An LMS online lecture course in mapping class groups.
Tag - Topology
Classically, heights are defined over number fields or transcendence degree one function fields. This is so that the Northcott property, which says that sets of points with bounded height are finite, holds. Here, expanding on work of Moriwaki and Yuan-Zhang, we show how to define arithmetic intersections and heights relative to any finitely generated field extension 𝐾/𝑘, and construct canonical heights for polarizable arithmetic dynamical systems 𝑓:𝑋→𝑋. These heights have a corresponding Northcott property when 𝑘 is ℚ or 𝔽𝑞. When 𝑘 is larger, we show that Northcott for canonical heights is conditional on the non-isotriviality of 𝑓:𝑋→𝑋, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. Additionally, we prove the Hodge Index Theorem for arithmetic intersections relative to 𝐾/𝑘. Since, when Northcott holds, pre-periodic points are the same as height zero points, this has applications to dynamical systems. By the Lefschetz principle, these results can be applied over any field.
An LMS online lecture course in Poisson structures.
An online lecture course by the University of Münster in L-theory of rings.
We will introduce Witt groups and various flavours of L-groups and discuss some examples. We will then discuss a process called algebraic surgery. This process permits, under suitable assumptions, to simplify representatives in L-groups, and we will touch on two flavours (surgery from below and surgery from above). We will indicate how these can be used to show that various comparison maps between different L-theories are isomorphisms (in suitable ranges). Then we will go on and discuss three methods that allow for more calculations: Localisation sequences, a dévissage theorem, and an arithmetic fracture square. Using those, we will calculate the L-groups of Dedekind rings whose fraction field is a global field.
An online lecture course by the University of Münster in K-theory of forms.
In this lecture series we will describe an approach to hermitian K-theory which sheds some new light on classical Grothendieck-Witt groups of rings, especially in the domain where 2 is not assumed to be invertible. Our setup is higher categorical in nature, and is based on the concept of a Poincaré ∞-category, first suggested by Lurie. We will explain how classical examples of interest can be encoded in this setup, and how to define the principal invariants of interest, consisting of the Grothendieck-Witt spectrum and L-theory spectrum, within it. We will then describe our main abstract results, including additivity, localization and universality statements for these invariants and their relation to each other and to algebraic K-theory via the fundamental fibre sequence.
We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bi-functor with respect to the bicategorical model structure of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings.
An LMS online lecture course in solitons.
The lectures will highlight some recent work on solvable models of topological solitons. The first involves generalisations of the U(1) Abelian-Higgs model whose integrability is intimately related to the geometry of constant curvature Riemann surfaces. The second piece of work is a study of magnetic skyrmions in chiral magnets. Recently a family of soluble models for magnetic skyrmions in chiral magnets was introduced. The energy functional for these models is bounded below by the topological charge, configurations which attain this bound solve first-order equations. The explicit solutions of these first-order equations are given in terms of arbitrary holomorphic functions. Finally I will explain how this model can be interpreted as a gauged non-linear sigma model.
Lecture 1: A primer on solitons. I will introduce the concept of a topological soliton through two prototypical examples, the φ4 and Sine-Gordon models in 1+1 dimensions. Next, we will meet Derrick's theorem and learn why solitons are hard to construct in higher dimensions. Finally, we will meet some examples of higher-dimensional models possessing soliton solutions.
Lecture 2: Solitons in chiral magnets. We will meet a specific model of 2-dimensional chiral magnetic systems which admits soliton solutions. For a special potential term exact, degree 1, skyrmion solutions can be constructed. This leads up to meeting a critically coupled version of the model where there is a whole zoo of analyitic skyrmion solutions.
The classical Pontryagin-Thom isomorphism equates manifold bordism groups with corresponding stable homotopy groups. This construction moreover generalizes to the equivariant context. I will discuss work which establishes a Pontryagin-Thom isomorphism for orbispaces (an orbispace is a 'space' that is locally modelled on Y/G for Y a space and G a finite group; examples of orbispaces include orbifolds and moduli spaces of pseudo-holomorphic curves). This involves defining a category of orbispectra and an involution of this category extending Spanier-Whitehead duality. Global homotopy theory also plays a key role.
The Liouville function is a multiplicative function that encodes important information related to distributional properties of the prime numbers. A conjecture of Chowla states that the values of the Liouville function fluctuate between plus and minus in such a random way, that all sign patterns of a given length appear with the same frequency. The Chowla conjecture remains largely open and in this talk we will see how ergodic theory combined with some feedback from number theory allows us to establish two variants of this conjecture. Key to our approach is an in-depth study of measure preserving systems that are naturally associated with the Liouville function. The talk is based on joint work with Bernard Host.
In this talk we will study the lax orthogonal factorization systems (LOFSs) of Clementino and Franco, with a particular focus on finding equivalent definitions of them.
In particular, we wish to define them as a pair of classes ℰ and ℳ subject to some conditions. To achieve this, we will reduce the definition of a LOFS in terms of algebraic weak factorization systems (defined as a KZ 2-comonad L and KZ 2-monad R on the 2-category of arrows [2, 𝒞] with a 2-distributive law LR ⇒ RL) to a more property-like definition (meaning a definition with less data but more conditions).
To do this, we replace strict KZ 2-monads with the property-like definition of KZ pseudomonads in terms of Kan-extensions due to Marmolejo and Wood. In addition, pseudo-distributive laws involving KZ pseudomonads have a property-like description which will be used. Thus one can deduce the conditions the classes ℰ and ℳ must satisfy.
We will also consider some similarities and differences between LOFSs and (pseudo-)orthogonal factorization systems, and will extend their definitions to include universal fillers for squares which only commute up to a comparison 2-cell.
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