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The puzzling phenonenon of wave localization refers to unexpected confinement of waves triggered by disorder in the propagating medium. Localization arises in many physical and mathematical systems and has many important implications and applications. A particularly important case is the Schrödinger equation of quantum mechanics, for which the localization behavior is crucial to the electrical properties of materials. Mathematically it is tied to exponential decay of eigenfunctions of operators instead of their expected extension throughout the domain. Although localization has been studied by physicists and mathematicians for the better part of a century, many aspects remain mysterious. In particular, the sort of deterministic quantitative results needed to predict, control, and exploit localization have remained elusive. This talk will focus on major strides made in recent years based on the introduction of the landscape function and its partner, the effective potential. We will describe these developments from the viewpoint of a computational mathematician who sees the landscape theory as a completely unorthodox sort of a numerical method for computing spectra.

Viruses encapsulate their genetic material into protein containers that act akin to molecular Trojan horses, protecting viral genomes between rounds of infection and facilitating their release into the host cell environment. In the majority of viruses, including major human pathogens, these containers have icosahedral symmetry. Mathematical techniques from group, graph and tiling theory can therefore be used to better understand how viruses form, evolve and infect their hosts, and point the way to novel antiviral solutions.

In this talk, I will present an overarching theory of virus architecture, that contains the seminal Caspar Klug theory as a special case and solves long-standing open problems in structural virology. Combining insights into virus structure with a range of different mathematical modelling techniques, such as Gillespie algorithms, I will show how viral life cycles can be better understood through the lens of viral geometry. In particular, I will discuss a recent model for genome release from the viral capsid. I will also demonstrate the instrumental role of the Hamiltonian path concept in the discovery of a virus assembly mechanism that occurs in many human pathogens, such as Picornaviruses – a family that includes the common cold virus– and Hepatitis B and C virus. I will use multi-scale models of a viral infection and implicit fitness landscapes in order to demonstrate that therapeutic interventions directed against this mechanism have advantages over conventional forms of anti-viral therapy. The talk will finish with a discussion of how the new mathematical and mechanistic insights can be exploited in bio-nanotechnology for applications in vaccination and gene therapy.

The subject of $p$-adic cohomologies is over fifty years old. Many new developments have recently occurred. I will mostly limit myself to discussing some pertaining to the de Rham-Witt complex. After recalling the historical background and the basic results, I will give an overview of the new approach of Bhatt, Lurie and Mathew.

Solidification processes of liquid metal alloys,  bubble dynamics (as in Taylor flows), pinch-offs of liquid-liquid jets, the formation of polymeric membranes, or the structure of high concentration photovoltaic cells are described by the dynamics of multiphase fluids. On the other hand, in applications such as mass spectrometry, lab-on-a-chip, and electro-fluidic displays, fluids on the micro-scale associated with a dielectric medium are of interest. Moreover, in many of these applications one is interested in influencing (or controlling) the underlying phenomenon in order to reach a desired goal. Examples for the latter could be the porosity structure of a polymeric membrane to achieve certain desired filtration properties of the membrane, or to optimize a microfluidic device for the transport of pharmaceutical agents.

A promising mathematical model for the behavior of multiphase flows associated with the applications mentioned above is given by a phase-field model of Cahn-Hilliard / Navier-Stokes (CHNS) type. Some strengths of phase field (or diffuse interface) approaches are due to their ability to overcome both, analytical difficulties of topological changes, such as, e.g., droplet break-ups or the coalescence of interfaces, and numerical challenges in capturing the interface dynamics between the fluid phases. Deep quenches in solidification processes of liquid alloys or rapid wall hardening in the formation of polymer membranes ask for non-smooth energies in connection with Cahn-Hilliard models. Analytically, this gives rise to a variational inequality coupled to the equations of hydrodynamics, thus yielding a non-smooth system (in the sense that the map associated with the underlying operator equation is not necessarily Frechet differentiable). In contrast to phase-field approaches,
one may consider sharp interface models. In view of this, our microfluidic applications alluded to above are formulated in terms of  sharp interface models and Hele-Shaw flows. In this context, we are particularly interested in applications of electrowetting on dielectric (EWOD) with contact line pinning. The latter phenomenon resembles friction, yields a variational inequality of the second kind, and – once again – it results in an overall nonsmooth mathematical model of the physical process.

   In both settings described above, optimal control problems are relevant in order to influence the underlying physical process to approach a desired system state.  The associated optimization problems are delicate as the respective constraints involve non-smooth structures which render the problems degenerate and prevent a direct application of sophisticated tools for the characterization of solutions. Such characterizations are, however, of paramount importance in the design of numerical solution schemes.

This talk addresses some of the analytical challenges associated with optimal control problems involving non-smooth structures, offers pathways to solutions, and it reports on numerical results for both problem classes introduced above.
 

We discuss a recent generalization of the classical 3-dimensional light bulb theorem to 4-dimensions. We connect this with fundamental questions about knotting of surfaces in 4-dimensional manifolds as well as new directions regarding knotting of 3-balls in 4-manifolds.

Aden Forrow
Optimal transport and cell differentiation

Abstract
Optimal transport is a rich theory for comparing distributions, with both deep mathematics and application ranging from 18th century fortification planning to computer graphics. I will tie its mathematical story to a biological one, on the differentiation of cells from pluripotency to specialized functional types. First the mathematics can support the biology: optimal transport is an apt tool for linking experimental samples across a developmental time course. Then the biology can inspire new mathematics: based on the branching structure expected in differentiation pathways, we can find a regularization method that dramatically improves the statistical performance of optimal transport.

Paul Ziegler
Geometry and Arithmetic

Abstract
For a family of polynomials in several variables with integral coefficients, the Weil conjectures give a surprising relationship between the geometry of the complex-valued roots of these polynomials and the number of roots of these polynomials "modulo p". I will give an introduction to this circle of results and try to explain how they are used in modern research.
 

Valérie Voorsluijs
Deterministic limit of intracellular calcium spikes
Abstract: In non-excitable cells, global calcium spikes emerge from the collective dynamics of clusters of calcium channels that are coupled by diffusion. Current modeling approaches have opposed stochastic descriptions of these systems to purely deterministic models, while both paradoxically appear compatible with experimental data. Combining fully stochastic simulations and mean-field analyses, we demonstrate that these two approaches can be reconciled. Our fully stochastic model generates spike sequences that can be seen as noise-perturbed oscillations of deterministic origin while displaying statistical properties in agreement with experimental data. These underlying deterministic oscillations arise from a phenomenological spike nucleation mechanism.

Matthias Nagel
Knots in dimensions three and four
Abstract: Knot theory studies the various embeddings of a circle into three-dimensional space. I will describe an equivalence relation on knots, called "concordance", which takes the fourth dimension into account. The study of concordance is intimately related with many problems at the heart of the topology of four-manifolds, such as the difference between the smooth and the topological category, and I will discuss results that illuminate this relation.

What is the point of giving a talk?  What is the point of going to a talk?  In this presentation, which is intended to have a lot of audience participation, I would like to explore how one should prepare talks for different audiences and different occasions, and what one should try to get out of going to a talk.

In a joint work with Matt Tointon, we study the fine structure of approximate groups. We deduce various applications on growth, isoperimetry and quantitative estimates for the the simple random walk on finite vertex transitive graphs.