Decomposition (aka unital 2-Segal) spaces are simplicial ∞-groupoids with a certain exactness property: they take pushouts of active (end-point preserving) along inert (distance preserving) maps in the simplicial category Δ to pullbacks. They encode the information needed for an 'objective' generalisation of the notion of incidence (co)algebra of a poset, and motivating examples include the decomposition spaces for (derived) Hall algebras, the Connes-Kreimer algebra of trees and Schmitt's algebra of graphs. In this talk I will survey recent activity in this area, including some work in progress on a categorification of (Hopf) bialgebroids.
This is joint work with Imma Gálvez and Joachim Kock.
 

Exponents of Diophantine approximation are defined to study specific sets of real numbers for which Dirichlet's pigeonhole principle can be improved. Khintchine stated a transference principle between the two exponents in the cases  of simultaneous approximation and approximation by linear forms. This shows that exponents of Diophantine approximation are related, and these relations can be studied via so called spectra. In this talk, we provide an optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation for both simultaneous approximation and approximation by linear forms. This is joint work with Nikolay Moshchevitin.

We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties of these basis vector fields and show how to approximate them. Then, we employ this basis to discretise shape Newton methods and investigate the impact of this discretisation on convergence rates.

Over the past decade, the randomized singular value decomposition (RSVD)
algorithm has proven to be an efficient, reliable alternative to classical
algorithms for computing low-rank approximations in a number of applications.
However, in cases where no information is available on the singular value
decay of the data matrix or the data matrix is known to be close to full-rank,
the RSVD is ineffective. In recent years, there has been great interest in
randomized algorithms for computing full factorizations that excel in this
regime.  In this talk, we will give a brief overview of some key ideas in
randomized numerical linear algebra and introduce a new randomized algorithm for
computing a full, rank-revealing URV factorization.

We introduce and analyze a discontinuous Galerkin method for the Oseen equations in two dimension spaces. The boundary conditions are mixed and they are assumed to be of three different types:
the vorticity  and the normal component of the velocity are given on a first part of the boundary, the pressure and the tangential component of the velocity are given on a second part of the boundary and the Dirichlet condition is given on the remainder part . We establish a priori error estimates in the energy norm for the velocity and in the L2 norm for the pressure. An a posteriori error estimate is also carried out yielding optimal convergence rate. The analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini.

We consider numerical methods for solving  time dependent partial differential equations with convection-diffusion terms and anti-diffusive fractional operator of order $\alpha \in (1,2)$. These equations are motivated by two distinct applications: a dune morphodynamics model and a signal filtering method. 
We propose numerical schemes based on local discontinuous Galerkin methods to approximate the solutions of these equations. Numerical stability and convergence of these schemes are investigated. 
Finally numerical experiments are given to illustrate qualitative behaviors of solutions for both applications and to confirme the convergence results. 

Several scholars of evolutionary biology have suggested that functional redundancy (also known as "biological degener-
acy") is important for robustness of biological networks. Structural redundancy indicates the existence of structurally
similar subsystems that can perform the same function. Functional redundancy indicates the existence of structurally
di erent subsystems that can perform the same function. For networks with Ornstein--Uhlenbeck dynamics, Tononi et al.
[Proc. Natl. Acad. Sci. U.S.A. 96, 3257{3262 (1999)] proposed measures of structural and functional redundancy that are
based on mutual information between subnetworks. For a network of n vertices, an exact computation of these quantities
requires O(n!) time. We derive expansions for these measures that one can compute in O(n3) time. We use the expan-
sions to compare the contributions of di erent types of motifs to a network's functional redundancy.