L1

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.

While scattering problems are posed on unbounded domains, volumetric discretizations typically require truncating the domain at a finite distance, closing the system with some sort of boundary condition.  These conditions typically suffer from some deficiency, such as perturbing the boundary value problem to be solved or changing the character of the operator so that the discrete system is difficult to solve with iterative methods.

We introduce a new technique for the Helmholtz problem, based on using the Green formula representation of the solution at the artificial boundary.  Finite element discretization of the resulting system gives optimal convergence estimates.  The resulting algebraic system can be solved effectively with a matrix-free GMRES implementation, preconditioned with the local part of the operator.  Extensions to the Morse-Ingard problem, a coupled system of pressure/temperature equations arising in modeling trace gas sensors, will also be given.

A 2d SCFT given as a non-linear sigma model of a Ricci-flat Kahler target 

space is not a rational CFT in general; only special points in the moduli 

space of the target-space metric, the 2d SCFTs are rational. 

Gukov-Vafa's paper in 2002 hinted at a possibility that such special points 

may be characterized by the property "complex multiplication" of the target space, 

which has its origin in number theory. We revisit the idea, refine the Conjecture, 

and prove it in the case the target space is T^4. 
 

This presentation is based on arXiv:2205.10299 and 2212.13028 .

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [Cohen-Cont-Rossier-Xu, "Scaling Properties of Deep Residual Networks”, 2021]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. Based on joint work with C. Bayer and P. K. Friz.
 

Topological data analysis (TDA) comprises a set of techniques of computational topology that has had enormous growth in the last decade, with applications to a wide variety of fields, such as images,  biological data, meteorology, materials science, time-dependent data, economics, etc. In this talk, we will first have a walk through a typical pipeline in TDA, to move later to its adaptation to a specific context: the topological characterization of the spatial distribution of regions in a segmented image

By definition, planktonic organisms drift with the water flows. But these millimetric organisms are not necessarily passive; many can swim and can sense the surrounding flow through mechanosensory hairs. But how useful can be flow sensing in a turbulent environment? To address this question, we show two examples of smart planktonic behavior: (1) we develop a model where plantkters choose a swimming direction as a function of the velocity gradient to "surf on turbulence" and move efficiently upwards; (2) we show how a plankter measuring the velocity gradient may track the position of a swimming target from its hydrodynamic signature. 

Ernst Haeckel, Kunstformen der Natur (1904), Copepoda 

Andrea Giudici: Multiple shapes from one elastomer sheet

Active soft materials, such as Liquid Crystal Elastomers (LCEs), possess a unique property: the ability to change shape in response to thermal or optical stimuli. This makes them attractive for various applications, including bioengineering, biomimetics, and soft robotics. The classic example of a shape change in LCEs is the transformation of a flat sheet into a complex curved surface through the imprinting of a spatially varying deformation field. Despite its effectiveness, this approach has one important limitation: once the deformation field is imprinted in the material, it cannot be amended, hindering the ability to achieve multiple target shapes.

In this talk, I present a solution to this challenge and discuss how modulating the degree of actuation using light intensity offers a route towards programming multiple shapes. Moreover, I introduce a theoretical framework that allows us to sculpt any surface of revolution using light.

Edwina Yeo: Modelling the onset of arterial blood clotting

Arterial blood clot formation (thrombosis) is the leading cause of both stroke and heart attack. The blood protein Von Willebrand Factor (VWF) is critical in facilitating arterial thrombosis. At pathologically high shear rates the protein unfolds and rapidly captures platelets from the flow.

I will present two pieces of modelling to predict the location of clot formation in a diseased artery. Firstly a continuum model to describe the mechanosensitive protein VWF and secondly a model for platelet transport and deposition to VWF. We interface this model with in vitro data of thrombosis in a long, thin rectangular microfluidic geometry. Using a reduced model, the unknown model parameters which determine platelet deposition can be calibrated.

We explore the dynamics of a compressible fluid bubble surrounded by an incompressible fluid of infinite extent in three-dimensions, constructing bubble solutions with finite time blowup under this framework when the equation of state relating pressure and volume is soft (e.g., with volume singularities that are locally weaker than that in the Boyle-Mariotte law), resulting in a finite time blowup of the surrounding incompressible fluid, as well. We focus on two families of solutions, corresponding to a soft polytropic process (with the bubble decreasing in size until eventual collapse, resulting in velocity and pressure blowup) and a cavitation equation of state (with the bubble expanding until it reaches a critical cavitation volume, at which pressure blows up to negative infinity, indicating a vacuum). Interestingly, the kinetic energy of these solutions remains bounded up to the finite blowup time, making these solutions more physically plausible than those developing infinite energy. For all cases considered, we construct exact solutions for specific parameter sets, as well as analytical and numerical solutions which show the robustness of the qualitative blowup behaviors for more generic parameter sets. Our approach suggests novel -- and perhaps physical -- routes to the finite time blowup of fluid equations.

Biological systems achieve their complexity by processing and exploiting information stored in molecular copolymers such as DNA, RNA and proteins. Despite the ubiquity and power of this approach in natural systems, our ability to implement similar functionality in synthetic systems is very limited. In this talk, we will first outline a new mathematical framework for analysing general models of colymerisation for infinitely long polymers. For a given model of copolymerisation, this approach allows for the extraction of key quantities such as the sequence distribution, speed of polymerisation and the rate of molecular fuel consumption without resorting to simulation. Subsequently, we will explore mechanisms that allow for reliable copying of the information stored in finite-length template copolymers, before touching on recent experimental work in which these ideas are put into practice.