Math Calendar
treatment. In this talk I will discuss two mathematical approaches to understand what we
can do to increase our understanding of resistance and how we exploit this when encountering resistant populations. I will focus on certain game theoretical models and affine control as simple models for resistance.
In the first part, we will introduce the fundamentals of b-geometry, including the notions of b-metrics and b-vector bundles on manifolds with boundaries.
The second part will shift to an analytical focus, exploring the asymptotic behavior of solutions to the Klein-Gordon equation on asymptotically de Sitter manifolds. We will introduce the essential analytical tools for this, such as b-differential operators and their normal operators, conormal and polyhomogeneous functions, and the Mellin transform.
Speaker: Boaz Moerman.
Content: Monoids, log structures, log schemes, morphisms of log schemes, examples: divisorial log structure, toric varieties (in particular Spec Z[P ] for a monoid P ), examples of a non-divisorial log structure (pullback of a log structure, example of a log curve (without definition of a log curve)).
This is joint work with Tim Browning.
Abstract: The Gromov-Wasserstein (GW) transport problem is a generalization of the classic optimal transport problem, which seeks a relaxed correspondence between two measures while preserving their internal geometry. Due to meeting this theoretical underpinning, it is a valuable tool for the analysis of objects that do not possess a natural embedding or should be studied independently of it. Prime applications can thus be found in e.g. shape matching, classification and interpolation tasks. To tackle the latter, one theoretically justified approach is the employment of GW barycenters, which are generalized Fréchet means with respect to the GW distance.
After giving a gentle and illustrative introduction to classic optimal transport theory we will thoroughly explore the GW transport problem. Subsequently we turn our attention to GW barycenters. Motivated by obtaining a numerically tractable method for their computation, we study the geometry of the induced GW space. Our theoretical results in this context allow us to lift a known fixpoint iteration for the computation of Fréchet means in Riemannian manifolds to the GW setting. The lifted iteration is simple to implement in practice and monotonically improves the quality of the barycenter. We provide numerical evidence of the potential of this method, including multi 3d shape interpolations.
Abstract. We study generalised polynomials, that is, functions that can be expressed using standard algebraic operations and the floor function. Generalised polynomials have long been studied (both explicitly and implicitly) in number theory and dynamics, often using dynamical and ergodic-theoretic methods (especially dynamics on nilmanifolds). These methods enable one to deduce precise information about the average behaviour of generalised polynomials, but allow for complicated behaviour on special sets of density zero. We will present some of the classical results and give examples of various interesting arithmetic and combinatorial behaviour occurring along sets of density zero as well as restrictions to what is possible. We will also state an analogue of Hadamard's quotient theorem for generalised polynomials (the classical version concerns linear recurrences). Several open problems will be formulated.
The talk is based on joint work with Jakub Konieczny (Oxford).
Abstract: Lang conjectured that varieties of general type over a number field do not have a dense set of rational points. In 2000, guided by Lang's conjecture and in search of a converse statement, Abramovich, Colliot-Thelene, Harris, and Tschinkel formulated the "Weakly Special Conjecture": every weakly special variety over a number field has a potentially dense set of rational points. In this talk I will explain how this conjecture contradicts the abc conjecture, and more precisely Campana's "Orbifold Mordell" conjecture. Indeed, starting from an Enriques surface over Q(t) constructed by Lafon, we give the first examples of smooth projective weakly special threefolds which fiber over the projective line in Enriques surfaces with nowhere reduced, but non-divisible, fibers. I will explain that the existence of these threefolds shows that the Weakly Special Conjecture contradicts the abc conjecture. The existence of such threefolds also shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised by Campana in 2005. This is joint work with Finn Bartsch, Frederic Campana, and Olivier Wittenberg.
Location: TBA.
Morning session:
Speaker: Marcin Lis (TU Wien)
Title: Around the planar Ising model
Abstract: In the first part I will talk about the classical result of Groeneveld, Boel and Kasteleyn that boundary spin correlations functions
in Ising models on planar graphs satisfy Pfaffian relations. I will consider the reverse question, and
show that any classical ferromagnetic spin model whose correlation functions satisfy Pfaffian relations must be
(up to local simplifications of the graph) an Ising model on a planar graph. The main tool is a new (coupled) version
of the Edwards—Sokal (Fortuin—Kasteleyn) representation of the Ising model applied to two independent copies of the spin model.
Joint work with Diederik van Engelenburg.
In the second part I will discuss a geometric formula for a certain set of complex zeros of the partition function of the planar Ising model recently proposed by Livine and Bonzom. Remarkably, the zeros depend locally on the geometry of an immersion of the graph in the three dimensional Euclidean space (different immersions give rise to different zeros). When restricted to the flat case, the weights become the critical weights on circle patterns. I will rigorously prove the formula by geometrically constructing a null eigenvector of the Kac-Ward matrix whose determinant is the squared partition function. The main ingredient of the proof is the realisation that the associated Kac-Ward transition matrix gives rise to an SU(2) connection on the graph, creating a direct link with rotations in three dimensions. The existence of a null eigenvector turns out to be equivalent to this connection being flat.
Location and time: JKH 2-3, Room 115 at 11-12.45
Afternoon session:
Speaker: Frank den Hollander (U Leiden)
Title: The Moran model with random resampling rates
Abstract: We consider the two-type Moran model with N individuals. Each individual is assigned a resampling rate, drawn independently from a probability distribution ℙ on ℝ+, and a type, either ♡ or ♢. Each individual resamples its type at its assigned rate, by adopting the type of an individual drawn uniformly at random. Let YN(t) denote the empirical distribution of the resampling rates of the individuals with type ♡ at time Nt. We show that if ℙ has countable support and satisfies certain tail and moment conditions, then in the limit as N→∞ the process (YN(t))t≥0 converges in law to the process (S(t)ℙ)t≥0, in the so-called Meyer-Zheng topology, where (S(t))t≥0 is the Fisher-Wright diffusion with diffusion constant D given by 1/D=∫ℝ+(1/r)ℙ(dr).
Location and time: JKH 2-3, Room 115, 14.15-16.00