In this talk I will present my work initiating the study of the C0 symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms, and briefly present the proofs of the first results regarding the topology of the group of symplectic homeomorphisms. For that purpose, we will introduce a method coming from Floer theory and barcodes theory. Applying this strategy to the Dehn-Seidel twist, a symplectomorphism of particular interest when studying the symplectic mapping class group, we will generalize to C0 settings a result of Seidel concerning the non-triviality of the mapping class of this symplectomorphism. We will indeed prove that the generalized Dehn twist is not in the connected component of the identity in the group of symplectic homeomorphisms. Doing so, we prove the non-triviality of the C0 symplectic mapping class group of some Liouville domains.
In 2016, Bao and Wang developed a general theory of canonical basis for quantum symmetric pairs (U,Ui), generalizing the canonical basis of Lusztig and Kashiwara for quantum groups and earning them the 2020 Chevalley Prize in Lie Theory. The i-divided powers are polynomials in a single generator that generalize Lusztig's divided powers, which are monomials. They can be similarly perceived as canonical basis in rank one, and have closed form expansion formulas, established by Berman and Wang, that were used by Chen, Lu and Wang to give a Serre presentation for coideal subalgebras Ui, featuring novel i-Serre relations when τ(i)=i. Quantum covering groups, developed by Clark, Hill and Wang, are a generalization that `covers' both the Lusztig quantum group and quantum supergroups of anisotropic type. In this talk, I will talk about how the results for i-divided powers and the Serre presentation can be extended to the quantum covering algebra setting, and subsequently applications to canonical basis for Uiπ, the quantum covering analogue of Ui, and quantum covering groups at roots of 1.
My recent work has involved taking questions asked for finite groups and considering them for infinite groups. There are various natural directions with this. In finite group theory, there exist many beautiful results regarding generation properties. One such notion is that of spread, and Scott Harper and Casey Donoven have raised several intriguing questions for spread for infinite groups. A group G has spread k if for every g1,…,gk we can find an h in G such that ⟨gi,h⟩=G. For any group we can say that if it has a proper quotient that is non-cyclic, then it has spread 0. In the finite world there is then the astounding result - which is the work of many authors - that this condition on proper quotients is not just a necessary condition for positive spread, but is also a sufficient one. Harper-Donoven's first question is therefore: is this the case for infinite groups? Well, no. But that’s for the trivial reason that we have infinite simple groups that are not 2-generated (and they point out that 3-generated examples are also known). But if we restrict ourselves to 2-generated groups, what happens? In this talk we'll see the answer to this question. The arguments will be concrete and accessible to a general audience.
In this talk, we will be interested in measure-preserving actions of countable groups on standard probability spaces, and more precisely in the partitions of the space into orbits that they induce, also called measure-preserving equivalence relations. In 2000, Gaboriau obtained a characterization of the ergodic equivalence relations which come from non-free actions of the free group on n > 1 generators: these are exactly the equivalence relations of cost less than n. A natural question is: how non-free can these actions be made, and what does the action on each orbit look like? We will obtain a satisfactory answer by showing that the action on each orbit can be made totipotent, which roughly means 'as rich as possible', and furthermore that the free group can be made dense in the ambient full group of the equivalence relation.
Let F be a finite subset of ℤd. We say that F is a translational tile of ℤd if it is possible to cover ℤd by translates of F with no overlaps. Given a finite subset F of ℤd, could we determine whether F is a translational tile in finite time? Suppose that F does tile, does it admit a periodic tiling? A well known argument of Wang shows that these two questions are closely related. In the talk, we will discuss this relation and present some new results, joint with Terence Tao, on the rigidity of tiling structures in ℤ2, and their applications to decidability.
The talk will focus on the question of whether existing symplectic methods can distinguish pseudo-rotations from rotations (i.e., elements of Hamiltonian circle actions). For the projective plane, in many instances, but not always, the answer is negative. Namely, for virtually every pseudo-rotation there exists a unique rotation with precisely the same fixed-point data. However, the hypothetical exceptions—ghost pseudo-rotations—suggest that the relation between the two classes of maps might be much weaker than previously thought, possibly leading to some unexpected consequences.
Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures. A prime example is Pila's modular Ax-Lindemann-Weierstrass (ALW) Theorem and its role in his proof of the André-Oort conjecture.
In this talk we will discuss how an entirely new approach, using the model theory of differential fields, can be used to prove the ALW Theorem with derivatives for Fuchsian automorphic functions - a direct generalization of Pila’s ALW theorem. We will also explain how new cases of the André-Pink conjecture can be obtained using this new approach.
This is joint work with G. Casale and J. Freitag.
A net is a subset of [0,1]d containing the expected number of points in every large dyadic subcube. Nets are one of the central objects in discrepancy theory, with numerous applications in numerical algorithms. In this talk, I will discuss a construction of sets that instead contain approximately correct number of points in every large dyadic subcube, and how these can be used to construct sets in ℝd without large convex holes.
Symplectic implosion was developed to solve the problem that the symplectic cross-section of a Hamiltonian K-space is usually not symplectic, when K is a compact Lie group. The symplectic implosion is a stratified symplectic space, introduced in a 2002 paper of the speaker with Guillemin and Sjamaar. I survey some examples showing how symplectic implosion has been used. I describe the universal imploded cross-section, which is the imploded cross-section of the cotangent bundle of a compact Lie group. Imploded cross-sections are normally not smooth manifolds. We describe some invariants (for example intersection homology) which replace homology for singular stratified spaces.
In characteristic p, the simple modules for the symmetric group Sn are the James modules Dλ, labelled by p-regular partitions of n. If we let sgn denote the 1-dimensional sign module, then for any p-regular λ, the module Dλ ⊗ sgn is also a simple module. So there is an involutory bijection mp on the set of p-regular partitions such that Dλ ⊗ sgn=Dmp(λ). The map mp is called the Mullineux map, and an important problem is to describe mp combinatorially. There are now several known solutions to this problem. I will describe the history of this problem and explain the known combinatorial solutions, and then give a new solution based on crystals and regularization.
Hilbert's Irreducibility Theorem shows that irreducibility over the field of rationals is 'often' preserved when one specializes a variable in some irreducible polynomial in several variables. I will present a version 'over the ring' for which the specialized polynomial remains irreducible over the ring of integers. The result also relates to the Schinzel Hypothesis about primes in value sets of polynomials: I will discuss a weaker 'relative' version for the integers and the full version for polynomials. The results extend to other base rings than the ring of integers; the general context is that of rings with a product formula.
Maass forms are a particular class of smooth functions defined on a hyperbolic Riemann surface. In the special case of Riemann surfaces associated with a congruence subgroup, it is often the case that results concerning Maass forms bear witness to the existence of profound arithmetic relations. Our main goal is to describe the problem of estimating the triple product functional, explain its significance, and illustrate the representation theoretical techniques employed by Bernstein and Reznikov to make progress. If time permits, we shall discuss non-Archimedean instances of the above theory. I will not be assuming familiarity with any of the abovementioned notions.
Motivated by a recent Diophantine transport problem about how to transport profitably a group of persons or objects, we survey classical facts about solving systems of linear Diophantine equations and inequalities in non-negative integers. We emphasize on the method of Elliott from 1903 and its further development by MacMahon in his 'Ω-Calculus' or Partition Analysis. Then we show how this approach can be used to solve problems in classical and non-commutative invariant theory and theory of algebras with polynomial identities.
We discuss some new lower bounds for the Erdős box problem, the problem of estimating the extremal number of the complete d-partite d-uniform hypergraph with two vertices in each part, thereby improving on work of Gunderson, Rödl and Sidorenko.
I will explain the construction of a functor from the exact symplectic cobordism category to a totally ordered set, which measures the complexity of the contact structure. Those invariants are derived from a bi-Lie infinity formalism of the rational SFT and a partial construction of the rational SFT. In this talk, I will focus on the construction and properties of the functor. Time permitting, I will explain applications, computations, and relations to the involutive bi-Lie infinity formalism of the full SFT. This is joint work with Agustin Moreno.
The global dimension of an associative algebra A over a a field is a measure of the complexity of its representations. It is 0 if A is a matrix algebra. It is 1 if A is a path algebras of quivers without directed cycles. It is infinite if A is the algebra of dual numbers.
I will give a brief introduction to Hochschild homology (1945), in order to explain Han's conjecture (2006): for finite-dimensional algebras, the Hochschild homology should control the finiteness of the global dimension.
Next, I will present some progress made in showing Han's conjecture, using the relative version of Hochschild homology (1956) with respect to a subalgebra B. This theory was little used until recently. Now we have a Jacobi-Zariski long nearly exact sequence which relates the usual and relative versions of Hochschild homology. Its gap to be exact is approximated by a spectral sequence which has Tor functors in its first page, of B-tensor powers of A/B. This tool enables to show, for instance, that the class of algebras verifying Han's conjecture is closed by bounded extensions of algebras.
Let λ be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.
The degree of a finite semigroup S is the minimum n such that S can be faithfully represented by transformations of an n-set. This talk will discuss some recent work calculating degrees of various classes of semigroups, including null semigroups, nilpotent semigroups, rectangular bands and sandwich semigroups. Some interesting combinatorial objects play an important role, including integer compositions and hypergraph colourings.