Tendon tissue engineering aims to grow functional tissue in the lab. Tissue is grown inside a bioreactor which controls both the mechanical and biochemical environment. As tendon cells alter their behaviour in response to mechanical stresses, designing suitable bioreactor loading regimes forms a key component in ensuring healthy tissue growth.  

Linking the forces imposed by the bioreactor to the shear stress experienced by individual cell is achieved by homogenisation using multiscale asymptotics. We will present a continuum model capturing fluid-structure interaction between the nutrient media and the fibrous scaffold where cells grow. Solutions reflecting different experimental conditions will be discussed in view of the implications for shear stress distribution experienced by cells across the bioreactor.  

I will be first introducing the Polyakov loop space formalism to
gauge theories. I will also discuss how the Polyakov loop space is modified
by Planck scale corrections.  The gauge theory will be deformed by the
Planck length as the minimum measurable length in the background spacetime.
This deformation will in turn deform the Polyakov loops space. It will be
observed that this deformation can have important consequences for
non-abelian monopoles in gauge theories.

Dark matter is known to exist, but no-one knows what it is or where it came
from.  We describe a new production mechanism of particle dark matter, which
hinges on a first-order cosmological phase transition.  We then show that
this mechanism can be slightly modified to produce primordial black holes.

While solar mass and supermassive black holes are now known to exist,
primordial black holes have not yet been seen but could solve a number of
problems in cosmology.  Finally, we demonstrate that if an evaporating
primordial black hole is observed, it will provide a unique window onto
Beyond the Standard Model physics.

We revisit rough paths and signatures from a geometric and "smooth model" perspective. This provides a lean framework to understand and formulate key concepts of the theory, including recent insights on higher-order translation, also known as renormalization of rough paths. This first part is joint work with C Bellingeri (TU Berlin), and S Paycha (U Potsdam). In a second part, we take a semimartingale perspective and more specifically analyze the structure of expected signatures when written in exponential form. Following Bonnier-Oberhauser (2020), we call the resulting objects signature cumulants. These can be described - and recursively computed - in a way that can be seen as unification of previously unrelated pieces of mathematics, including Magnus (1954), Lyons-Ni (2015), Gatheral and coworkers (2017 onwards) and Lacoin-Rhodes-Vargas (2019). This is joint work with P Hager and N Tapia.

An insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalues or linearity, it would be “matrix factorizations.”  We will discuss a blueprint to generate  53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wigner (1968) discovered factorizations such as the svd in passing in a way that was buried and only eight authors have referenced that work. Ironically Wigner referenced Sigurður Helgason (1962) but Wigner did not recognize his results in Helgason's book. This work also extends upon and completes open problems posed by Mackey²&Tisseur (2003/2005).

Classical results of Random Matrix Theory concern exact formulas from the Hermite, Laguerre, Jacobi, and Circular distributions. Following an insight from Freeman Dyson (1970), Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensembles. We present a completed theory based on the generalized Cartan distribution. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampling algorithms and the derivation of the joint probability density of A.

Joint work with Sungwoo Jeong.

Consider an n by n random matrix A with i.i.d entries. In this talk, we discuss some results on the magnitude of the smallest singular value of A, and, in particular, the problem of estimating the singularity probability when the entries of A are discrete.

For five decades, mathematicians have exploited the beauties of random matrix theory (RMT) in the hope of discovering principles which govern complex ecosystems. While RMT lies at the heart of the ideas, their translation toward biological reality requires some heavy lifting: dynamical systems theory, statistics, and large-scale computations are involved, and any predictions should be challenged with empirical data. This can become very awkward.

In addition to a morose journey through some of my personal failures to make RMT meet reality, I will try to sketch out some more constructive future perspectives. In particular, new methods for microbial community composition, dynamics and evolution might allow us to apply RMT ideas to the treatment of cystic fibrosis. In addition, in fisheries I will argue that sometimes the very absence of an empirical dataset can add to the practical value of models as tools to influence policy.

This is joint work with Paul Bourgade and Benjamin McKenna (Courant Institute, NYU).

The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations), on the one hand, and elastic self-interactions (preferring ordered configurations), on the other. The elastic manifold model is interesting because it displays a depinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the high-dimensional limit by Mézard and Parisi (1992) using the replica method. 

We study the topology of the energy landscape of this model in the Mézard-Parisi setting, and compute the (annealed) topological complexity both of total critical points and of local minima. Our main result confirms the recent formulas by Fyodorov and Le Doussal (2020) and allows to identify the boundary between simple and glassy phases. The core argument relies on the analysis of the asymptotic behavior of large random determinants in the exponential scale.

There has been a lot of interest in recent years in understanding the multifractality of characteristic polynomials of random matrices. In this talk I shall consider the study of moments of moments from the probabilistic perspective of Gaussian multiplicative chaos, and in particular establish exact asymptotics for the so-called critical-subcritical regime in the context of large Haar-distributed unitary matrices. This is based on a joint work with Jon Keating.

In recent years important progress has been made in the understanding of random tilings of large Aztec diamonds with doubly periodic weights. Due to the double periodicity a new phase appears that  has not been observed in tiling models with uniform weights. One of the challenges is to find expressions of for the correlation functions that are amenable for asymptotic studies. In the case of the uniform weight the model is an example of a Schur process and, consequently,  such expressions for the correlation functions are known and well-studied in that case. In a joint work with Tomas Berggren we studied a more  general  integrable structure that includes certain doubly periodic weightings planar domains, such as the Aztec diamond.  A key feature is a dynamical system hiding in the background. In case of a periodic orbit, explicit double integrals for the correlation function can be found, paving the way for an asymptotic study using saddle point methods.