Past

We will give a general overview of how one gets groups to act on CAT(0) cube complexes, how compatible such actions are and how this plays out in the setting of Coxeter groups.

I will discuss the large N behavior of partition functions of the ABJM theory on compact Euclidean manifolds. I will pay particular attention to the S^3 free energy and the topologically twisted index for which I will present closed form expressions valid to all order in the large N expansion. These results have important implications for holography and the microscopic entropy counting of AdS_4 black holes which I will discuss. I will also briefly discuss generalizations to other SCFTs arising from M2-branes.

Conformal defects can be characterised by their contributions to the Weyl anomaly. The coefficients of these terms, often called defect central charges, depend on the particular defect insertion in a given conformal field theory. I will review what is currently known about defect central charges across dimensions, and present novel results. I will discuss many examples where they can be computed exactly without requiring any approximations or limits. Particular emphasis will be placed on recently developed tools for superconformal defects as well as defects in free theories.

2:00 Ziheng Wang, EPSRC CDT in Mathematics of Random Systems Student

Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

Abstract: We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using a stochastic estimate for the gradient of the stationary distribution. The gradient estimate satisfies an SDE and is simultaneously updated, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of our online algorithm for dissipative SDE models and present numerical results for other nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm.

2:45 Ian Melbourne,  Professor of Mathematics, University of Warwick

Interpretation of stochastic integrals, and the Levy area

Abstract: An important question in stochastic analysis is the appropriate interpretation of stochastic integrals. The classical Wong-Zakai theorem gives sufficient conditions under which smooth integrals converge to Stratonovich stochastic integrals. The conditions are automatic in one-dimension, but in higher dimensions it is necessary to take account of corrections stemming from the Levy area. The first part of the talk covers work with Kelly 2016, where we justified the Levy area correction for large classes of smooth systems, bypassing any stochastic modelling assumptions. The second part of the talk addresses a much less studied question: is the Levy area zero or nonzero for systems of physical interest, eg Hamiltonian time-reversible systems? In recent work with Gottwald, we classify (and clarify) the situations where such structure forces the Levy area to vanish. The conclusion of our work is that typically the Levy area correction is nonzero.

3:45 Break

4:15 Sara Franceschelli, Associate Professor,  École Normale Supérieure de Lyon

When is a model is a good model? Epistemological perspectives on mathematical modelling

When a model is a good model? Must it represent a specific target system? Allow to make predictions? Provide an explanation for observed behaviors?  After a brief survey of general epistemological questions on modelling, I will consider examples of mathematical modelling in physics and biology from the perspective of dynamical systems theory. I will first show that even if it has been little noticed by philosophers, dynamical systems theory itself as a mathematical theory has been a source of questions and criteria in order to assess the goodness of a model (notions of stability, genericity, structural stability). I will then discuss the theoretical fruitfulness of arguments of (in)stability in the mathematical modelling of morphogenesis.

Recent advances in Deep Learning have enabled us to explore new application domains in molecular biology and drug discovery - including those driven by complex processes that defy analytical modelling.  However, despite the combined forces of increased data, improving compute resources and continuous algorithmic innovation all bringing previously intractable problems into the realm of possibility, many advances are yet to make a tangible impact for life science discovery.  In this talk, Dr Marcin J. Skwark will discuss the challenge of bringing machine learning innovation to tangible real-world impact.  Following a general introduction of the topic, as well as newly available methods and data, he will focus on the modelling of COVID-19 variants and, in particular, the DeepChain Early Warning System (EWS) developed by InstaDeep in collaboration with BioNTech.  With thousands of new, possibly dangerous, SARS-CoV-2 variants emerging each month worldwide, it is beyond humanities combined capacity to experimentally determine the immune evasion and transmissibility characteristics of every one.  EWS builds on an experimentally tested AI-first computational biology platform to evaluate new variants in minutes, and is capable of risk monitoring variant lineages in near real-time.  This is done by combining an AI-driven protein structure prediction framework with large, spike protein sequence-oriented Transformer models to allow for rapid simulation-free assessment of the immune escape risk and expected fitness of new variants, conditioned on the current state of the world.  The system has been extensively validated in cooperation with BioNTech, both in terms of host cell infection propensity (including experimental assays of receptor binding affinity), and immune escape (pVNT assays with monoclonal antibodies and real-life donor sera). In these assessments, purely unsupervised, data-first methods of EWS have shown remarkable accuracy. EWS flags and ranks all but one of the SARS-CoV-2 Variants of Concern (Alpha, Beta, Gamma, Delta… Omicron), discriminates between subvariants (e.g. BA.1/BA.2/BA.4 etc. distinction) and for most of the adverse events allows for proactive response on the day of the observation. This allows for appropriate response on average six weeks before it is possible for domain experts using domain knowledge and epidemiological data. The performance of the system, according to internal benchmarks, improves with time, allowing for example for supporting the decisions on the emerging Omicron subvariants on the first days of their occurrence. EWS impact has been notable in general media [2, 3] for the system's applicability to a novel problem, ability to derive generalizable conclusions from unevenly distributed, sparse and noisy data, to deliver insights which otherwise necessitate long and costly experimental assays.

Peel Ports operate a number of locks that allow ships to enter and leave the port. The lock gates comprise a single caisson structure which blocks the waterway when closed and retracts into the dockside as the gate opens. Build up of silt ahead of the opening lock gate can prevent it from fully opening or requiring excessive power to move. If the lock is not able to fully open, ships are unable to enter the port, leading to significant operational impacts for the whole port. Peel ports are interested in understanding, and mitigating, this silt build up. 

In joint work with Jacob Tsimerman we study when the primitive
of a given algebraic function can be constructed using primitives
from some given finite set of algebraic functions, their inverses,
algebraic functions, and composition. When the given finite set is just {1/x}
this is the classical problem of "elementary integrability".
We establish some results, including a decision procedure for this problem.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

The design and analysis of finite element methods for high Reynolds flow remains a challenging task, not least because of the difficulties associated with turbulence. In this talk we will first revisit some theoretical results on interior penalty methods using equal order interpolation for smooth solutions of the Navier-Stokes’ equations at high Reynolds number and show some recent computational results for turbulent flows.

Then we will focus on so called pressure robust methods, i.e. methods where the smoothness of the pressure does not affect the upper bound of error estimates for the velocity of the Stokes’ system. We will discuss how convection can be stabilized for such methods in the high Reynolds regime and, for the lowest order case, show an interesting connection to turbulence modelling.

Numerous applications in geometric, visual, and scientific computing rely on the ability to nicely distribute points in space according to a repulsive potential.  In contrast, there has been relatively little work on equidistribution of higher-dimensional geometry like curves and surfaces—which in many contexts must not pass through themselves or each other.  This talk explores methods for optimization of curve and surface geometry while avoiding (self-)collision. The starting point is the tangent-point energy of Buck & Orloff, which penalizes pairs of points that are close in space but distant with respect to geodesic distance. We develop a discretization of this energy, and introduce a novel preconditioning scheme based on a fractional Sobolev inner product.  We further accelerate this scheme via hierarchical approximation, and describe how to incorporate into a constrained optimization framework. Finally, we explore how this machinery can be applied to problems in mathematical visualization, geometric modeling, and geometry processing.

When considering compatifications of heterotic string theory down to 3D, the heterotic $G_2$ system arises naturally. It is a system for both geometric fields and gauge fields over a manifold with a $G_2$-structure. In particular, it asks for the $G_2$-structure to be coclosed. We will begin this talk defining this system and giving a description of the geometry of cohomogeneity one manifolds. Then, we will look for coclosed $G_2$-structures in the cohomogeneity one setting. We will end up by proving the existence of a family of coclosed $G_2$-structures which are invariant under a cohomogeneity one action of $\text{SU}(2)^2$ on certain seven-dimensional simply connected manifolds.

One of the most important constructions in operator algebras is the tracial ultrapower for a tracial state on a C*-algebra. This tracial ultrapower is a finite von Neumann algebra, and it appears in seminal work of McDuff, Connes, and more recently by Matui-Sato and many others for studying the structure and classification of nuclear C*-algebras. I will talk about how to generalise this to unbounded traces (such as the standard trace on B(H)). Here the induced tracial ultrapower is not a finite von Neumann algebra, but its multiplier algebra is a semifinite von Neumann algebra.

The Jacobi Ensembles of random matrices have joint distribution of eigenvalues proportional to the integration measure in the Selberg integral. They can also be realised as the singular values of principal submatrices of random unitaries. In this talk we will review some old and new results concerning the distribution of the largest and smallest eigenvalues.

The introduction to the talk will review basics of $G_{2}$ geometry in seven dimensions, and associative and co-associative submanifolds. In one part of the talk we will explain how fibrations with co-associative fibres, near the “adiabatic limit” when the fibres are very small,  give insights into various questions about moduli spaces of $G_{2}$ structures and singularity formation. This part is mostly speculative. In the other part of the talk we discuss some analysis questions which enter when setting up the foundations of this adiabatic theory. These can be seen as codimension 2 analogues of free boundary problems and related questions have arisen in a number of areas of differential geometry recently.

In this talk we discuss solution of the well-known problem of Erdős and Sauer from 1975 which asks for the maximum number of edges an $n$-vertex graph can have without containing a $k$-regular subgraph, for some fixed integer $k\geq 3$. We prove that any $n$-vertex graph with average degree at least $C_k\log\log n$ contains a $k$-regular subgraph. This matches the lower bound of Pyber, Rödl and Szemerédi and substantially
improves an old result of Pyber, who showed that average degree at least $C_k\log n$ is enough.

Our method can also be used to settle asymptotically a problem raised by Erdős and Simonovits in 1970 on almost regular subgraphs of sparse graphs and to make progress on the well-known question of Thomassen from 1983 on finding subgraphs with large girth and large average degree.

Joint work with Oliver Janzer

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.

Motivated by the physical relevance of many Hodge Laplace-type PDEs from the finite element exterior calculus, we analyse the Hodge Laplacian boundary value problem arising from the strain space in the linear elasticity complex, an exact sequence of function spaces naturally arising in several areas of continuum mechanics. We propose a discretisation based on the adaptation of discontinuous Galerkin FEM for the incompatibility operator $\mathrm{inc} := \mathrm{rot}\circ\mathrm{rot}$, using the symmetric-tensor-valued Regge finite element to discretise  the strain field; via the 'Regge calculus', this element has already been successfully applied to discretise another metric tensor, namely that arising in general relativity. Of central interest is the characterisation of the associated Sobolev space $H(\mathrm{inc};\mathbb{R}^{d\times d}_{\mathrm{sym}})$. Building on the pioneering work of van Goethem and coauthors, we also discuss promising connections between functional analysis of the $\mathrm{inc}$ operator and Kröner's theory of intrinsic elasticity in the presence of defects.

This is based on ongoing work with Dr Kaibo Hu.

In this talk, I review some recent results obtained for black holes using
effective field theory methods applied to quantum gravity, in particular the
unique effective action. Black holes are complex thermodynamical objects
that not only have a temperature but also have a pressure. Furthermore, they
have quantum hair which provides a solution to the black hole information
paradox.

The aim of this talk is to present some novel inequalities for the k-plane transform acting on the modulus square of solutions of the linear time-dependent Schrodinger equation. Our motivation for studying these tomographic expressions comes for virial identities in the context of Schrodinger equations, where tomographic Strichartz estimates of the type we will discuss here appear naturally.

I'll introduce the classical arithmetic topology dictionary of Mumford-Manin-Mazur-Morishita-etc. I'll then present an interesting instance of parallel phenomena related to symplectic structures on moduli spaces of certain bundles. The arithmetic side turns out to be an application of Poitou-Tate duality. Depending on time, I'll delve into the delicate details which make the analogy useful for Diophantine geometers.

Noether’s theorem plays a fundamental role in modern physics by relating symmetries of a Lagrangian to conserved quantities of the Euler-Lagrange equations. In ideal fluid dynamics, the theorem relates the particle labeling symmetry to a Kelvin circulation law. Circulation is conserved for incompressible flows and, otherwise, is generated by advected variables through the momentum map due to a broken symmetry. We will introduce variational principles for fluid dynamics that constrain advection to be the sum of a smooth and geometric rough-in-time vector field. The corresponding rough Euler-Poincare equations satisfy a Kelvin circulation theorem and lead to a natural framework to develop parsimonious non-Markovian parameterizations of subgrid-scale dynamics.

I will describe joint work with Tyler Kelly and Ran Tessler. FJRW (Fan-Jarvis-Ruan-Witten) theory is an enumerative theory of quasi-homogeneous singularities, or alternatively, of Landau-Ginzburg models. It associates to a potential W:C^n -> C given by a quasi-homogeneous polynomial moduli spaces of (orbi-)curves of some genus and marked points along with some extra structure, and these moduli spaces carry virtual fundamental classes as constructed by Fan-Jarvis-Ruan. Here we specialize to the case W=x^r+y^s and construct an analogous enumerative theory for disks. We show that these open invariants provide perturbations of the potential W in such a way that mirror symmetry becomes manifest. Further, these invariants are dependent on certain choices of boundary conditions, but satisfy a beautiful wall-crossing formalism.

Predicting a protein’s structure from its primary sequence has been a grand challenge in biology for the past 50 years, holding the promise to bridge the gap between the pace of genomics discovery and resulting structural characterization. In this talk, we will describe work at DeepMind to develop AlphaFold, a new deep learning-based system for structure prediction that achieves high accuracy across a wide range of targets. We demonstrated our system in the 14th biennial Critical Assessment of Protein Structure Prediction (CASP14) across a wide range of difficult targets, where the assessors judged our predictions to be at an accuracy “competitive with experiment” for approximately 2/3rds of proteins. The talk will cover both the underlying machine learning ideas and the implications for biological research as well as some promising further work.

In this talk, I will investigate the origin of Euclidean wormholes in the gravitational part integral in the context of AdS/CFT. These geometries are confusing since they prevent products of partition functions to factorize, as they should in any quantum mechanical system. I will briefly review the different proposals for the origin of these wormholes, one of which is that one should consider ensemble of average of boundary systems instead of a fixed quantum system with a fixed Hamiltonian. I will explain that it seems unlikely that one can average over CFTs and present a new idea: averaging over approximate CFTs, which I will define. I will then study the variance of the crossing equation in an ensemble relevant for 3d gravity. Based on work in progress with de Boer, Jafferis, Nayak and Sonner.

I will describe recent work with Zhibeodov on the boundary interpretation of orbits around an AdS black hole. When the orbits are far away from the black hole, these orbits describe heavy-light double-twist operators on the boundary. I will discuss how the dimensions of these operators can be computed exactly in terms of quasinormal modes in the bulk, using techniques from a paper to appear soon with Grassi, Iossa, Lichtig, and Zhiboedov. Then I will explain how these results are related to the concept of quantum scars, which are eigenstates that do not obey ETH. 

Melanie Weber 

Title: Geometric Methods for Machine Learning and Optimization

Abstract: A key challenge in machine learning and optimization is the identification of geometric structure in high-dimensional data. Such structural understanding is of great value for the design of efficient algorithms and for developing fundamental guarantees for their performance. Motivated by the observation that many applications involve non-Euclidean data, such as graphs, strings, or matrices, we discuss how Riemannian geometry can be exploited in Machine Learning and Optimization. First, we consider the task of learning a classifier in hyperbolic space. Such spaces have received a surge of interest for representing large-scale, hierarchical data, since they achieve better representation accuracy with fewer dimensions. Secondly, we consider the problem of optimizing a function on a Riemannian manifold. Specifically, we will consider classes of optimization problems where exploiting Riemannian geometry can deliver algorithms that are computationally superior to standard (Euclidean) approaches.

Francesca Panero

Title: A general overview of the different projects explored during my DPhil in Statistics.

Abstract: In the first half of the talk, I will present my work on statistical models for complex networks. I will propose a model to describe sparse spatial random graph underpinned by the Bayesian nonparametric theory and asymptotic properties of a more general class of these models, regarding sparsity, degree distribution and clustering coefficients.

The second half will be devoted to the statistical quantification of the risk of disclosure, a quantity used to evaluate the level of privacy that can be achieved by publishing a microdata file without modifications. I propose two ways to estimate the risk of disclosure, using both frequentist and Bayes nonparametric statistics.

Directed graphs are a model for various phenomena in the
sciences. In topological data analysis particularly the advent of
applying topological tools to networks of brain neurons has spawned
interest in constructing topological spaces out of digraphs, developing
computational tools for obtaining topological information, and using
these to understand networks. At the end of the day, (homological)
computations of the spaces reveal something about the geometric
realisation, thereby losing the directionality information.

However, digraphs can also be associated with path algebras. We can now
consider applying Hochschild homology to extract information, hopefully
obtaining something more refined in terms of the combinatorics of the
directed edges and paths in the digraph. Unfortunately, Hochschild
homology tends to vanish beyond degree 1. We can overcome this by
considering different higher paths of simplices, and thus introduce
Hochschild homology of digraphs in higher degrees. Moreover, this
procedure gives an implementable persistence pipeline for network
analysis. This is a joint work with Luigi Caputi.

In the early 2010s, Riche and Williamson proposed new character formulas for simple and indecomposable tilting modules over reductive algebraic groups $G$ in positive characteristic. Even better, they showed their formulas would follow from a conceptually satisfying "categorical conjecture", which they were able to prove for $G = GL_n$. Our first goal in this talk will be to explain the statement of the categorical conjecture, indicating its connection to representation theory and assuming minimal background knowledge. Subsequently, we will introduce Smith–Treumann theory and outline how it may be applied to prove the categorical conjecture in general. Time permitting, we will conclude with remarks on future directions of study.

Experimental biologists study diseases mostly through their abnormal molecular or cellular features. For example, they investigate genetic abnormalities in cancer, hormonal imbalances in diabetes, or an aberrant immune system in vascular diseases. Moreover, many diseases also have a mechanical component which is critical to their deadliness. Most notably, cancer kills typically through metastasis, where the cancer cells acquire the capability to remodel their adhesions and to migrate. Solid tumours are also characterised by physical changes in the extracellular matrix – the material surrounding the cells. While such physical changes are long known, only relatively recent research revealed that cells can sense altered physical properties and transduce them into chemical information. An example is the YAP/TAZ signalling pathway that can activate in response to altered matrix mechanics and that can drive tumour phenotypes such as the rate of cell proliferation.
Systems-biology models aim to study diseases holistically. In this talk, I will argue that physical signatures are a critical part of many diseases and therefore, need to be incorporated into systems-biology. Crucially, physical disease signatures bi-directionally interact with molecular and cellular signatures, presenting a major challenge to developing such models. I will present several examples of recent and ongoing work aimed at uncovering the relations between mechanical and molecular/cellular signatures in health and disease. I will discuss how blood vessel cells interact mechano-chemically with each other to regulate the passage of cells and nutrients between blood and tissue and how cancer cells grow and die in response to mechanical and geometrical stimuli.

Alumina is a raw material for aluminium production, and Attila Kovacs made mathematical models for alumina feeding, including heating, melt infiltration, and dissolution. One of his assumptions is that when several alumina particle stick together to form a "raft", these will stay together even if initial frozen cryolite inside this "raft" melts, and even if almost all alumina in the "raft" is dissolved. In reality, the "raft" will break up, either from one of the two mechanisms already mentioned, or from the expansion of gas or water vapor stuck within the "raft". We would therefore like to investigate mathematically when and under which circumstances this splitting up will take place. 

In the aftermath of the Thirty Years War (1618–1648), the mining regions of Central Europe underwent numerous technical and political evolutions. In this context, the role of underground geometry expanded considerably: drawing mining maps and working on them became widespread in the second half of the seventeenth century. The new mathematics of subterranean surveyors finally realized the old dream of “seeing through stones,” gradually replacing alternative tools such as written reports of visitations, wood models, or commented sketches.

I argue that the development of new cartographic tools to visualize the underground was deeply linked to broad changes in the political structure of mining regions. In Saxony, arguably the leading mining region, captain-general Abraham von Schönberg (1640–1711) put his weight and reputation behind the new geometrical technology, hoping that its acceptance would in turn help him advance his reform agenda. At-scale representations were instrumental in justifying new investments, while offering technical road maps to implement them.

The spread of a matrix is defined as the diameter of its spectrum. In this talk, we consider the problem of maximizing the spread of a symmetric non-negative matrix with bounded entries and discuss a number of recent results. This optimization problem is closely related to a pair of conjectures in spectral graph theory made by Gregory, Kirkland, and Hershkowitz in 2001, which were recently resolved by Breen, Riasanovsky, Tait, and Urschel. This talk will give a light overview of the approach used in this work, with a strong focus on ideas, many of which can be abstracted to more general matrix optimization problems.

Consider the injection of a fluid onto an impermeable surface for an infinite length of time... Does the injected fluid reach a finite height, or does it keep on growing forever? The classical theory of gravity currents suggests that the height remains finite, causing the radius to grow outwards like the square root of time. When the fluid resides within a porous medium, the same is thought to be true. However, recently I used some small scale experiments and numerical simulations, spanning 12 orders of magnitude in dimensionless time, to demonstrate that the height actually grows very slowly, at a rate ~t^(1/7)*(log(t))^(1/2). This strange behaviour can be explained by analysing the flow in a narrow "inner region" close to the source, in which there are significant vertical velocities and non-hydrostatic pressures. Analytical scalings are derived which match closely with both numerics and experiments, suggesting that the blob of fluid is in fact ever-growing, and therefore becomes unbounded with time.

11:30 Refreshments (tea, coffee and homemade biscuits)

12:00 Talks (main room)

13:15 Buffet Style Lunch (incl. tea, coffee and homemade cakes)

15:00 End

After a general introduction to the study of random walks on groups, we discuss the relationship between limit theorems for random walks on Lie groups and Diophantine properties of the underlying distribution. Indeed, we will discuss the classical abelian case and more recent results by Bourgain-Gamburd for compact simple Lie groups such as SO(3). If time permits, we discuss some new results for non-compact simple Lie groups such as SL_2(R). We hope to touch on the relevant methods from harmonic analysis, number theory and additive combinatorics. The talk is aimed at a general audience. 

We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will present recent progress on this topic. Based on joint work with Huy Tuan Pham.

 I will discuss ongoing work with Toke Carlsen and Aidan Sims on ideal structure of C*-algebras of commuting local homeomorphisms. This is one aspect of a general attempt to bridge C*-algebras with multidimensional (symbolic) dynamics.

Social, biological and physical systems are widely studied through the modeling of dynamical processes over networks, and one commonly investigates the interplay between structure and dynamics. I will discuss how cyclic patterns in networks can influence models for collective and diffusive processes, including generalized models in which dynamics are defined over simplicial complexes and multiplex networks. Our approach relies on homology theory, which is the subfield of mathematics that formally studies cycles (and more generally, k-dimensional holes). We will make use of techniques including persistent homology and Hodge theory to examine the role of cycles in helping organize dynamics onto low-dimensional manifolds. This pursuit represents an emerging interface between the fields of network-coupled dynamical systems and topological data analysis.

Let G(R) be the real points of a complex reductive algebraic group G. There are many difficult questions about admissible representations of real reductive groups which have (relatively) easy answers in the case of complex groups. Thus, it is natural to look for a relationship between representations of G and representations of G(R). In this talk, I will introduce a functor from admissible representations of G to admissible representations of G(R). This functor interacts nicely with many natural invariants, including infinitesimal character, associated variety, and restriction to a maximal compact subgroup, and it takes unipotent representations of G to unipotent representations of G(R).

We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

Hydrodynamics allow for efficient computation of many-body dynamics and have been successfully used in the study of black hole horizons, collective behaviour of QCD matter in heavy ion collisions, and non-equilibrium behaviour in strongly-interacting condensed matter systems.
In this talk, I will present the application of hydrodynamics to quantum field theory with an infinite number of local conservation laws. Such an integrable system can be described within the recently developed framework of generalised hydrodynamics. I will present the key assumptions of generalised hydrodynamics as well as summarise some recent developments in this field. In particular, I will concentrate on the study of the SU(3)_2-Homogeneous sine-Gordon model. Thanks to the hydrodynamic approach, we were able to identify the key dynamical signatures of unstable excitations in this integrable quantum field theory and simulate the real time RG-flow of the theory between interacting and free conformal regimes.
The talk is based on joint work with Olalla Castro-Alvaredo, Cecilia De Fazio and Benjamin Doyon.

We consider an oriented acyclic version of the Erdős-Rényi random graph: the set of vertices is {1,...,n}, and for each pair i < j, an edge from i to j is independently added to the graph with probability p. The length of the longest path in such a graph grows linearly with the number of vertices in the graph, and its growth rate is a deterministic function C of the probability p of presence of an edge.
Foss and Konstantopoulos introduced a coupling between these graphs and a particle system called the "Infinite-bin model". By using this coupling, we prove some properties of C, that it is analytic on (0,1], its development in series at point 1 and its asymptotic behaviour as p goes to 0.

I will introduce a new parabolic system for the flow of nematic liquid crystals, enjoying a free boundary condition. After recent works related to the construction of blow-up solutions for several critical parabolic problems (such as the Fujita equation, the heat flow of harmonic maps, liquid crystals without free boundary, etc...), I will  construct a physically relevant weak solution blowing-up in finite time. We make use of  the so-called inner/outer parabolic gluing. Along the way, I will present a set of optimal estimates for the Stokes operator with Navier slip boundary conditions. I will state several open problems related to the partial regularity of the system under consideration. This is joint work with F.-H. Lin (NYU), Y. Zhou (JHU) and J. Wei (UBC).