L3

Recent years have seen many new ideas appearing in the solution theories of singular stochastic partial differential equations. An exciting application of SPDEs that is beginning to emerge is to the construction and analysis of quantum field theories. In this talk, I will describe how stochastic quantisation of Parisi–Wu can be used to study QFTs, especially those arising from gauge theories, the rigorous construction of which, even in low dimensions, is largely open.

In their seminal 2012 paper, Elekes and Szabó found that a certain weak combinatorial non-expansion property of an algebraic relation suffices to trigger the group configuration theorem, showing that only (approximate subgroups of) algebraic groups can be responsible for it. I will discuss some more recent variations and elaborations on this result, focusing on the case of ternary relations on varieties of dimension >1.

The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of “vectors” or by bi-operads and another based on “matrices” or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

Tensor decomposition express a tensor as a linear combination of elementary tensors. They have applications in chemometrics, computer science, machine learning, psychometrics, and signal processing. Their uniqueness properties render them suitable for data analysis tasks in which the elementary tensors are the quantities of interest. However, in applications, the idealized mathematical model is corrupted by measurement errors. For a robust interpretation of the data, it is therefore imperative to quantify how sensitive these elementary tensors are to perturbations of the whole tensor. I will give an overview of recent results on the condition number of tensor decompositions, established with my collaborators C. Beltran, P. Breiding, and N. Dewaele.

Recently Linares-Otto-Tempelmayr have unveiled a very interesting algebraic structure which allows to define a new class of rough paths/regularity structures, with associated applications to stochastic PDEs or ODEs. This approach does not consider trees as combinatorial tools but their fertility, namely the function which associates to each integer k the number of vertices in the tree with exactly k children. In a joint work with J-D Jacques we have studied this algebraic structure and shown that it is related with a general and simple class of so-called post-Lie algebras. The construction has remarkable properties and I will try to present them in the simplest possible way.

Placeholder entry; date+time TBC. 

Abstract for talk: In this talk, we present a systematic discretization of the Bernstein-Gelfand-Gelfand (BGG) diagrams and complexes over cubical meshes of arbitrary dimension via the use of tensor-product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and div-div complexes as examples for our construction.

Let $k,l>1$ be two multiplicatively independent integers. A subset $X$ of $\mathbb{N}^n$ is $k$-recognizable if the set of $k$-ary representations of $X$ is recognized by some finite automaton. Cobham's famous theorem states that a subset of the natural numbers is both $k$-recognizable and $l$-recognizable if and only if it is Presburger-definable (or equivalently: semilinear). We show the following strengthening. Let $X$ be $k$-recognizable, let $Y$ be $l$-recognizable such that both $X$ and $Y$ are not Presburger-definable. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable. Our work strengthens and depends on earlier work of Villemaire and Bès. The essence of Cobham's theorem is that recognizability depends strongly on the choice of the base $k$. Our results strengthens this: two non-Presburger definable sets that are recognizable in multiplicatively independent bases, are not only distinct, but together computationally intractable over Presburger arithmetic. This is joint work with Christian Schulz.

Linear poroelasticity models have important applications in biology and geophysics. In particular, the well-known Biot consolidation model describes the coupled interaction between the linear response of a porous elastic medium saturated with fluid and a diffusive fluid flow within it, assuming small deformations. This is the starting point for modeling human organs in computational medicine and for modeling the mechanics of permeable
rock in geophysics. Finite element methods for Biot’s consolidation model have been widely studied over the past four decades.
In the first part of the talk, we discuss a posteriori error estimators for locking-free mixed finite element approximation of Biot’s consolidation model. The simplest of these is a conventional residual-based estimator. We establish bounds relating the estimated and true errors, and show that these are independent of the physical parameters. The other two estimators require the solution of local problems. These local problem estimators are also shown to be reliable, efficient and robust. Numerical results are presented that
validate the theoretical estimates, and illustrate the effectiveness of the estimators in guiding adaptive solution algorithms.
In the second part of talk, we discuss a novel locking-free stochastic Galerkin mixed finite element method for the Biot consolidation model with uncertain Young’s modulus and hydraulic conductivity field. After introducing a five-field mixed variational formulation of the standard Biot consolidation model, we discuss stochastic Galerkin mixed finite element approximation, focusing on the issue of well-posedness and efficient linear algebra for the discretized system. We introduce a new preconditioner for use with MINRES and
establish eigenvalue bounds. Finally, we present specific numerical examples to illustrate the efficiency of our numerical solution approach.

Finally, we discuss some remarks related to non-conforming approximation of Biot’s consolidation model.

References:
1. A. Khan, D. J. Silvester, Robust a posteriori error estimation for mixed finite
element approximation of linear poroelsticity, IMA Journal of Numerical Analysis, Oxford University Press, 41 (3), 2021, 2000-2025.
2. A. Khan, C. E. Powell, Parameter-robust stochastic Galerkin approxination for linear poroelasticity with uncertain inputs, SIAM Journal on Scientific Computing (SISC), 43 (4), 2021, B855-B883.
3. A. Khan, P. Zanotti, A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot’s model. Mathematics of Computation, 91 (335), 2022, 1143-1170.
4. V. Anaya, A. Khan, D. Mora, R. Ruiz-Baier, Robust a posteriori error analysis for rotation-based formulations of the elasticity/poroelasticity coupling, SIAM Journal
on Scientific Computing (SISC), 2022.

We study the positional information conferred by the morphogens Sonic Hedgehog and BMP in neural tube patterning. We use the mathematics of information theory to quantify the information that cells use to decide their fate. We study the encoding, recoding and decoding that take place as the morphogen gradient is formed, triggers a nuclear response and determines cell fates using a gene regulatory network.

When a biological system is modelled using a mathematical process, the next step is often to estimate the system parameters. Although computational and statistical techniques have been developed to estimate parameters for complex systems, this can be a difficult task. As a result, researchers have focused on revealing parameter-independent dynamical properties of a system. In this talk, we will discuss the study of qualitative behaviors of stochastic biochemical systems using reaction networks, which are graphical configurations of biochemical systems. The goal of this talk is to (1) introduce the basic modelling aspects of stochastically-modelled reaction networks and (2) discuss important results in this literature, including the random time representation, relationships between stochastic and deterministic models, and derivation of stability via network structures.