C4

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).
Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.
In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Araklev theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

Complexity Science aims to understand what is that makes some systems to be "more than the sum of their parts". A natural first step to address this issue is to study networks of pairwise interactions, which have been done with great success in many disciplines -- to the extend that many people today identify Complexity Science with network analysis. In contrast, multivariate complexity provides a vast and mostly unexplored territory. As a matter of fact, the "modes of interdependency" that can exist between three or more variables are often nontrivial, poorly understood and, yet, are paramount for our understanding of complex systems in general, and emergence in particular. 
In this talk we present an information-theoretic framework to analyse high-order correlations, i.e. statistical dependencies that exist between groups of variables that cannot be reduced to pairwise interactions. Following the spirit of information theory, our approach is data-driven and model-agnostic, being applicable to discrete, continuous, and categorical data. We review the evolution of related ideas in the context of theoretical neuroscience, and discuss the most prominent extensions of information-theoretic metrics to multivariate settings. Then, we introduce the O-information, a novel metric that quantify various structural (i.e. synchronous) high-order effects. Finally, we provide a critical discussion on the framework of Integrated Information Theory (IIT), which suggests an approach to extend the analysis to dynamical settings. To illustrate the presented methods, we show how the analysis of high-order correlations can reveal critical structures in various scenarios, including cellular automata, Baroque music scores, and various EEG datasets.

References:
[1] F. Rosas, P.A. Mediano, M. Gastpar and H.J. Jensen, ``Quantifying High-order Interdependencies via Multivariate Extensions of the Mutual Information'', submitted to PRE, under review.
https://arxiv.org/abs/1902.11239
[2] F. Rosas, P.A. Mediano, M. Ugarte and H.J. Jensen, ``An information-theoretic approach to self-organisation: Emergence of complex interdependencies in coupled dynamical systems'', in Entropy, vol. 20 no. 10: 793, pp.1-25, Sept. 2018.
https://www.mdpi.com/1099-4300/20/10/793

 

Social surveys are widely used in today's society as a method for obtaining opinions and other information from large groups of people. The questions in social surveys are usually presented in either multiple-choice or free-response formats. Despite their advantages, free-response questions are employed less commonly in large-scale surveys, because in such situations, considerable effort is needed to categorise and summarise the resulting large dataset. This is the so-called coding problem. Here we propose a survey framework in which, respondents not only write down their own opinions, but also input information characterising the similarity between their individual responses and those of other respondents. This is done in much the same way as ``likes" are input in social network services. The information input in this simple procedure constitutes relational data among opinions, which we call the opinion graph. The diversity of typical opinions can be identified as a modular structure of such a graph, and the coding problem is solved through graph clustering in a statistically principled manner. We demonstrate our approach using a poll on the 2016 US presidential election and a survey given to graduates of a particular university.

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however often these are not metrics (rendering standard data-mining techniques infeasible), or are computationally infeasible for large graphs. In this work, we define a new metric based on the spectrum of the non-backtracking graph operator and show that it can not only be used to compare graphs generated through different mechanisms but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the non-backtracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.

The world of quantum invariants began in 1983 with the discovery of the Jones polynomial. Later on, Reshetikhin and Turaev developed an algebraic machinery that provides knot invariants. This algebraic construction leads to a sequence of quantum generalisations of this invariant, called coloured Jones polynomials. The original Jones polynomial can be defined by so called skein relations. However, unlike other classical invariants for knots like the Alexander polynomial, its relation to the topology of the complement is still a mysterious and deep question. On the topological side, R. Lawrence defined a sequence of braid group representations on the homology of coverings of configuration spaces. Then, based on her work, Bigelow gave a topological model for the Jones polynomial, as a graded intersection pairing between certain homology classes. We aim to create a bridge between these theories, which interplays between representation theory and low dimensional topology. We describe the Bigelow-Lawrence model, emphasising the construction of the homology classes. Then, we show that the sequence of coloured Jones polynomials can be seen through the same formalism, as topological intersection pairings of homology classes in coverings of the configuration space in the punctured disc.

I will explain how Lurie‘s approach to L-theory via Poincaré categories can be extended to yield cobordism categories of Poincaré objects à la Ranicki. These categories can be delooped by an iterated Q-construction and the resulting spectrum is a derived version of Grothendieck-Witt-theory.  Its homotopy type can be described in terms of K- and L-theory as conjectured by Hesselholt-Madsen. Furthermore, it has a clean universal property analogous to that of K-theory, localisation sequences in much greater generality than classical Grothendieck-Witt theory, gives a cycle description of Weiss-Williams‘ LA-theory and allows for maps from the geometric cobordism category, refining and unifying various known invariants.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.

The large-scale features of groups and spaces are recorded by asymptotic invariants. Examples of asymptotic invariants are the asymptotic cone and, for hyperbolic groups, the Gromov boundary.
In his study of asymptotic cones of connected Lie groups, Yves Cornulier introduced a class of maps called sublinearly biLipschitz equivalences. Like the more traditionnal quasiisometries, sublinearly biLipschitz equivalences are biLipschitz on the large-scale, but unlike quasiisometries, they are generally not coarse. Sublinearly biLipschitz equivalences still induce biLipschitz homeomorphisms between asymptotic cones. In this talk, I will focus on Gromov-hyperbolic groups and show how the Gromov boundary can be used to produce invariants distinguishing them up to sublinearly biLipschitz equivalences when the asymptotic cones do not. I will especially give applications to the large-scale sublinear geometry of hyperbolic Lie groups.
 

Gauge-theoretic invariants such as Donaldson or Seiberg–Witten invariants of 4-manifolds, Casson invariants of 3-manifolds, Donaldson–Thomas invariants of Calabi–Yau 3- and 4-folds, and putative Donaldson–Segal invariants of G_2 manifolds are defined by constructing a moduli space of solutions to an elliptic PDE as a (derived) manifold and integrating the (virtual) fundamental class against cohomology classes. For a moduli space to have a (virtual) fundamental class it must be compact, oriented, and (quasi-)smooth. We first describe a general framework for addressing orientability of gauge-theoretic moduli spaces due to Joyce–Tanaka–Upmeier. We then show that the moduli stack of perfect complexes of coherent sheaves on a Calabi–Yau 4-fold X is a homotopy-theoretic group completion of the topological realisation of the moduli stack of algebraic vector bundles on X. This allows one to extend orientations on the locus of algebraic vector bundles to the boundary of the (compact) moduli space of coherent sheaves using the universal property of homotopy-theoretic group completions. This is a necessary step in constructing Donaldson–Thomas invariants of Calabi–Yau 4-folds. This is joint work with Yalong Cao and Dominic Joyce.

My goal for the talk is to give a "from the ground-up" introduction to symplectic topology. We will cover the Darboux lemma, pseudo-holomorphic curves, Gromov-Witten invariants, quantum cohomology and Floer cohomology.

In this talk, I will sketch a geometrically flavoured proof of the 
Madsen-Weiss theorem based on work by Eliashberg-Galatius-Mishachev.
In order to prove the triviality of appropriate relative bordism groups, 
in a first step a variant of the wrinkling theorem shows
that one can reduce to consider fold maps (with additional structure). 
In a subsequent step, a geometric version of the Harer stability
theorem is used to get rid of the folds via surgery. I will focus on 
this second step.