Junior strings is a seminar series where DPhil students present topics of comment interest that do not necessarily overlap with their own research areas. This is primarly aimed at PhD students and post-docs but everyone is welcome.
The study of gravity currents has long been of interest due to their prevalence in industry and in nature, one such example being the spreading of viscoplastic (yield-stress) fluid films. When a viscoplastic fluid is extruded onto a flat plate, the resulting gravity current expands axisymmetrically when the surface is dry and rough. In this talk, I will discuss two instabilities that arise when (1) the no-slip surface is replaced by a free-slip surface; and (2) the flat plate is wet by a thin coating of water.
The injection of CO2 into porous subsurface reservoirs is a technological means for removing anthropogenic emissions, which relies on a series of complex porous flow properties. During injection of CO2 small-scale heterogeneities, often in the form of sedimentary layering, can play a significant role in focusing the flow of less viscous CO2 into high permeability pathways, with large-scale implications for the overall motion of the CO2 plume. In these settings, capillary forces between the CO2 and water preferentially rearrange CO2 into the most permeable layers (with larger pore space), and may accelerate plume migration by as much as 200%. Numerous factors affect overall plume acceleration, including the structure of the layering, the permeability contrast between layers, and the playoff between the capillary, gravitational and viscous forces that act upon the flow. However, despite the sensitivity of the flow to these heterogeneities, it is difficult to acquire detailed field measurements of the heterogeneities owing to the vast range of scales involved, presenting an outstanding challenge. As a first step towards tackling this uncertainty, we use a simple modelling approach, based on an upscaled thin-film equation, to create ensemble forecasts for many different types and arrangements of sedimentary layers. In this way, a suite of predictions can be made to elucidate the most likely scenarios for injection and the uncertainty associated with such predictions.
The question asked in the title is addressed from two points of view: First, we show that providing enough (term to be explained) spectral data, suffices to reconstruct uniquely generic (term to be explained) matrices. The method is well defined but requires somewhat cumbersome computations. Second, restricting the attention to banded matrices with band-width much smaller than the dimension, one can provide more spectral data than the number of unknown matrix elements. We make use of this redundancy to reconstruct generic banded matrices in a much more straight-forward fashion where the “cumbersome computations” can be skipped over. Explicit criteria for a matrix to be in the non-generic set are provided.
Hermitian matrix models with non-trivial covariance will be introduced. The Kontsevich Model is the prime example, which was used to prove Witten's conjecture about the generating function of intersection numbers of the moduli space $\overline{\mathcal{M}}_{g,n}$. However, we will discuss these models in a different direction, namely as a quantum field theory. As a formal matrix model, the correlation functions of these models have a unique combinatorial/perturbative interpretation in the sense of Feynman diagrams. In particular, the additional structure (in comparison to ordinary quantum field theories) gives the possibility to compute exact expressions, which are resummations of infinitely many Feynman diagrams. For the easiest topologies, these exact expressions (given by implicitly defined functions) will be presented and discussed. If time remains, higher topologies are discussed by a connection to Topological Recursion.
In this talk, we use tools from representation theory and symmetric function theory to compute correlations of eigenvalues of unitary invariant ensembles. This approach provides a route to write exact formulae for the correlations, which further allows us to extract large matrix asymptotics and study universal properties.
Free fermion chains are particularly simple exactly solvable models. Despite this, typically one can find closed expressions for physically important correlators only in certain asymptotic limits. For a particular class of chains, I will show that we can apply Day's formula and Gorodetsky's formula for Toeplitz determinants with rational generating function. This leads to simple closed expressions for determinantal order parameters and the characteristic polynomial of the correlation matrix. The latter result allows us to prove that the ground state of the chain has an exact matrix-product state representation.
For a family $A$ in $\{0,...,k\}^n$, its deletion shadow is the set obtained from $A$ by deleting from any of its vectors one coordinate. Given the size of $A$, how should we choose $A$ to minimise its deletion shadow? And what happens if instead we may delete only a coordinate that is zero? We discuss these problems, and give an exact solution to the second problem.