C1

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 Arakelov theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

Given a morphism of schemes of characteristic p, we can construct a morphism from the exterior algebra of Kahler differentials to the cohomology of De Rham complex, which is an isomorphism when the original morphism is smooth.

The goal of the talk is to present a proof of the following statement:
Let (K,v) be an algebraic extension of (Q_p,v_p) whose completion is perfectoid. We show that K is relatively decidable to its tilt K^♭, i.e. if K^♭ is decidable in the language of valued fields, then so is K. 
In the first part [of the talk], I will try to cover the necessary background needed from model theory and the theory of perfectoid fields.

I will give an overview of the localization technique: a powerful dimension-reduction tool for proving geometric and functional inequalities.  Having its roots in a  pioneering work of Payne-Weinberger in the 60ies about sharp Poincare’-Wirtinger inequality on Convex Bodies in Rn, recently such a technique found new applications for a range of sharp geometric and functional inequalities in spaces with Ricci curvature bounded below.

It is known that many real-world networks exhibit geometric properties.  Brain networks, social networks, and wireless communication networks are a few examples.  Storage and transmission of the information contained in the topologies and structures of these networks are important tasks, which, given their scale, is often nontrivial.  Although some (but not much) work has been done to characterize and develop compression limits and algorithms for nonspatial graphs, little is known for the spatial case.  In this talk, we will discuss an information theoretic formalism for studying compression limits for a fairly broad class of random geometric graphs.  We will then discuss entropy bounds for these graphs and, time permitting, local (pairwise) connection rules that yield maximum entropy properties in the induced graph distribution.

In 1941, Chabauty gave a way to compute the set of rational points on specific curves. In 2004, Minhyong Kim showed how to extend Chabauty's method to a bigger class of curves using anabelian methods. In the talk, I will explain Chabauty's method and give an outline of how Kim extended those methods.

Knotoids are a generalisation of knots that deals with open curves. In the past few years, they’ve been extensively used to classify entanglement in proteins. Through a double branched cover construction, we prove a 1-1 correspondence between knotoids and strongly invertible knots. We characterise forbidden moves between knotoids in terms of equivariant band attachments between strongly invertible knots, and in terms of crossing changes between theta-curves. Finally, we present some applications to the study of the topology of proteins. This is based on joint works with D.Buck, H.A.Harrington, M.Lackenby and with D. Goundaroulis.

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

Many soft materials, such as elastomers and hydrogels, are made of long chain molecules crosslinked to form a three-dimensional network. Their mechanical properties depend on network parameters such as chain density, chain length distribution and the functionality of the crosslinks. Understanding the relationships between the topology of polymer networks and their mechanical properties has been a long-standing challenge in polymer physics.

In this work, we focus on so-called “near-ideal” networks, which are produced by the cross-coupling of star-like macromolecules with well-defined chain length. We developed a computational approach based on random discrete networks, according to which the polymer network is represented by an assembly of non-linear springs connected at junction points representing crosslinks. The positions of the crosslink points are determined from the conditions of mechanical equilibrium. Scaling relations for the elastic modulus and maximum extensibility of the network were obtained. Our scaling relations contradict some predictions of classical estimates of rubber elasticity and have implications for the interpretation of experimental data for near-ideal polymer networks.

Reference: G. Alame, L. Brassart. Relative contributions of chain density and topology to the elasticity of two-dimensional polymer networks. Soft Matter 15, 5703 (2019).

We aim to generate gene coexpression networks from gene expression data. In our networks, nodes represent genes and edges depict high positive correlation in their expression across different samples. Methods based on Pearson correlation are the most commonly used to generate gene coexpression networks. We propose the use of distance correlation as an effective alternative to Pearson correlation when constructing gene expression networks. Our methodology pipeline includes a thresholding step which allows us to discriminate which pairs of genes are coexpressed. We select the value of the threshold parameter by studying the stability of the generated network, rather than relying on exogenous biological information known a priori.