-Lodron-University Salzburg, Austria)

Hölder continuity for doubly nonlinear equations

The prototype of the partial differential equations considered in this talk is

$$

\partial_t \big( |u|^{q-1} u \big) – \operatorname{div} \big( |Du|^{p-2} Du \big) = 0

\quad \text{in } E_T = E \times (0,T]

$$

with parameters $q \in (0,\infty)$ and $p \in (1,\infty)$.

Here, $E \subset \mathbb{R}^N$ denotes an open set and $0<T<\infty$.

This doubly nonlinear equation is a generalization of the porous medium equation (case $p=2$), the parabolic $p$-Laplace equation (case $q=1$) and Trudinger’s equation (case $q=p-1$).

I will present Hölder continuity results for possibly sign-changing weak solutions to equations of this type.

The talk is based on joint work with Verena Bögelein, Frank Duzaar and Naian Liao.

**Organizers**:

- Antonella Nastasi
- Cintia Pacchiano Camacho
- Riikka Korte

## Minisymposia on Wednesday

- Algebraic methods in life sciences (Room: Y228a)
- CA18232: Mathematical models for interacting dynamics on networks I (Room: Y228b)
- Real algebraic geometry in action with a view toward optimization (Room: Y229c)
- Trends in ring theory and related topics (Room: Y307)
- Variational Methods and Nonlinear Analysis (Room: Y229a)

**09:00-09:25** Organizers

Algebraic methods in life sciences: introductory talk

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.

**09:30-09:55 **Gillian Grindstaff (University of Oxford, UK)

Representations of partial leaf sets in phylogenetic tree space

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.

**10:30-10:55 **Marta Casanellas (Universitat Politècnica de Catalunya, Spain)

Incorporating semi-algebraic constraints to improve phylogenetic reconstruction

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.

**11:00-11:25** Annachiara Korchmaros (University of Leipzig)

Shape and linearity of neural manifolds

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.

**11:30-11:55** Eliana Duarte (Universidade do Porto, Portugal)

Representation of Context-Specific Causal models in the Life Sciences

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.

**Organizers**:

- Marina Garrote-López
- Beatriz Pascual
- Roser Homs

**09:00-09:25 **Marjeta Kramar Fijavz (University of Ljubljana, Slovenia), Eszter Sikolya (Eötvös Loránd University, Budapest, Hungary)

Mat-Dyn-Net in a Nutshell

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.

**09:30-09:55 **Petra Csomós (Eötvös Loránd University, Budapest, Hungary)

Operator semigroups for convergence analysis of numerical methods

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.

**10:30-10:55 **Setenay Akduman (Izmir Democracy University, Turkey)

Nonlinear Schrödinger Equation on Discrete Graphs

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).

**11:00-11:25 **Julie Valein (Université de Lorraine, France)

On the boundary controllability of the Korteweg-de Vries equation on a tree-shaped network

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).

**11:30-12:30 **Discussion

**Organizers**:

- Marjeta Kramar Fijavž
- Eszter Sikolya

**09:00-09:25 **Anna-Laura Sattelberger (MPI-MiS, Leipzig, Germany)

Toric Varieties as Probability Spaces

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.

**09:30-09:55 **Moritz Schick (University of Konstanz, Germany)

The Minkowski of the SOS and SONC cones - an inner approximation of the PSD cone

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.

**10:30-10:55 **Sarah Tanja Hess (University of Konstanz, Germany)

Intermediate Cones between the SOS and PSD Cone

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.

**11:00-11:25 **Cordian Riener (UiT The Arctic university of Norway)

Symmetries and sums of squares

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)

**11:30-11:55 **Ada Boralevi (Politecnico di Torino, Italy)

Uniform determinantal representations and spaces of singular matrices

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.

**12:00-12:25 **Markus Schweighofer (University of Konstanz, Germany)

Pure states and the stability number of graphs

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.

**Organizers**:

- Charu Goel
- Salma Kuhlmann

**09:00-09:25 **Ana Cristina Barroso (University of Lisbon, Portugal) & Elvira Zappale (Sapienza – Università di Roma, Italy)

Variational methods and nonlinear analysis

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.

**09:30-09:55 **Peter Hästö (University of Turku, Finland)

Elliptic PDE in quasi-isotropic generalized Orlicz spaces

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/.

**10:30-10:55 **Ana Margarida Ribeiro (Universidade Nova de Lisboa, Portugal)

On the convexity notions related to lower semi-continuity of supremal functionals

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.

**11:00-11:25 **Roberta Marziani (University of Dortmund, Germany)

A non-parametric Plateau problem with partial free boundary

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.

**11:30-11:55 **Roberto Paroni (DICI, Università di Pisa, Italy)

Microscopically piece-wise rigid plates inspired by graphene

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.

**Organizers**:

- Ana Cristina Barroso
- Elvira Zappale

## Minisymposia on Friday

**09:00-09:25 **Ana Jacinta Pereira Costa Soares (Centre of Mathematics – University of Minho, Portugal)

A coupled human-landscape model: stability analysis, network dynamics and numerical tests

We develop a mathematical model by coupling the human activity with an ecological model for a network of Landscape Units (LUs). Each LU is endowed with a system of ODEs for two variables relevant to the percentage of green areas and to the production of bio-energy, with an additional equation for the dynamics of the fraction of environmentalists in the population. The resulting model constitutes a network of interacting dynamical systems, each of them referring to a single node of the network but all dynamical systems sharing the same qualitative structure. The interaction among the various dynamical systems is described by a linear term defining the coupling between each LU of the mosaic and its neighbours in terms of the exchanging bio-energy fluxes. We study the dynamics in each LU, with reference to equilibria and their stability, and prove the possible occurrence of Hopf bifurcations with consequent periodic oscillations of environmental and human variables. The numerical investigation shows that such oscillations may disappear by global heteroclinic bifurcations. Then, the connectivity between the LUs is considered, with the aim of pointing out the effects of the single LU dynamics on the network landscape model. Numerical simulations of different scenarios are performed in a sample model of an environmental system in Northern Italy.

[Work in collaboration with R. Della Marca and M. Groppi]

**09:30-09:55 **Federica Gregorio (University of Salerno, Italy)

Schrödinger and polyharmonic operators on infinite graphs

We analyze properties of semigroups generated by Schrödinger operators or polyharmonic operators, on metric graphs both on Lp-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators’ spectra coincide on all relevant function spaces.

**10:30-10:55 **Alessia Elisabetta Kogoj (Università di Urbino Carlo Bo, Italy)

Liouville-type theorems for Kolmogorov and Ornstein–Uhlenbeck operators

We collect Liouville-type properties that hold true for Kolmogorov operators with constant coefficients and for their time-stationary counterpart, the Ornstein-Uhlenbeck operators. In particular, we discuss uniqueness results for solutions and sub-solutions in $L^p$-spaces, for solutions in the whole space or in halfspaces bounded just from one-side. Polynomial Liouville properties and a Liouville theorem “at $t = -\infty$” are also presented.

**11:00-11:25 **Dušanka Perisic (University of Novi Sad, Serbia)

Cost Actions Act as a Pillar of Support for Female Mathematicians

In the mini-symposium you were able to hear about scientific results of COST Action 18232: Mathematical Models for Interacting Dynamics on Networks. I will present data on the participation of female scientists in various COST Actions and point out examples of good practice in supporting female scientists.

**11:30-12:30 **Final discussion

**Organizers**:

- Marjeta Kramar Fijavž
- Eszter Sikolya

**09:00-09:25** Rhoslyn Coles (University of Potsdam, Germany)

Ideal embeddings of knotted and linked curves.

I will present an experimental approach to investigating part of the embedding space of knotted or linked curves. In this work, the knotted curve, referred to as the thickened curve, is considered a physical object, an embedded tube of a certain cross–sectional diameter and length, a machine whose mechanism is obstructed or coordinated by the underlying knot type of the curve. Geometrically interesting embeddings are investigated computationally, by minimising an energy of the thickened curve and deforming the curve trajectory towards a so–called ideal shape of low energy.

Our main motivation is towards understanding the interplay between form and function of long potentially complicated entangled materials like proteins in solution, for which thickened knotted curves are a simple model. The types of energy functionals considered here are derived from this intention. These functionals give us a way of surveying and generating a range of ideal embeddings of a thickened knot of a given length, providing a uniquely physical perspective of knots and links.

This is joint work with my PhD supervisor Prof. Myf Evans of Potsdam Universitaet and our collaborator Prof. Roland Roth of the Universitaet Tuebingen.

**09:30-09:55** Viveka Erlandsson (University of Bristol, UK)

Mirzakhani’s curve counting theorem

It is a classical result that on a hyperbolic surface, the number of closed geodesics of length bounded by L grow exponentially in L. A famous result by Mirzakhani shows that if we restrict to curves that do not self-intersect (or more generally, simple curves of a fixed type) then the asymptotic growth is polynomial. In this talk I will discuss this result and various generalizations, among other things explain why it holds when we replace the hyperbolic metric with any “nice” length function.

**10:30-10:55** Anna Maria Hartkopf (FU Berlin, Germany)

Science Communication in Mathematics

Amidst the discrete geometry working group at Freie Universität Berlin, a subgroup of researchers concerned with mathematical science communication has emerged, originating with the project “Adopt a Polyhedron” that prompted a scientific analysis of mathematical outreach activities which have been done rather pragmatically prior to this study. Subsequently, we have founded the MIP.labor, a research project for science journalism in mathematics, computer science and physics. It is funded by Klaus Tschira Stiftung.

In my talk, I want to highlight the characteristics of science communication in mathematics, the importance of quality science journalism and a very brief agenda for the next steps to build a relationship between (scientific) mathematics and society that is based on mutual trust.

**11:00-11:25** Nina Smeenk (TU Berlin, Germany)

Exploring the World of 3D Visualization in Geometry

Major advances in computer graphics have been achieved during the past years in terms of the visualization of 3D objects. Accordingly, mathematical research topics and methods have developed, adapted and been extended by computer-based methods. Currently, a part of the SFB Transregio 109 “Discretization in Geometry and Dynamics” is engaged with the implementation of a (soon to be published) python based programming library. The library is dedicated to simplifying computer-based work with mathematical and geometric objects. It possesses various interfaces for the visualization of given objects. One of such is using Blender, a free and open-source 3D computer graphics software for high-end production of images and animations. During the talk we will explore the usage of the python library in Blender (in real-time) on the example of energy minimization of a surface. Among other things, we will see non-planar faces of a surface deform to planar ones and further deform to circular ones. Moreover, we will look at some finished renderings that were created using the library and appeared in mathematical publications.

**11:30-11:55** Katrin Leschke (University of Leicester, UK)

Integrable System methods in Surface Theory

Recently the field of discrete differential geometry has experienced a surge in interest. Discretisation in curve and surface theory has a number of applications via polygonal meshes: for example, freeform architecture benefits from analysis of polygonal meshes, or computer graphics uses triangular meshes to represent 3D models. For the study of discrete curves and surfaces, integrable system methods have been crucial, e.g., when studying discrete pseudo-spherical or discrete isothermic surfaces. In my talk, we will discuss with the example of polarised curves how integrable system methods can be used to approach global aspects in both the smooth and discrete theory.

**Organizers**:

- Hana Dal Poz Kourimska
- Gudrun Szewieczek

**09:00-09:25 **Alessandra Bernardi (University of Trento, Italy)

On the dimension of tensor network varieties

In this talk I will introduce tensor network varieties and an approach to study their dimensions.

**09:30-09:55 **Emanuele Ventura (Politecnico di Torino, Italy)

Implicitisation and parameterisation in polynomial functors

Recently, Draisma showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In a follow-up work, Bik-Draisma-Eggermont-Snowden found a deep description of the structure of these closed subsets. Varieties of interest in the context of tensors may be thought in terms of polynomial functors. In this talk,

we indicate how these results can be made algorithmic. This is based on a joint work with Blatter and Draisma.

**10:30-10:55 **Weronika Buczyńska (Warsaw University, Poland)

Border apolarity

For a given polynomial f the classical apolarity lemma connects the ideals of points contained in the apolar ideal of f with the linear spans of points that contain f. As a consequence, the apolarity lemma allows to compute or bound the rank of a tensor or a polynomial. For many years there did not exist any analogue for border rank of tensors. The main issue was that the definition of border rank involves limits. By considering the ideals that are limits of ideals of points in the multigraded Hilbert scheme, we were able to come up with an analogous statement, which is useful for studying the border rank. This method is now called the border apolarity. It has already been used to compute the rank of the 3×3 determinant, and some matrix multiplication tensors.

**11:00-11:25 **Luca Sodomaco (Aalto University, Finland)

Log-concave density estimation in undirected graphical models

We study the problem of maximum likelihood estimation of densities that have a log-concave factorization according to a given undirected graph G. We show that the maximum likelihood estimate (MLE) exists and is unique with probability one as long as the data sample size is larger than a parameter that depends uniquely on G. We describe the support of the MLE in terms of projections of the convex hull of the data sample. Furthermore, we show that the MLE is the product of the exponentials of several tent functions, one for each maximal clique of the graph. The talk is based on a joint paper with Kaie Kubjas, Olga Kuznetsova, Elina Robeva, and Pardis Semnani.

**11:30-11:55** Kathlén Kohn (KTH Royal Institute of Technology, Sweden)

Invariant theory of maximum likelihood estimation

The task of fitting data to a model is fundamental in statistics. For this, a widespread approach is finding a maximum likelihood estimate (MLE), where one maximizes the likelihood of observing the data as we range over the model. For two common statistical settings (log-linear models and Gaussian transformation families), this approach is equivalent to a capacity problem in invariant theory: finding a point of minimal norm in an orbit under a corresponding group action. The existence of the MLE can then be characterized by stability notions under the action. Moreover, algorithms from statistics can be used in invariant theory, and vice versa. This talk provides an introduction to this dictionary between invariant theory and statistics, which has already led to the solution of long-standing questions concerning the MLE of matrix normal models. This talk is based on joint work with Carlos Améndola, Philipp Reichenbach, and Anna Seigal.

**Organizers**:

- Ada Boralevi
- Elisa Postinghel

**09:00-09:25** Laura Thesing (LMU Munich, Germany)

Which networks can be learned by an algorithm? - Expressivity meets Turing in Deep Learning

Deep neural networks show great performance in a variety of applications and are now also used in security and safety-sensitive areas like self-driving cars and medical health care. However, besides all the success stories of the last decade, there is overwhelming empirical evidence suggesting that modern AI is often non-robust (unstable), may generate hallucinations, and thus can produce nonsensical output with high levels of prediction confidence. When we look at the collection of approximation results, we see that the problem does not lie in the expressivity of the networks. However, recent results demonstrate that stable networks cannot be computed by an algorithm from point samples. In this talk, we discuss deep learning in the light of Turing computability and present a first step to extend the results from expressivity theory to computability. The main result is in a similar spirit to the universal approximation theorem and demonstrates that there are no limitations for computable neural networks to approximate computable functions.

**09:30-09:55** Francesca Bartolucci (ETH Zurich, Switzerland)

Understanding neural networks with reproducing kernel Banach spaces

Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this talk we discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, we prove a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite width. Further, we show that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure, with norm given by the total variation norm of the measure.

**10:30-10:55** Selina Drews (TU Darmstadt, Germany)

On the universal consistency of an overparameterised deep neural network estimate learned by gradient descent

While neural networks are often heavily overparameterised in practice, leading to

better optimisation and generalisation, this seems to contradict classical learning

theory. So why do these networks generalise well? In this talk we will analyse

overparameterised networks from a statistical point of view. In a non-parametric

regression setting, we analyse networks trained by gradient descent. In particular,

we show that overparameterised deep neural networks are universally consistent under

appropriate conditions for the step size, the number of gradient descent steps and a

suitable random initialisation of the starting weights.

**11:00-11:25** Fanny Yang (ETH Zurich, Switzerland)

Prospects and Perils of Interpolating Models

In this talk, I will discuss recent work from our group studying interpolating high-dimensional linear models. On the bright side, we show that for sparse ground truths, minimum-norm interpolators (including max-margin classifiers) can achieve high-dimensional asymptotic consistency and fast rates for isotropic Gaussian covariates. However, we also prove some caveats of such interpolating solutions in the context of robustness that are also observed for neural network learning: when performing adversarial training, interpolation can hurt robust test accuracy as compared to regularized solutions. Further, in the low-sample regime, the adversarially robust max-margin solution surprisingly can achieve lower robust accuracy than the standard max-margin classifier.

**11:30-11:55** Maria Skoularidou (University of Cambridge, UK)

On the inversion of Generative Adversarial Networks

TBA

**Organizers**:

- Sophie Langer
- Nicole Mücke