Past

We investigate certain (hopefully new) arithmetic aspects of abelian varieties defined over function fields of curves over finitely generated fields. One of the key ingredients in our investigation is a new specialisation theorem a la N\'eron for the first Galois cohomology group with values in the Tate module, which generalises N\'eron specialisation theorem for rational points. Also, among other things, we introduce a discrete version of Selmer groups, which are finitely generated abelian groups. We also discuss an application of our investigation to anabelian geometry (joint work with Akio Tamagawa).

Familiar species; humans, mammals, fish, reptiles and plants represent only a razor’s edge of the Earth’s immense biodiversity. Most of the Earth’s multicellular species lie buried in soil, inside of plants, and in the undergrowth, and include millions of unknown species, almost half of which are thought to be fungi. Part of the amazing success of fungi may be the elegant solutions that they have evolved to the problems of dispersing, growing and adapting to changing environments. I will describe how we using both math modeling and experiments to discover some of these solutions. I will focus on (i) how cytoplasmic mixing enables some species to tolerate internal genetic diversity, making them better pathogens and more adaptable, and (ii) how self-organization of these flows into phases of transport and stasis enables cells to function both as transport conduits, and to perform other functions like growth and secretion.

Adaptive finite element methods (AFEMs) with Dörflers marking strategy are known to converge with

optimal asymptotical rates. Practical experiences show that AFEMs with a maximum marking strategy

produces optimal results thereby being less sensitive to choices of the marking parameter.

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In this talk, we prove that an AFEM with a modified maximum strategy is even instance optimal for the

total error, i.e., for the sum of the error and the oscillation. This is a non-asymptotical optimality result.

Our approach uses new techniques based on the minimisation of the Dirichlet energy and a newly

developed tree structure of the nodes of admissible triangulations.

I will discuss free-by-cyclic groups and cases where they can and cannot act on CAT(0) spaces. I will specifically go into a construction building CAT(0) 2-complexes on which free of rank 2-by-cyclic act. This is joint work with Martin Bridson and Martin Lustig.

I will discuss some very well studied cohomology groups that turn out to be captured by the machinery of critical CoHAs, for example the compactly supported cohomology of singular quiver varieties and untwisted character varieties. I will explain the usefulness of this extra CoHA structure on these groups, starting with a new proof of the Kac conjecture, and discuss a conjectural form for the CoHA associated to untwisted character varieties that provides a new way to think about the conjectures of Hausel and Rodriguez-Villegas. Finally I will discuss an approach to purity for the compactly supported cohomology of quiver varieties and a related approach to a conjecture of Shiffmann and Vasserot, analogous to Kirwan surjectivity for the stack of commuting matrices.

Nature and the world of human technology are full of
networks. People like to draw diagrams of networks: flow charts,
electrical circuit diagrams, signal flow diagrams, Bayesian networks,
Feynman diagrams and the like. Mathematically-minded people know that
in principle these diagrams fit into a common framework: category
theory. But we are still far from a unified theory of networks.

The cohomological Hall algebra of vanishing cycles associated to a quiver with potential is a categorification of the refined DT invariants associated to the same data, and also a very powerful tool for calculating them and proving positivity and integrality conjectures. This becomes especially true if the quiver with potential is "self dual" in a sense to be defined in the talk. After defining and giving a general introduction to the relevant background, I will discuss the main theorem regarding such CoHAs: they are free supercommutative.

State-space models are a very popular class of time series models which have found thousands of applications in engineering, robotics, tracking, vision,  econometrics etc. Except for linear and Gaussian models where the Kalman filter can be used, inference in non-linear non-Gaussian models is analytically intractable.  Particle methods are a class of flexible and easily parallelizable simulation-based algorithms which provide consistent approximations to these inference problems. The aim of this talk is to introduce particle methods and to present the most recent developments in this area.

Pluripotency is a key feature of embryonic stem cells (ESCs), and is defined as the ability to give rise to all cell lineages in the adult body. Currently, there is a good understanding of the signals required to maintain ESCs in the pluripotent state and the transcription factors that comprise their gene regulatory network. However, little is known about how ESCs exit the pluripotent state and begin the process of differentiation. We aim to understand the molecular events associated with this process via an experiment-model cycle.

The revolution in molecular biology within the last few decades has led to the identification of multiple, diverse inputs into the mechanisms governing the measurement and regulation of organ size. In general, organ size is controlled by both intrinsic, genetic mechanisms as well as extrinsic, physiological factors. Examples of the former include the spatiotemporal regulation of organ size by morphogen gradients, and instances of the latter include the regulation of organ size by endocrine hormones, oxygen availability and nutritional status. However, integrated model platforms, either of in vitro experimental systems amenable to high-resolution imaging or in silico computational models that incorporate both extrinsic and intrinsic mechanisms are lacking. Here, I will discuss collaborative efforts to bridge the gap between traditional assays employed in developmental biology and computational models through quantitative approaches. In particular, we have developed quantitative image analysis techniques for confocal microscopy data to inform computational models – a critical task in efforts to better understand conserved mechanisms of crosstalk between growth regulatory pathways. Currently, these quantitative approaches are being applied to develop integrated models of epithelial growth in the embryonic Drosophila epidermis and the adolescent wing imaginal disc, due to the wealth of previous genetic knowledge for the system. An integrated model of intrinsic and extrinsic growth control is expected to inspire new approaches in tissue engineering and regenerative medicine.

We study liquid crystal point defects in 2D domains. We employ Landau-de

Gennes theory and provide a simplified description of global minimizers

of Landau- de Gennes energy under homeothropic boundary conditions. We

also provide explicit solutions describing defects of various strength

under Lyuksutov's constraint.

Open 2d TCFTs correspond to cyclic A-infinity algebras, and Costello showed

that any open theory has a universal extension to an open-closed theory in

which the closed state space (the value of the functor on a circle) is the

Hochschild homology of the open algebra.  We will give a G-equivariant

generalization of this theorem, meaning that the surfaces are now equipped

with principal G-bundles.  Equivariant Hochschild homology and a new ribbon

graph decomposition of the moduli space of surfaces with G-bundles are the

principal ingredients.  This is joint work with Ramses Fernandez-Valencia.

Abstract: Polynomial preserving processes are defined as time-homogeneous Markov jump-diffusions whose generator leaves the space of polynomials of any fixed degree invariant. The moments of their transition distributions are polynomials in the initial state. The coefficients defining this relationship are given as solutions of a system of nested linear ordinary differential equations. Polynomial processes include affine processes, whose transition functions admit an exponential-affine characteristic function. These processes are attractive for financial modeling because of their tractability and robustness. In this work we study approximations of polynomial preserving processes with finite-state Markov processes via a moment-matching methodology. This approximation aims to exploit the defining property of polynomial preserving processes in order to reduce the complexity of the implementation of such models. More precisely, we study sufficient conditions for the existence of finite-state Markov processes that match the moments of a given polynomial preserving process. We first construct discrete time finite-state Markov processes that match moments of arbitrary order. This discrete time construction relies on the existence of long-run moments for the polynomial process and cubature methods over these moments. In the second part we give a characterization theorem for the existence of a continuous time finite-state Markov process that matches the moments of a given polynomial preserving process. This theorem illustrates the complexity of the problem in continuous time by combining algebraic and geometric considerations. We show the impossibility of constructing in general such a process for polynomial preserving diffusions, for high order moments and for sufficiently many points in the state space. We provide however a positive result by showing that the construction is possible when one considers finite-state Markov chains on lifted versions of the state space. This is joint work with Damir Filipovic and Martin Larsson.