### Mathematisches Forschungskolloquium 2019

**Daniel Gerike**(Otto-von-Guericke University of Magdeburg)**"On the Cycle Structure of Permutation Polynomials of Shape $x^t + \gamma \Tr_{q^n/q}(x^k)$"**

Abstract**20.11.2019, 15:15 Uhr, HS 228 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Gohar Kyureghyan

**Prof. Dr. Wolfgang Willems**(Otto-von-Guericke Universität Magdeburg)**"Group codes are asymptotically good"**

23.10.2019, 15:15 Uhr, HS 228 (Ulmenstr. 69, Haus 3)

Kolloquiumsleiter: Prof. Dr. Gohar Kyureghyan

**Anton Evgrafov**(Department of Mechanical Engineering, Technical University of Denmark)**"Nonlocal optimal control in the conduction coefficients: a bit of theory and practice"**

Abstract: One of the most popular methods for optimal design in continuum mechanics is topology optimization. Within this framework, which has been made particularly relevant owing to the recent advances in additive manufacturing, at each point in the design domain one takes a decision about which material should be placed there. Computationally one solves a relaxed version of this problem, which reduces to optimal control in the coefficients of partial differential equations. In most interesting and relevant cases the problem is non-convex and lacks optimal solutions. There are essentially two strategies for dealing with this issue. One is to restrict the set of allowable controls by applying external regularization tools to the problem, thereby only allowing designs satisfying certain explicit or implicit regularity criteria. The alternative is to extend the set of allowable controls by also considering limits of highly oscillatory designs, interpreting them as composite materials in the sense defined by the homogenization process.

We present a completely different approach to this optimal design problem. Namely we replace the governing partial differential equations of the continuum mechanical model with a non-local, integral, peridynamic model. The control variables still enter the problem as coefficients of the governing non-local equation. On the one hand, we verify that in the limit of small interaction horizons we recover the governing PDE with coefficients we control. On the other hand the resulting control problem, in spite of its non-convexity, attains its infimum and therefore admits globally optimal solutions without the need for any external regularization techniques. We will illustrate our results with numerical examples related to steady state heat conduction.

This is joint work with Jose C. Bellido (Department of Mathematics, University of Castilla-La Mancha, Spain).**23.10.2019, 15:00 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Jens Starke

**Prof. Dr. Enno Mammen**(Universität Heidelberg)**"STATISTICAL INFERENCE ON PARTY POSITIONS FROM TEXTS: STATISTICAL MODELING, BOOTSTRAP AND ADJUSTING FOR TIME EFFECTS"**

Abstract: In this talk we discuss Poisson reduced rank models for low-dimensional summaries of high-dimensional Poisson vectors that allow for inference on the location of individuals in a low-dimensional space. We show that under weak dependence assumptions the locations can be consistently estimated by Poisson maximum likelihood estimation. Using our theory, we develop a rule of determining the proper dimension of the positions. Additionally, we consider a novel model which allows the parameters to change over time and develop an estimation procedure based on LASSO and fused LASSO penalization techniques to address high-dimensionality via significant dimension reduction. Bootstrap confidence intervals are proposed based on integer-valued time series processes.

Our main motivation for studying Poisson reduced rank models came from applications to political text data where we model word counts in a political document as Poisson random variables. We use party manifesto data from German parties over seven federal elections after German reunification and apply our approach for statistical inference on the evolution of multi-dimensional party positions.

The talk reports on joint work with C. Jentsch (Dortmund) and Eun Ryung Lee (Seoul).**21.10.2019, 17:00 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Alexander Meister

**Prof. Dr. Andrzej Zuk**(Université Paris 7)**"From PDEs to Groups"**

03.09.2019, 11:00 Uhr, HS 326 (Ulmenstr. 69, Haus 3)

Kolloquiumsleiter: Prof. Dr. Jan-Christoph Schlage-Puchta

**M.Sc. Theresa Lange**(TU Berlin)**"On the continuous time limit of ensemble-based filtering algorithms"**

Abstract: The stochastic filtering problem consists in estimating the current state of an unknown stochastic system based on its computational model and measurement data retrieved from it. In the last decades, a huge amount of filtering algorithms was proposed and improved in accordance to the increase in modelling and observational complexity. Especially in the nonlinear, high-dimensional case, ensemble-based algorithms numerically proved to work really well. When investigating their algorithmic properties, an alternative viewpoint is to analyse their continuous time limit, i.e. the continuous-time formulation of the algorithm. In this analysis, different tools coming from the theory of continuous-time dynamical systems may be applied and potentially yield different insights into the algorithmic structure.

In my talk, I will present recent results on the existence of a continuous time limit of a selection of the above ensemble-based algorithms, especially the sense of convergence together with concrete convergence rates. These results give the consistent derivation of continuous-time filtering algorithms and form the basis for future property analyses.**28.08.2019, 14:00 Uhr, HS 125 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Jens Starke

**PD Dr. Aleksandra Zimmermann**(Universität Duisburg-Essen und Universität Rostock)**"Renormalized solutions for a stochastic p-Laplace equation with L^1-initial data"**

Abstract**14.08.2019, 15:00 Uhr, HS 125 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Peter Takac, Ph. D.

**Dr. Annalisa Iuorio**(Technische Universität Wien)**"Interplay of water and toxicity in vegetation pattern formation"**

Abstract: Vegetation patterns are largely present in nature. Their formation, however, is still far from being fully understood. A prevailing view is to treat water availability as the main causal factor for the emergence of vegetation patterns, as they are mostly observed in arid and semiarid regions. While successful, this hypothesis fails to explain the presence of vegetation patterns in humid environments. We present here a novel toxicity-mediated model for vegetation pattern formation and investigate its rich structure. This model consists of three PDEs accounting for a dynamic balance between biomass, water, and toxic compounds. Different (ecologically feasible) regions of the model’s parameter space give rise to stable spatial vegetation patterns in Turing and non-Turing regimes. Strong negative feedback gives rise to dynamic spatial patterns that continuously move in space while retaining their stable topology.**12.07.2019, 13:15 Uhr, Raum 427 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Jens Starke

**Dr. Leo Perrin**(INRIA-Paris)**"How to Analyse an S-box, and, in the Process, Prove the Russian Standardizing Agency is Wrong"**

Abstract: S-boxes are functions mapping {0,1}^n to {0,1}^m (for m, n small, typically 4 or 8) that are the only source of non-linearity in many symmetric cryptographic primitives. In this tutorial, I will explain what the key properties of these objects are in terms of security and how to analyse these properties in practice using SAGE.

Then, I will describe the state of the art on how to recover the algorithm used to generate an S-box using only its truth table (a process called "S-box reverse-engineering"). In particular, I will walk you through a line of research that allowed me and my co-authors to eventually uncover the structure hidden inside the S-box used by the latest Russian standards, namely the hash function Streebog and the block cipher Kuznyechik. As I will show, the presence of this structure is incompatible with the explanations provided by its designers.

The corresponding techniques have applications far beyond the investigation of this specific S-box. They are in fact closely related to the method used by Dillon et al. to investigate the big APN problem, and more generally to the understanding of CCZ-equivalence.**26.06.2019, 13:30 Uhr, HS 228 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Gohar Kyureghyan

**Dr. Shuxing Li**(Otto-von-Guericke Universität Magdeburg)**"Intersection distribution and its application"**

Abstract: For a point set of the classical projective plane PG(2,q), we introduce the concept of intersection distribution, which reflects how this point set interacts the lines of PG(2,q). For point set with q+1 points, which usually has a compact polynomial representation, the intersection dsitribution turns out to be very informative. Indeed, the intersection distribution measures how close a (q+1)-set to an oval or equivalently, how close the corresponding polynomial to an o-polynomial (when q is even) or to x^2 (when q is odd). These two closely related aspects, the geometric one and the polynomial one, of the intersection distribution, offer an advantage viewpoint to appreciate the connection between (q+1)-sets of PG(2,q) and their polynomials.

By characterizing the point sets with certain special intersection distributions, we show that the information of intersection distribution is crucial. While the computation of intersection distribution is in general difficult, via the polynomial viewpoint, we succeed in computing the intersection distribution for several classes of monomials. In fact, the intersection distribution provides a new angle to distinguish polynomials, especially monomials.

Finally, we display a natural application of intersection distribution to the Kakeya set on classical affine planes. While determining the size of a Kakeya set is hard, we can achieve this by considering several infinite families of Kakeya sets with simple polynomial representations, in which their sizes follow immediately from the intersection distributions.**19.06.2019, 15:15 Uhr, HS 228 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Gohar Kyureghyan

**Dipl. Math. Alexandr Polujan**(Otto-von-Guericke Universität Magdeburg)**"Homogeneous Cubic Bent Functions: From Known Examples to New Constructions"**

Abstract**15.05.2019, 13:15 Uhr, HS 228 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Gohar Kyureghyan

**PD Dr. Wolfram Just**(Queen Mary University of London)**"Transfer operator technique for analytic maps - Or: Chaos in the Hilbert space"**

11.04.2019, 16:00 Uhr, Raum 427 (Ulmenstr. 69, Haus 3)

Kolloquiumsleiter: Prof. Dr. Jens Starke

**Prof. Dr. Eduard Feireisl**(Akademie der Wiss., Prag)**"On weak--weak uniqueness for the isentropic Euler system"**

Abstract: We discuss uniqueness of certain weak solutions in the class of general (measure--valued) solutions for the isentropic Euler system. The weak solution is unique provided it belongs to a certain Besov class and its symmetric gradient satisfies a one--sided Lipschitz condition.**10.04.2019, 15:00 Uhr, HS 125 (Ulmenstr. 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. Peter Takac, Ph. D.

**Prof. John Hogan**(University of Bristol, UK)**"The rough with the smooth; regularization of the Painlevé paradox"**

Abstract: When a piece of chalk is pushed across a blackboard, it can jump and judder, often accompanied by an unpleasant screeching noise. This unwanted phenomenon occurs in many engineering applications, including robotic manipulators. In 1905, Painlevé showed that the governing rigid body equations can produce a paradox: when the coefficient of friction exceeds a critical value, the rod appears to be driven the rigid surface. In an attempt to resolve this paradox, a jump in vertical velocity called impact without collision (IWC), has been proposed. But this approach can itself produce contradictions. A more consistent approach is to relax the rigid body assumption in the neighbourhood of the contact point, by assuming compliance there. This regularization produces a singularly perturbed problem. In this talk, I will show that the compliant problem has three major phases: the frictional torque compresses the compliant surface until the rod stops sliding, then the rod sticks, before finally leaving the surface. The rigid body problem has one non-hyperbolic point. Using blowup, I will show that the compliant problem has a canard at this point.

Joint work with Kristian Kristiansen (DTU).**21.03.2019, 13:00 Uhr, HS 326 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Jens Starke

**Dr. Konstantinos Spiliotis**(National Technical University of Athens, Greece)**"Equation Free Computations on Neuronal Networks: From Neuronal Interactions To Emergent Brain Dynamics."**

Abstract**14.03.2019, 10:00 Uhr, Raum 427 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Jens Starke

**Prof. Dr. Faruk Gologlu**(Charls University Prague)**"Families which are not equivalent to APN permutations"**

Abstract: Almost Perfect Nonlinear (APN) functions are important in cryptography. We will present theoretical results on inequivalence of APN functions to permutations including Gold and Kasami families. We also present results which explain why it wasn't possible to generalize the Kim function which is equivalent to the only known APN permutation on an even extension degree.**12.03.2019, 14:00 Uhr, HS 228 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Gohar Kyureghyan

**Dr. Annalisa Iuorio**(TU Wien)**"Geometric Singular Perturbation Analysis of a Model for Micro-Electro Mechanical Systems (MEMS)"**

Abstract**07.03.2019, 14:00 Uhr, HS 326 (Ulmenstr. 69, Haus 3)**

Kolloquiumsleiter: Prof. Dr. Jens Starke

**Prof. Dr. Thomas Sunn Pedersen (Max-Planck-Institut für Plasmaphysik, IPP)**

"Neural Networks for the W7-X Experiment"

23.01.2019, 15:00 Uhr, HS 228 (Ulmenstr. 69, Haus 3)

Kolloquiumsleiter: Prof. Dr. Roger Labahn