Last passage percolation is a model of random planar geometry which captures notions of distances and geodesics. It admits a rich scaling limit, called the directed landscape, that is conjectured to be universal for all geometric models in the so called KPZ universality class. A closely related notion is the KPZ fixed point, which represents the scaling limit of certain growing interfaces. A variational formula links the evolution of the KPZ fixed point to the directed landscape. The optimizer of the variational formula is akin to the polymer endpoint of a point-to-line last passage percolation problem. I will explain how to compute the law of this endpoint using the integrability of the KPZ fixed point. Joint work with Jeremy Quastel and Sourav Sarkar.

Scattering amplitudes provide crucial theoretical input in collider and gravitational wave physics, and at the same time exhibit a remarkable mathematical structure. These lectures will introduce essential concepts and modern techniques exploiting this structure so as to efficiently compute amplitudes and their building blocks, Feynman integrals, in perturbation theory. We will start by decomposing gauge theory amplitudes into simpler pieces based on colour and helicity information. Focusing on tree level, we will then show how these may be determined from their analytic properties with the help of Britto-Cachazo-Feng-Witten recursion. Moving on to loop level, we will define the the class of polylogarithmic functions amplitudes and integrals often evaluate to, and explain their properties as well as relate them to the universal framework for predicting their singularities, known as the Landau equations. Time permitting, we will also summarise the state of the art in the calculation of the aforementioned singularities, and their intriguing relation to mathematical objects known as cluster algebras.

Gromov used convex integration to prove that any closed
orientable three-manifold equipped with a volume form admits three
divergence-free vector fields which are linearly independent at every
point. We provide an alternative proof of this using geometric
properties of eigenspinors in three dimensions. In fact, our proof
shows that for any Riemannian metric, one can find three
divergence-free vector fields such that at every point they are
orthogonal and have the same non-zero length.

We are interested in the recurrence and transience of a branching random walk in Z^d indexed by a critical Galton-Watson tree conditioned to survive. When the environment is homogeneous, deterministic, and if the offspring distribution has a finite third moment, it is known to be recurrent for d at most 4, and transient for d larger than 4. In this talk we consider an environment made of random conductances, and we prove that, if the conductances satisfy suitable technical assumptions, the same result holds. The argument is based on the combination of a 0-1 law and a truncated second moment method, which only requires to have good estimates on the quenched Green's function of a (non-branching) random walk in random conductances.

Scattering amplitudes provide crucial theoretical input in collider and gravitational wave physics, and at the same time exhibit a remarkable mathematical structure. These lectures will introduce essential concepts and modern techniques exploiting this structure so as to efficiently compute amplitudes and their building blocks, Feynman integrals, in perturbation theory. We will start by decomposing gauge theory amplitudes into simpler pieces based on colour and helicity information. Focusing on tree level, we will then show how these may be determined from their analytic properties with the help of Britto-Cachazo-Feng-Witten recursion. Moving on to loop level, we will define the the class of polylogarithmic functions amplitudes and integrals often evaluate to, and explain their properties as well as relate them to the universal framework for predicting their singularities, known as the Landau equations. Time permitting, we will also summarise the state of the art in the calculation of the aforementioned singularities, and their intriguing relation to mathematical objects known as cluster algebras.

Talk 1 (15:00): Introduction to Graph-based Learning
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Matthew will give an introduction to graph-based learning, touching on variational problems on graphs, techniques from optimal transport, the calculus of variations, and large data limits.

Talk 2 (16:00): Discrete-To-Continuum Limits in Graph-Based Semi-Supervised Learning

Semi-supervised learning (SSL) is the problem of finding missing labels from a partially labelled data set. The heuristic one uses is that ���similar feature vectors should have similar labels���. The notion of similarity between feature vectors explored in this talk comes from a graph-based geometry where an edge is placed between feature vectors that are closer than some connectivity radius. A natural variational solution to the SSL is to minimise a Dirichlet energy built from the graph topology. And a natural question is to ask what happens as the number of feature vectors goes to infinity? In this talk I will give results on the asymptotics of graph-based SSL using an optimal transport topology. The results will include a lower bound on the number of labels needed for consistency.

Classical statistical mechanics describes the macroscopic properties of large numbers of particles. It has a hidden weakness: it assumes that the microscopic forces derive from a Hamiltonian. The same mathematical object then controls both the equations of motion, and the Boltzmann distribution. This is why quantities like pressure are not only time averages of forces (on a wall), but also thermodynamic state functions (which exist independently of any wall). Active matter systems are different. Their particles take energy out of the environment, and use it for dissipative self-propulsion, violating Hamiltonian dynamics. Examples include swimming micro-organisms, and synthetic colloids propelled by optical or chemical energy. The absence of a Hamiltonian-derived detailed balance principle requires a rebuild of statistical mechanics, with some surprising outcomes. For example: (i) the pressure of an active fluid on a wall is not a state function -- it depends on the type of wall\DSEMIC (ii) various interfacial phenomena, governed in equilibrium by a single surface tension, now involve different tensions, some of which can be negative. I will survey these among other surprises and, if time allows, say how they affect kinetic questions such as nucleation rates.