This will be a conference on the wide range of topics aimed to inspire future directions and new applications of microlocal methods. 

I will introduce ideal webs = ideal bipartite graphs on decorated surfaces. They are the simplest possible higher rank analogs of ideal triangulations of decorated surfaces.  

I will explain how to use them to define cluster atlases on several flavors of character varieties for the groups of type A_m, equivariant under the action of the mapping class group, and how to define a related family of 3d Calabi-Yau categories, also equivariant under the action of the mapping class group.

It is an expository talk, mostly following arXiv:1607.05228

I will introduce ideal webs = ideal bipartite graphs on decorated surfaces. They are the simplest possible higher rank analogs of ideal triangulations of decorated surfaces.  

I will explain how to use them to define cluster atlases on several flavors of character varieties for the groups of type A_m, equivariant under the action of the mapping class group, and how to define a related family of 3d Calabi-Yau categories, also equivariant under the action of the mapping class group.

It is an expository talk, mostly following arXiv:1607.05228

We will use the microlocal analysis to describe and compare the monostatic Synthetic aperture radar (SAR) imaging and the  Doppler SAR imaging.   In these problems, the forward operator F, which maps the image to the data is a  FIO with singularities: folds, cusps, blowdowns. We will focus on the composition calculus of these operators and we will  consider the normal operator F*F.  This method produces artifacts  which can be classified as smooth Lagrangians like canonical graphs or two-sided folds or as singular Lagrangians like open umbrellas.

I will discuss a stability estimate for the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature.

The estimate is of the form $L^2\mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry). Only tangential derivatives at the boundary are used. The proof is based on the observation that the Pestov identity with boundary term localizes in frequency. This estimate answers a question raised by Boman and Sharafutdinov.

I will discuss a collection of recent results on both linear and nonlinear results on discrete and quantum graphs. First I will discuss results related to spectral theory for the (possibly magnetic) Laplacian on a variety of graphs that relates to work with Berkolaiko, Canzani, Cox, Weinstein and others. I will also discuss nonlinear equations on graphs that comprises work with Bandres, Berkolaiko, Osting, Pelinovsky, Rechtsman,  and others. 

In this talk, we consider the following question: Given the source to solution map for a relativistic Boltzmann equation on a known open set $V$ of a Lorentzian spacetime $(\mathbb{R}\times N,g)$, can we use this data to uniquely determine the spacetime metric on an unknown region of $\mathbb{R}\times N$?

We will show that the answer is yes. Precisely, we determine the metric on the domain of causal influence for the set $V$. Key to our proof is that the nonlinear term in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linearized problem. We use this relationship together with an analysis of the behaviour of particle collisions by classical microlocal techniques to determine the set of locations in $V$ where we first receive light signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.

The technique of using the nonlinearity in a partial differential equation as a feature with which to gain knowledge of a related linearized problem first appeared in [1] in the context of a wave equation with a quadratic nonlinearity. We will briefly survey this and related work as they provide context for our result. 

The new results presented in this talk is joint work with Antti Kujanapää, Matti Lassas, and Tony Liimatainen (University of Helsinki).

[1] Kuylev Y., Lassas M., Uhlmann G., Inventiones mathematicae 212.3 (2018): 781-857.