Past

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.

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.

 

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).

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.

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). 

We show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon. The proof makes use of representation varieties of the knot groups to S O(N) and relations between them induced by a ribbon concordance.

(Joint with Y. Lekili) If someone gives you a variety with a singular point, you can try and get some understanding of what the singularity looks like by taking its “link”, that is you take the boundary of a neighbourhood of the singular point. For example, the link of the complex plane curve with a cusp $y^2 = x^3$ is a trefoil knot in the 3-sphere. I want to talk about the links of a class of 3-fold singularities which come up in Mori theory: the compound Du Val (cDV) singularities. These links are 5-dimensional manifolds. It turns out that many cDV singularities have the same 5-manifold as their link, and to tell them apart you need to keep track of some extra structure (a contact structure). We use symplectic cohomology to distinguish the contact structures on many of these links.

Cryo-Electron Microscopy (cryo-EM) is an imaging technology that is revolutionizing structural biology. Cryo-electron microscopes produce many very noisy two-dimensional projection images of individual frozen molecules; unlike related methods, such as computed tomography (CT), the viewing direction of each particle image is unknown. The unknown directions and extreme noise make the determination of the structure of molecules challenging. While other methods for structure determination, such as x-ray crystallography and NMR, measure ensembles of molecules, cryo-electron microscopes produce images of individual particles. Therefore, cryo-EM could potentially be used to study mixtures of conformations of molecules. We will discuss a range of recent methods for analyzing the geometry of molecular conformations using cryo-EM data.

Hydrodynamic excitations corresponding to sound and shear modes in fluids are characterized by gapless dispersion relations. In the hydrodynamic gradient expansion, their frequencies are represented by power series in spatial momenta. In this talk we will discuss the convergence properties of the hydrodynamic series by studying the associated spectral curve in the space of complexified frequency and complexified spatial momentum. For the N=4 supersymmetric Yang-Mills plasma at infinite 't Hooft coupling, we will use the holographic methods to demonstrate that the derivative expansions have finite non-zero radii of convergence. Obstruction to the convergence of hydrodynamic series arises from level-crossings in the quasinormal spectrum at complex momenta. We will discuss how finiteness of 't Hooft coupling affects the radius of convergence. We will show that the purely perturbative calculation in terms of inverse 't Hooft coupling gives the increasing radius of convergence when the coupling is decreasing. Applying the non-perturbative resummation techniques will make radius of convergence piecewise continuous function that decreases after the initial increase. Finally, we will provide arguments in favour of the non-perturbative approach and show that the presence of nonperturbative modes in the quasinormal spectrum can be indirectly inferred from the analysis of perturbative critical points.

In this talk, we will present a new class of invariants of multi-parameter persistence modules : \emph{projected barcodes}. Relying on Grothendieck's six operations for sheaves, projected barcodes are defined as derived pushforwards of persistence modules onto $\R$ (which can be seen as sheaves on a vector space in a precise sense). We will prove that the well-known fibered barcode is a particular instance of projected barcodes. Moreover, our construction is able to distinguish persistence modules that have the same fibered barcodes but are not isomorphic. We will present a systematic study of the stability of projected barcodes. Given F a subset of the 1-Lipschitz functions, this leads us to define a new class of well-behaved distances between persistence modules, the  F-Integral Sheaf Metrics (F-ISM), as the supremum over p in F of the bottleneck distance of the projected barcodes by p of two persistence modules. 

In the case where M is the collection in all degrees of the sublevel-sets persistence modules of a function f : X -> R^n, we prove that the projected barcode of M by a linear map p : R^n \to R is nothing but the collection of sublevel-sets barcodes of the post-composition of f by p. In particular, it can be computed using already existing softwares, without having to compute entirely M. We also provide an explicit formula for the gradient with respect to p of the bottleneck distance between projected barcodes, allowing to use a gradient ascent scheme of approximation for the linear ISM. This is joint work with François Petit.