# Mathematics

### General study plan for PhD studies in mathematics

## Projects for PhD positions

### Stochastic control with applications to mean-field games

Contact: Astrid Hilbert

Stochastic control theory combines the dynamics of states, given by a stochastic differential equation, with an optimization problem for a so called control parameter. It leads to solving a system of two coupled differential equations. In the setting of dynamic game theory the control parameters to be optimized are the preferences of the players. An interesting and challenging setting relates to the situation, when the dynamics and/or the cost function of the optimization problem depend on the law of the dynamics or if the number of states is very large and the empirical mean appears as a parameter. Such mathematical models are said to be of mean-field type. Recently these kind of mathematical studies have been attracting immense interest based on conditions of our contemporary society. One particular however typical setting, in which this type of modelling appears is the setting of networks.

Informal networks are abundant in our society. Be it physical networks like the following: communication, traffic, electric, water infrastructure; financial networks like banks, companies, investment, e-commerce; social networks like facebook, twitter; medical networks like disease spreading. Even though the areas are different, they all can be modelled by the same mathematical model consisting of three components: The agents and the environment they act in, the goal to be achieved, the dynamics, i.e., how the agents can act. Most of the "interesting" networks are very large, and consist of indistinguishable agents or groups of such, so that an exact analysis or simulation is out of scope. Understanding, analyzing, and predicting such networks will have a substantial impact on the areas named above. This project we will take a step towards the analysis of such networks.

### P-adic and ultrametric analysis and its applications

Contact: Andrei Khrennikov

Tree-like structures are widely present in nature. Mathematically they are modeled with the aid of ultrametric spaces; p-adic numbers give us the simplest, but mathematically very rich model. The project is developed to applications of ultrametric and especially p-adic analysis: dynamical systems, cognition, pseudo-differential operators, distribution theory, diffusion-reaction equation, geophysics, modeling of petroleum reservoirs.

### Applications of the mathematical apparatus of quantum theory to theory of decision making

Contact: Andrei Khrennikov

Nowadays the mathematical formalism of quantum theory is actively applied to cognition, psychology, economics, and finances. This is the basis of the novel decision making theory (DMT). In general, decision making (DM) is characterized by a high degree of risk and uncertainty. The creation of DMT for risky and uncertain contexts is a complex and challenging problem. From the very beginning DMT, which is based on classical probability theory, has been suffering of numerous paradoxes (the most famous are Allais, Ellsberg and Machina paradoxes). The new, quantum decision making theory (QDMT) is based on the quantum formalism, and has shown to resolve some of these paradoxes. QDMT is a consistent model which respects basic cognitive effects such as conjunction. disjunction, and order effects. QTDM is based on the quantum measurement theory, the theory of open quantum systems, quantum field theory (QFT), and quantum game theory. So far, only simplest models of QDMT have been elaborated. However, already QDMT in its present form is used in economics to model the behaviour of traders and in cognitive psychology to model irrationality. QDMT's predictions match experimental data from cognitive science, psychology, and behavioural economics, violating predictions of classical DMT. Roughly speaking, the quantum model of uncertainty matches agents' behaviour better than the classical one. However, only the first steps in the experimental comparison of classical and quantum models have been done. The aim of the PHD-project is to apply the existing quantum apparatus in its full power and develop a general QDMT as well as apply it to a variety of concrete DM-problems. In short, we plan to make important steps towards the development of quantum information and quantum probability based DM under risk and uncertainty.

More concretely the following working directions can be chosen:

- WP1 is devoted to dynamical (Markovian and non-Markovian) modelling of the DM-process by using open quantum systems and QFT.
- WP2 should resolve the paradoxes of DMT and create models for the basic cognitive effects (e.g., conjunction or order effects) and problems of behavioural economics and finances. WP3 is devoted to applications of quantum game theory to economics.

### Periodic points and nonlinear phenomena in arithmetical dynamical systems

Contact: Karl-Olof Lindahl

Periodic phenomena or cycles arise in many systems that evolve with time; physical, biological, financial and technical. The theory of dynamical systems concerns mathematical models of such systems. Of particular interest is characterization of cycles and the border between periodic, quasi-periodic, and non-periodic behaviour.

In the simplest mathematical models the dynamics is modeled by function iteration; a given state of the system is transformed into a new state by a function. From the work of Henri Poincaré in the late nineteenth century it is known that such models can mimic the behaviour of very complex systems.

As a model, we focus on ADS where the evolution in time is modeled by iterations of functions defined over fields like the rational numbers, p-adic fields, or fields of positive characteristic. Proposed research questions concern linearizability near periodic points, normal forms, and geometric distribution of periodic points.

Outcomes of expected results of the project could serve as the foundation of a geometric approach for solving Poonen's conjecture, stating that quadratic polynomials with rational coefficients cannot have rational periodic points of exact period greater than three. The conjecture is a fundamental case of the Morton-Silverman conjecture that is driving the field of ADS.

### Computational algebra and dynamical systems over finite rings

Contact: Karl-Olof Lindahl

Since the introduction of the BBS pseudorandom number generator in the 1980's there has been an increasing interest in dynamical systems defined by function iteration over finite rings. Other motivations is security in ICT systems and Pollard's rho-algorithm for prime factorization.

Given a map f over a finite ring we can associate with f a directed graph over the elements in the ring. We propose to study periodic points, the number of invariant components of the corresponding graph, and statistical aspects of such systems. This will involve tools from computational algebra, (analytic) number theory, graph theory, statistics and probability theory.

### Dynamical systems in population dynamics

Contact: Torsten Lindström

In this project we use dynamical systems theory in order to analyze various models and phenomena that arise in ecology. The models analyzed may be mechanistically formulated, fitted to data, deterministic, or stochastic. Deterministic models may include ordinary differential equations, discrete dynamical systems, and delayed differential equations. Various relations between such models that arise in different modeling approaches and under different simplifying assumptions can be analyzed. In discrete models the modelling part is not straightforward and the arising models might possess different dynamical behavior depending on how the dependencies between different processes are taken into account. In the theoretically difficult delay differential equation case results regarding both nearby ordinary differential equation models and discrete models can be of importance for the analysis.

Important questions are possible dynamical consequences of the specific modelling assumptions included in the model. Dynamical consequences can range from persistence of species to oscillatory and initial value dependent behavior and all these results are of importance for the understanding of concepts like biodiversity. In several quite simple cases the possibilities for initial value dependent behavior has not been fully explored and one such example is proving uniqueness of limit cycles for the chemostat. Here, uniqueness of limit cycles have been proved for several nearby and limiting cases but not for the case where the resource level has been explicitly taken into account.

### Approximations for stochastic differential equations

Contact: Roger Pettersson

The approximations are of different kind somewhat depending on what type of equations are considered. For systems describing epidemics the model is a birth-death process which, suitably scaled, is approximated by a diffusion process. Backwards stochastic partial differential equations, written as infinite dimensional systems, are approximated in space and time. Convergence rates of Yosida approximations is also in focus. Extensions to systems which allow reflections and/or jumps is of major interest.

### Microlocal analysis and evolution equations

Contacts: Patrik Wahlberg, Joachim Toft

This project concerns the interplay between evolution equations, harmonic and complex analysis, time-frequency analysis and pseudodifferential operators. Evolution equations are partial differential equations involving a parameter that can be interpreted as time. There is a close connection to mathematical physics where many fundamental equations are evolution equations, e.g. the heat equation and the Schrödinger equation. Microlocal analysis is also related to mathematical physics and may be described as phase space analysis. Phase space originates from classical mechanics and is also used in harmonic analysis.

A non-exhaustive list of possible subprojects follows.

- Refine and extend the theory for evolution equations of quadratic type based on Bargmann type transforms.
- Refine and extend results on propagation of Gabor type singularities for evolution equations.
- Work out results on propagation of Gabor singularities for evolutions equations of quadratic type, with continuous time-variability of the quadratic operator.
- Study well-posedness for the inverse problem of evolution equations for periodic and non-periodic initial data in the scales of Gevrey and Gelfand-Shilov regularity.
- Work out results on one-parameter groups of symbols for pseudodifferential operators inspired by work of Bony and Chemin, relaxing the polynomially bounded weights to subexponential and exponential weights.
- Carry over results on one-parameter groups of symbols for pseudodifferential operators to complex pseudodifferential operators using the Bargmann transform.