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Transitions between weak-noise-induced resonance phenomena in a multiple timescale neural systemMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We consider a stochastic slow-fast nonlinear dynamical system derived from computational neuroscience. Independently, we uncover the mechanisms that underlie two different forms of weak-noise-induced resonance phenomena, namely, self-induced stochastic resonance (SISR) and inverse stochastic resonance (ISR) in the system. We then show that SISR and ISR are actually mathematically related through the relative geometric positioning (and stability) of the fixed point and the generic folded singularity of the system's critical manifold. This result could explain the experimental observation in which real biological neurons with identical physiological features and synaptic inputs sometimes encode different information.

Kreisel’s philosophy of mathematics021 (Ludwigstr. 31, Raum 021 , 80539 München)

Kreisel has described how his interest in foundations of mathematics arose early: “Since my school days I had had those interests in foundations that force themselves on beginners when they read Euclid's Elements (which was then still done at school in England), or later when they are introduced to the differential calculus.” At the same time, he had a mathematician’s distrust of philosophers of mathematics, though he was one himself, in the way in which other mathematicians such as Cantor, Dedekind, Hilbert, Brouwer, Weyl, and Gödel were philosophers of mathematics, motivating and justifying the way in which they did their mathematics. Among Kreisel’s more than 200 publications, a relatively small number are explicitly philosophical, but these grow out of and at the same time inform the whole body of his work. Even when we recognize Kreisel as a philosopher of mathematics, it’s not easy to say which philosophy of mathematics is his. My talk will be a preliminary attempt to do this.

Algebraic Systems BiologyMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Signalling pathways can be modelled as a biochemical reaction network. When the kinetics are to follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system. I will overview approaches to analyse these models using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability, specifically focusing on parameter identifiability for parameter inference. I will briefly mention how topological data analysis can help distinguish models and data.

From PDEs to data science: an adventure with the graph LaplacianGebäude 33, Raum 1431 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

In this talk we briefly review some basic PDE models that are used to model phase separation in materials science. They have since become important tools in image processing and over the last years semi-supervised learning strategies could be implemented with these PDEs at the core. The main ingredient is the graph Laplacian that stems from a graph representation of the data. This matrix is large and typically dense. We illustrate some of its crucial features and show how to efficiently work with the graph Laplacian. In particular, we need some of its eigenvectors and for this the Lanczos process needs to be implemented efficiently. Here, we suggest the use of the NFFT method for evaluating the matrix vector products without even fully constructing the matrix. We illustrate the performance on several examples.

Partial differential equations on surfaces: Analysis, numerical methods and applicationsMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

In this presentation we give an overview of several aspects related to the analysis and numerical simulation of elliptic and parabolic PDEs on (evolving) surfaces. We consider both scalar valued and vector valued PDEs. Surface partial differential equations are used in models from, for example, computational fluid dynamics and computational biology. Two examples of such applications are briefy addressed. Concerning the analysis of surface PDEs, a few results related to well-posedness of certain weak formulations are explained. We discuss a class of recently developed finite element discretization methods for an accurate numerical simulation of surface PDEs. The key idea of these methods is explained and numerical simulation results are presented.

The many-worlds interpretation and the Born rule021 (Ludwigstr. 31, Raum 021 , 80539 München)

I will describe my version of the many-worlds interpretation. I will explain how one can see our three-dimensional world, in spite of the fact that the basic ontology is the wave function in 3N dimensions. I will explain our experience of probability, in spite of the fact that the theory is deterministic and we have no ignorance about relevant wave function ontology in the measurement-type situations. The main novel element in the talk will be my understanding that explanation of empirically observed Born rule requires additional postulate beyond description of the unitary evolution of the wave function of the Universe.

On the geometry of flat isometric immersions with Hölder type regularityRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

Smooth isometric immersions of flat domains into ٰEuclidean spaces are known to enjoy a rigidity property referred to as developability, provided that the dimension of the target space is not too high. In particular, the images of such isometries from 2d domains into the 3d space are locally ruled surfaces. It has been known since a few years that this rigidity property survives if the second fundamental form of the immersion is merely L^2 integrable. On the other hand, convex integration methods à la Nash and Kuiper show that this would no more be the case for C^1α immersions if α < 1/5. We will show that a geometrically meaningful second fundamental form can be defined for such immersions if α is large enough in order to prove their developability for α > 2/3.

Charles S. Pierce as mathematical philosopher: the case of the Existential Graphs021 (Ludwigstr. 31, Raum 021 , 80539 München)

The aim of this presentation is to relate the philosophy of Charles S. Peirce (1839-1914) to the current idea of mathematical philosophy (at least in some aspects). In this sense Peirce could be seen as a forerunner of this idea. At many opportunities he expressed his aim at an exact or scientific philosophy, where problems should be treated as mathematically as possible. The “pragmatic maxim” and his theory of signs were two main bases of his thinking. Now, Peirce argued for a particular conception of mathematics where the special kind of signs called icons played an essential role. In this talk I will discuss the case of his diagrammatic system, the Existential Graphs, as a mathematical philosophical reconstruction of deductive logic. Moreover, I will focus on the way the notion of identity is formulated in the “Beta Graphs” and to show some problems arising from it.

Reaction-Diffusion Models: Dynamics, Control, and NumericsRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

Reaction-diffusion equations are ubiquitous in a variety of fields including combustion and population dynamics. There is an extensive mathematical literature addressing the analysis of steady state solutions, traveling waves, and their stability, among other properties. Control problems involving these models arise in many applications . Often times, control and/or state constraints emerge as intrinsic requirements of the processes under consideration. There is also a broad literature on the control of those systems, addressing, in particular, issues such as the possibility of driving the system to a given final configuration in finite time. But, the necessity of preserving the natural constraints of the process are rarely taken into account. In this lecture we shall present the recent work of our team on the Fisher-KPP and Allen-Canh or bistable model, showing results of two different types depending of the initial and final states under consideration. First, the fact that, in some cases, the target can be reached for large enough time horizons, but that there is a minimal waiting time for this property to hold. And, second, negative results showing the existence of threshold effects, making some targets unreachable. We shall also present some numerical experiments showing that optimal trajectories are often quite complex, and hard to guess from purely analytical arguments.

The Modal Logic of Kripkean Truth 021 (Ludwigstr. 31, Raum 021 , 80539 München)

(joint work with Carlo Nicolai) We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model M, or an axiomatization S thereof, we find a modal logic M such that a modal sentence A is a theorem of M if and only if the sentence A* obtained by translating the modal operator with the truth predicate is true in M or a theorem of S under all such translations. To this end, we introduce a novel possible world semantics featuring both classical and subclassical worlds and establish the completeness of a familiy of non-classical modal logics (in the sense of Segerberg) modal logics, whose internal logic is subclassical, with respect to this semantics. In a second step we show how to emulate the models of the modal logic within the lattice of Kripkean fixed-point models.

Takeuti’s finitism in the context of the Kyoto schoolA017 (Geschwister-Scholl-Pl. 1, 80539 München)

Gaisi Takeuti (1926-2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He furthered the realization of Hilbert's program by formulating Gentzen's sequent calculus for higher-oder logics, conjecturing the cut-elimination theorem holds for it (Takeuti's conjecture), and obtaining several stunning results in the 1950--60's towards the solution of his conjecture. In this talk, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, we point out that there is a crucial difference between Takeuti's program and Hilbert's program, which is based on the fact that Takeuti's philosophical thinking goes back to Nishida's philosophy in Japan.

This is joint work with Andrew Arana.

Takeuti’s argument of the well-foundedness of ordinals up to the epsilon_0021 (Ludwigstr. 31, Raum 021 , 80539 München)

Gaisi Takeuti (1926-2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. In 1950—60’s, he provided several partial solutions to his conjecture that the cut-elimination theorem holds for it. For the crucial part of the proof, he invented a new system of ordinals called “ordinal diagrams” and presented several proofs of the well-foundedness of it. From a philosophical point of view, such proofs of the well-foundedness must be examined carefully since this is exactly the point which must be beyond Hilbert’s finitism. In this talk, we focus on his argument of the well-foundedness of ordinals up to the epsilon_0 in his well-known book “Proof Theory” (1975, 1987) as a starting point. The structure of the argument is deserved to be examined even though Takeuti says his argument is more clear than Gentzen’s well-known argument in 1943 since some parts of it lack precise formulations. Hence, we address this issue in the detail and try to specify a suitable meta-theory for it.

This is joint work with Andrew Arana.

Optimal Kidney Exchange with ImmunosuppressantsMI 01.10.033 (Boltzmannstr. 3, 85748 Garching)

Potent immunosuppressant drugs suppress the body’s ability to reject a transplanted organ up to the point that a transplant across blood- or tissue-type incompatibility becomes possible. In contrast to the standard kidney exchange problem, our setting also involves the decision about which patients receive from the limited supply of immunosuppressants that make them compatible with originally incompatible kidneys. We firstly present a general computational framework to model this problem. Our main contribution is a range of efficient algorithms that provide flexibility in terms of meeting meaningful objectives. We also show that these algorithms satisfy desirable axiomatic and strategic properties.

Tipping points of nonautonomous dynamical systems: from theory to applicationMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Dynamical systems have been immensely successful in describing and predicting behaviours of a wide range of applications especially in cases where they are well-modelled by a closed (unforced or autonomous) dynamical system. However, not all systems are closed; for example, elements in the climate system respond to changes in greenhouse gas forcing in quite a nontrivial manner over a range of timescales. For such open (forced or nonautonomous) systems with time-varying inputs, it is hard to get much insight out of dynamical systems theory and usually one resorts to numerical simulations. This is especially the case when inputs vary on a similar timescale to the system itself. I will discuss some recent progress in understanding critical transitions or tipping points in classes of nonautonomous dynamical systems, and an application to modelling a specific climate tipping point.