In this talk, we propose a model describing the growth of tree stems and vine, taking into account also the presence of external obstacles. The system evolution is described by an integral differential equation which becomes discontinuous when the stem hits the obstacle. The stem feels the obstacle reaction not just at the tip, but along the whole stem. This fact represents one of the main challenges to overcome, since it produces a cone of possible reactions which is not normal with respect to the obstacle. However, using the geometric structure of the problem and optimal control tools, we are able to prove existence and uniqueness of the solution for the integral differential equation under natural assumptions on the initial data.

An interval exchange transformation is a map  of an 
interval to 
itself that rearranges a finite number of intervals by translations.  They 
appear among other places in the 
subject of rational billiards and flows of translation surfaces. An 
interesting phenomenon is that an IET may have dense orbits that are not 
uniformly distributed, a property known as non unique ergodicity.  I will 
talk about this phenomenon and present some new results about how common 
this is. Joint work with Jon Chaika.

The structure of the state of art of scientific research is an important object of study motivated by the understanding of how research evolves and how new fields of study stem from existing research. In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks, in particular keyword co-occurrence networks proved useful to provide maps of knowledge inside a scientific domain. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose to adopt a simplicial complex approach to co-occurrence relations, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the shape of concepts in mathematical research, focusing on homological cycles, regions with low connectivity in the simplicial structure, and we discuss their role in the understanding of the evolution of scientific research. In addition, we map authors’ contribution to the conceptual space, and explore their role in the formation of homological cycles.

Authors: Daniele Cassese, Vsevolod Salnikov, Renaud Lambiotte
 

The number of solutions of a given algebro-geometric configuration, when it is finite, does not change upon a small perturbation of the parameters; this persists 
even upon specializations that change the topology.    The precise formulation of this principle of Poncelet and Schubert   required, i.a., the notions of   algebraically closed fields, flatness, completenesss, multiplicity.     I will explain a model-theoretic version, presented in   quite different terms.  It applies notably to difference equations involving the Galois-Frobenius automorphism $x^p$, uniformly in a prime $p$.   In fixed positive characteristic, interesting technical problems arise that I will discuss if time permits.  

We are all familiar with the need for continuum mechanics-based models in physical applications. In this case, we are interested in large-scale water-wave problems, such as coastal flows and dam breaks.
When modelling these problems, we inevitably wish to solve them on a finite domain, and require boundary conditions to do so. Ideally, we would recreate the semi-infinite nature of a coastline by allowing any generated waves to flow out of the domain, as opposed to them reflecting off the far-field boundary and disrupting the remainder of our simulation. However, applying an appropriate boundary condition is not as straightforward as we might think.
In this talk, we aim to evaluate alternatives to so-called 'active boundary condition' absorption. We will derive a toy model of a shallow-water wavetank, and consider the implementation and efficacy of two 'passive' absorption techniques.
 

We study how the synchronizability of a diffusive network increases (or decreases) when we add some links in its underlying graph. This is of interest in the context of power grids where people want to prevent from having blackouts, or for neural networks where synchronization is responsible of many diseases such as Parkinson. Based on spectral properties for Laplacian matrices, we show some classification results obtained (with Tiago Pereira and Philipp Pade) with respect to the effects of these links.
 

There are a huge number of nonlinear partial differential equations that do not have analytic solutions.   Often one can find similarity solutions, which reduce the number of independent variables, but still leads, generally, to a nonlinear equation.  This can, only sometimes, be solved analytically.  But always the solution is independent of the initial conditions.   What role do they play?   It is generally stated that the similarity  solution agrees with the (not determined) exact solution when (for some variable say t) obeys t >> t_1.   But what is  t_1?   How does it depend on the initial conditions?  How large must  t be for the similarity solution to be within 15, 10, 5, 1, 0.1, ….. percent of the real solution?   And how does this depend on the parameters and initial conditions of the problem?   I will explain how two such typical, but somewhat different, fundamental problems can be solved, both analytically and numerically,  and compare some of the results with small scale laboratory experiments, performed during the talk.  It will be suggested that many members of the audience could take away the ideas and apply them in their own special areas.

Stable commutator length (scl) is a well established invariant of group elements g  (write scl(g)) and  has both geometric and algebraic meaning.

It is a phenomenon that many classes of non-positively curved groups have a gap in stable commutator length: For every non-trivial element g, scl(g) > C for some C>0. Such gaps may be found in hyperbolic groups, Baumslag-solitair groups, free products, Mapping class groups, etc. 
However, the exact size of this gap usually unknown, which is due to a lack of a good source of “quasimorphisms”.

In this talk I will construct a new source of quasimorphisms which yield optimal gaps and show that for Right-Angled Artin Groups and their subgroups the gap of stable commutator length is exactly 1/2. I will also show this gap for certain amalgamated free products.