I'll give an introduction to spaces of geometric structures, and the power of considering such spaces as geometric objects in their own right. We'll start with some classical sets that are actually metric spaces with interesting geometry: the space of inner products on a real vector space; the space of flat n-tori; the space of norms on a p-adic vector space; and the space of compact metric spaces (!!).

I'll then focus on two sporadic but important spaces that arise only in low dimensions: "moduli space", the space of hyperbolic surfaces (or complex curves); and "Outer space", the space of metric graphs. A common thread throughout will be the presence and consequences of negative / non-positive curvature, both coarse and fine.

This talk should be accessible to all graduate students, including first-year grad students, who are familiar with the definition of a metric space; no other background is required.

Dynamics of the infectious disease transmission is often best understood taking into account the structure of population with respect to specific features, in example age or immunity level. Practical utility of such models depends on the appropriate calibration with the observed data. Here, we discuss the Bayesian approach to data assimilation in case of two-state age-structured model. This kind of models are frequently used to describe the disease dynamics (i.e. force of infection) basing on prevalence data collected at several time points. We demonstrate that, in the case when the explicit solution to the model equation is known, accounting for the data collection process in the Bayesian framework allows to obtain an unbiased posterior distribution for the parameters determining the force of infection. We further show analytically and through numerical tests that the posterior distribution of these parameters is stable with respect to cohort approximation (Escalator Boxcar Train) to the solution. Finally, we apply the technique to calibrate the model based on observed sero-prevalence of varicella in Poland.

The group of symmetries of the Icosahedron has fascinated mathematicians since the antiquity. I will review some current work about realization of this group as a group of symmetries of algebraic surfaces and as a group of birational automorphisms of projective space.

Cartan-Fubini type extension theorems are generalizations of Liouville's Theorem in conformal geometry. They are formulated as extension theorems of holomorphic maps in the setting of complex algebraic geometry. I will survey background history and the recent progress on this topic.

Given an algebraic variety, finding non-trivial morphisms to other varieties is one of the fundamental problems in algebraic geometry. Mori and others solved this problem in the 1980s for varieties whose canonical class is not non-negative on all curves. The question remains open for other varieties. However, in this talk I will report on a recent progress on this problem in a joint work with Thomas Peternell. I will concentrate on varieties with trivial canonical class and on Calabi-Yau manifolds in particular.

The nonlinear Beltrami equation is a planar nonlinear PDE that both describes all first-order elliptic nonlinear equations and has important relations to the theory of Quasiconformal mappings. We show that if the equation has Hölder continuous coefficients, then any homeomorphic solution must have a positive Jacobian. The proof is based on new Schauder-type estimates for nonlinear Beltrami equations, as well as the key observation that the inverse of a homeomorphic solution also solves a similar equation. This is a joint work with Kari Astala, Albert Clop, Daniel Faraco and Jarmo Jääskeläinen.

It is an old theorem in topological dynamics that to every topological group, one can associate a unique universal minimal flow (UMF): a flow that maps onto every minimal flow of the group. For some groups (for example, the locally compact ones), this flow is not metrizable and does not admit a concrete description. However, somewhat surprisingly, for many "large" Polish groups, the UMF is metrizable, can be computed, and carries interesting combinatorial information. Examples of extremely amenable groups (groups for which the UMF is a point) have been known since the 70s but more recently, Kechris, Pestov, and Todorcevic, inspired by previous results of Pestov and Glasner and Weiss, developed a systematic approach connecting metrizable UMFs with structural Ramsey theory and many new examples were found. I will describe the state of the art in the area and also discuss some new results that give a complete characterization of metrizable UMFs. The new results are jonit with I. Ben Yaacov, J. Melleray and L. Nguyen Van Thé.

Property (T) was introduced by Kazhdan in 1967 and since then has found many important applications. Among them is for instance the first explicit construction of expanders by Margulis, solution to the Ruziewicz problem by Margulis and Sullivan and many rigidity results for group actions and operator algebras. In this talk I will give an overview of property (T), discuss some examples, applications and open questions.

Warped cones are metric spaces that enable the study of a group action on a compact space via tools of large scale geometry. The compact space is blown to form an infinite cone and then the metric is modified according to the action.

To avoid being too specific, I will try to put my PhD research into a broader landscape of large scale geometry and geometric group theory. If time permits, we will prove how a spectral gap of the action makes the geometry of the associated warped cone incompatible with the geometry of any $L_p$ space ($p<\infty$).

I will introduce quiver varieties, which provide an interesting class of spaces including many classical examples of symplectic spaces, for example Grassmannians or partial flag manifolds. I will demonstrate on examples how to realize such spaces as symplectic reductions (a short introduction to symplectic geometry will also be given).

Starting from the definition of a (Gromov) hyperbolic group, I will discuss the properties of its Gromov boundary, which is a certain compactification of its Cayley graph. It turns out that such boundary features "local topological self-similarity", which can be perceived as a topological analogue of the concept of a finite-state automaton.

An isometric immersion of co-dimension-$k$ is a mapping from a domain $\mathbb R^n$ to $\mathbb R^{n+k}$ that preserves the angle between any two curves passing through each point of the domain, as well as their lengths. It has been well-known that any $C^2$ isometric immersions of a flat domain is developable, that is, passing through each point there is at least one line segment on which the map is affine. As a surprising contrast, Nash and Kuiper established the existence of $C^1$ isometric immersions of any flat domain into balls of any higher dimension and of arbitrarily small radius. In particular, it cannot be affine on any line segments. It is then natural to consider isometric immersion in the intermediate class $W^{2,p}$. Actually, the study of $W^{2,p}$ isometric immersions is also important in nonlinear elasticity. In this talk, we will demonstrate that the critical Sobolev exponent which guarantees developability depends on the co-dimension rather than the dimension of the domain. We will also discuss $W^{2,p}$ isometric immersions when $p$ is below the critical exponent. In particular, we show that in this case, the Gaussian curvature vanishes in the almost everywhere sense, but may remain a singular measure, as contrast to the critical case where the Gaussian curvature vanishes as a measure. This result thus connects isometric immersion to the $W^{2,p}$ solution of the Monge-Ampere equation $\det D^2u=0$ a.e. Indeed, if $p$ is below the same critical exponent, the solution can be nowhere affine, but once $p$ is above the critical exponent, it must be developable.

Given a group G, the classifying space EG is a homotopical invariant of G which carries information about certain actions of G on topological spaces. As a convenient generalization, given a family F of subgroups of G, one can consider the so-called classifying space of G for the family F. It is in a certain sense the universal G–space with stabilizers in F, and when F contains only the trivial subgroup it recovers the usual classifying space EG. In this talk I will define this notion, discuss its basic properties and present some examples of such classifying spaces arising from geometry and topology. Also, I will try to give some motivation for studying these spaces, and indicate possible applications of the theory.

A Galton-Watson process is a simple probabilistic model of population reproduction. We start with one particle and in each step every particle splits into a number of particles who behave in the same way as the original one. Depending on the expected value of the offspring distribution, the model falls into three cases: subcritical, critical and supercritical. I will present some classical results and a new result from my master thesis concerning the supercritical Galton-Watson process in continuous time.

Many operators appearing in harmonic analysis turn out to be bounded as operators over $L^p$, $1<p<\infty$, but not over $L^1$. In my talk I will present classical Calderón-Zygmund theory. The Marcinkiewicz interpolation theorem will be used as a tool. I will also speak about connections between a.e. convergence and weak type estimates and mention the alternative version of estimates for the critical case $p=1$.

Linkages are very simple mechanical systems constructed from rods and pivots which date back to the industrial revolution and the need to convert one form of motion to another (e.g., linear and circular). Understanding their configurations spaces is an interesting mathematical problem and their dynamical behaviour can be understood in terms of the associated geodesic flow.