Inspired by the symplectic Banach-Mazur distance, proposed by Ostrover and Polterovich in the setting of non-degenerate starshaped domains of Liouville manifolds, we define a distance on the space of contact forms supporting a given contact structure on a closed contact manifold and we use it to bi-Lipschitz embed part of the 2-dimensional Euclidean space into the space of overtwisted contact forms supporting a given contact structure on a smooth closed manifold.
A classical branching theorem of Weyl describes how an irreducible representation of compact U(n+1) decomposes when restricted to U(n). The local Gan-Gross-Prasad conjecture provides a conjectural extension to the setting of representations of noncompact unitary groups lying in a generic L-packet. We prove this conjecture. Previously Beuzart-Plessis proved the "multiplicity one in a Vogan packet" part of the conjecture for tempered L-packets using the local trace formula approach initiated by Waldspurger. Our proof uses theta lifts instead, and is independent of the trace formula argument.
Let G be a group and let V be a finite-dimensional vector space over a field K. We equip VG = {x : G → V} with the prodiscrete uniform structure, the G-shift action ((gx)(h) := x(g-1h)), and the natural structure of a K-vector space. A G-invariant closed subspace X ≤ VG is called a linear subshift. A linear subshift X ⊆ VG is said to be of finite type provided that there exists a finite subset Ω ⊆ G and a subspace W ⊆ VΩ such that
X = X(Ω,W) := {x ∈ VG : (gx)|Ω ∈ W for all g ∈ G}.
The group G is said to be of linear-Markov type if for every finite-dimensional vector space V over any field K, every linear subshift X ⊆ VG is of finite type. A uniformly continuous and G-equivariant K-linear map τ : VG → VG is called a cellular automaton. The group G is said to be linearly surjunctive provided that for every finite-dimensional vector space V over any field K the following holds: every injective linear cellular automaton τ : VG → VG is surjective.
THEOREM 1 (CS-Coornaert 2007) Sofic groups are linearly surjunctive.
COROLLARY 1 (Elek-Szabo 2004; CS-Coornaert 2007) Group rings of sofic groups are stably finite.
THEOREM 2 (CS-Coornaert-Phung 2020) A group is of linear-Markov type if and only if the group ring K[G] is left-Noetherian for any field K.
COROLLARY 2 (CS-Coornaert-Phung 2020) Polycyclic-by-finite groups are of linearly-Markov type.
An exponential function is a homomorphism from the additive group of a field to its multiplicative group. The most important examples are the real and complex exponentials, and these are naturally studied analytically.
However, one can also study the algebra of exponential fields and their logical theory. It turns out that the natural ways to do this take one outside the usual finitary classical logic of model theory and into positive/coherent logic, geometric logic, or other infinitary logics, or to the more algebraic and abstract setting of accessible categories.
I will describe some of this story, focusing on the more algebraic aspects of existentially closed exponential fields.
Consider the following three properties of a general group G:
Algebra: G is abelian and torsion-free.
Analysis: G is a metric space that admits a 'norm', namely, a translation-invariant metric d( . , . ) satisfying: d(1,gn) = |n| d(1,g) for all g ∈ G and integers n.
Geometry: G admits a length function with 'saturated' subadditivity for equal arguments: l(g2) = 2 l(g) for all g ∈ G.
While these properties may a priori seem different, in fact they turn out to be equivalent (and also to G being isometrically and additively embedded in a Banach space, hence inheriting its norm). The non-trivial implication amounts to saying that there does not exist a non-abelian group with a 'norm'. We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and finally, the logistics of how the problem was solved, via a PolyMath project that began on a blog post of Terence Tao.
The Duflo-Serganova functors DS are tensor functors relating representations of different Lie superalgebras. In this talk I will consider the behaviour of various invariants, such as the defect, the dual Coxeter number, the atypicality and the cores, under the DS-functor. I will introduce a notion of depth playing the role of defect for algebras and atypicality for modules. I will mainly concentrate on examples of symmetrizable Kac-Moody and Q-type superalgebras.
The set of T-stable curves in a Schubert variety through a T-fixed point is relatively easy to understand, but the tangent space is more difficult. In this talk we describe some new relations between the tangent space and the set of T-fixed curves. This is joint work with Victor Kreiman.
We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, showing in particular that this group is not quasi-isometric to a line. This answers a well-known question of Kapovich-Polterovich from 2006. We show that these flats in Ham(S2)Ham(S2) stabilize to certain product four-manifolds, prove constraints on Lagrangian packing, and find new instances of Lagrangian PoincarÉ recurrence. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. This is joint work with Leonid Polterovich.
Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension n, defined over a fixed number field, with good reduction outside a fixed finite set of primes, up to isomorphism. In joint work with Brian Lawrence, we prove an analogous finiteness statement for hypersurfaces in a fixed abelian variety with good reduction outside a finite set of primes. I will give a broad introduction to some of the ideas in the proof, which builds on p-adic Hodge theory techniques from work of Lawrence and Venkatesh as well as sheaf convolution in algebraic geometry.
A recent advancement in the theory of left-orderable groups is the discovery of finitely generated left-orderable simple groups by Hyde and Lodha. We will discuss a construction that extends this result by showing that every countable left-orderable group is a subgroup of such a group. In conjunction with this construction, we will also discuss computability properties of left-orders in groups. Based on a joint work with M. Steenbock.
I will discuss a construction of a new model structure on simplicial objects in a countably lextensive category (i.e., a category with well-behaved finite limits and countable coproducts). This builds on previous work on a constructive model structure on simplicial sets, originally motivated by modelling Homotopy Type Theory, but now applicable in a much wider context.
In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.
In 1977, Kac classified simple Lie superalgebras over ℂ and showed they play an analogous role to simple Lie algebras over the complex numbers. For simple algebraic groups and their Lie algebras, the notions of a maximal torus, Borel subgroups and the Weyl groups provide a uniform method to treat the structure and representation theory for these groups and Lie algebras. Historically, much of the work for simple Lie superalgebras has involved dealing with these objects using a case by case analysis.
Fifteen years ago, Boe, Kujawa and the speaker introduced the important concept of detecting subalgebras for classical Lie superalgebras. These algebras were constructed by using ideas from geometric invariant theory. More recently, D. Grantcharov, N. Grantcharov, Wu and the speaker introduced the BBW parabolic subalgebras. Given a Lie superalgebra 𝔤, one has a triangular decomposition 𝔤=𝔫- ⨁ 𝔣 ⨁ 𝔫+ with 𝔟=𝔣 ⨁ 𝔫- where 𝔣 is a detecting subalgebra and 𝔟 is a BBW parabolic subalgebra. This holds for all classical 'simple' Lie superalgebras, and one can view 𝔣 as an analogue of the maximal torus, and 𝔟 like a Borel subalgebra. This setting also provide a useful method to define semisimple elements and nilpotent elements, and to compute various sheaf cohomology groups R• indBG (-).
The goal of my talk is to provide a survey of the main ideas of this new theory and to give indications of the interconnections within the various parts of this topic. I will also indicate how our ideas can further unify the study of the representation theory of classical Lie superalgebras.
The classical Turán problem asks: given a graph H, how many edges can an n-vertex graph have while containing no isomorphic copy of H? By viewing (k+1)-uniform hypergraphs as k-dimensional simplicial complexes, we can ask a topological version of this, first posed by Nati Linial: given a k-dimensional simplicial complex S, how many facets can an n-vertex k-dimensional simplicial complex have while containing no homeomorphic copy of S? Until recently, little was known for k greater than 2. In this talk, we give an answer for general k using dependent random choice - a method that has produced powerful results in extremal graph theory, additive combinatorics, and Ramsey theory.
A classical construction in topology associates to a space X and prime p, a new 'localized' space Xp whose homotopy and homology groups are obtained from those of X by inverting p. In this talk, I will discuss a symplectic analogue of this construction, extending work of Abouzaid-Seidel and Cieliebak-Eliashberg on flexible Weinstein structures. Concretely, I will produce prime-localized Weinstein subdomains of high-dimensional Weinstein domains and also show that any Weinstein subdomain of a cotangent bundle agrees Fukaya-categorically with one of these special subdomains. The key will be to classify which objects of the Fukaya category of T∗M – twisted complexes of Lagrangians – are quasi-isomorphic to actual Lagrangians.
The purpose of my talk is to discuss the following results recently obtained in collaboration with A.Masuoka (Tsukuba University, Japan). First, we prove that a certain category of Harish-Chandra pairs is equivalent to the category of (not necessary affine) locally algebraic group superschemes. Using this fundamental equivalence we superize the famous Barsotti-Chevalley theorem and prove that the sheaf quotient of an algebraic group superscheme over its group super-subscheme is again a superscheme of finite type. I will also formulate some open problems whose solving would bring significant progress in the supergroup theory.
Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and finish with my recent counterexample to the unit conjecture.
In this talk I will show that the simplicial Moore path functor, first defined by Van den Berg and Garner, is a polynomial functor. This result, which surprised us a bit at first, has helped a great deal in developing effective Kan fibrations for simplicial sets.
I will address two problems about recognizing surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension 2. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognizing surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.
