Let SO3(ℝ) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure μ. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO3(ℝ) with sufficiently small measure, then μ(A2) > 3.99 μ(A).
Étale algebraic groups over a field k are equivalent to finite groups with a continuous action of the absolute Galois group of k. The difference version of this well-known result asserts that étale difference algebraic groups over a difference field k (i.e., a field equipped with an endomorphism) are equivalent to profinite groups equipped with an expansive endomorphism and a certain compatible difference Galois action. In any case, understanding the structure of expansive endomorphisms of profinite groups seems a worthwhile endeavour and that's what this talk is about.
Difference algebraic groups are a generalization of algebraic groups. Instead of just algebraic equations, one allows difference algebraic equations as the defining equations. Here one can think of a difference equation as a discrete version of a differential equation. Besides their intrinsic beauty, one of the main motivations for studying difference algebraic groups is that they occur as Galois groups in certain Galois theories. This talk will be an introduction to difference algebraic groups.
Modular fusion categories (MFCs) arise naturally in many areas of mathematics and physics. Associated with an MFC is a pair of complex matrices, called modular data, which are arguably the most important invariants of an MFC. The modular data of an MFC generate some uncanonical congruence representations of SL2(ℤ). In this talk, we will discuss how modular data could be reconstructed or discovered from congruence representations of SL2(ℤ). The talk is based on a joint work with Eric Rowell, Zhenghan Wang and Xiao-Gang Wen.
Rota-Baxter operators on Lie algebras were first studied by Belavin, Drinfeld and Semenov-Tian-Shansky as operator forms of the classical Yang-Baxter equation. Integrating the Rota-Baxter operators on Lie algebras, we introduce the notion of Rota-Baxter operators on Lie groups and more generally on groups. Then the factorization theorem can be achieved directly on groups. We introduce the notion of post-Lie groups, whose differentiations are post-Lie algebras. A Rota-Baxter operator on a group naturally induces a post-group. Post-groups are also closely related to operads, braces, Lie-Butcher groups and various structures.
K. Brown introduced in 1975 the p-subgroup complex of a finite group G. It is the simplicial complex whose vertices are the nontrivial p-subgroups of G, where a collection of subgroups spans a simplex if it is a chain. This complex was further studied by Quillen, who observed that for a finite group of Lie type G with defining characteristic p, this complex is homotopy equivalent to the building of G. He also conjectured that the p-subgroup complex is contractible if and only if G contains a nontrivial normal p-subgroup and proved his conjecture for solvable groups. The Quillen conjecture remains open but was proved for almost simple groups by Aschbacher and Kleidman, and strong reduction theorem was obtained by Aschbacher and Smith.
The group G acts on its p-subgroups by conjugation and hence acts simplicially on the p-subgroup complex. Webb conjectured in 1987 that the orbit space of the p-subgroup complex is always contractible. He proved that its mod-p homology vanishes using methods from group cohomology. Webb's conjecture was first proved by Symonds in 1998, and a number of other proofs have since appeared. All the proofs I am aware of go through Robinson's subcomplex, which is G-homotopy equivalent to Brown's. None of the proofs are explicit. Symonds computes the fundamental group and integral homology and uses the Hurewicz and Whitehead theorems. Bux gave an inductive proof using a variant of Bestvina-Brady style discrete Morse theory. In this talk, I will use Brown's theory of collapsing schemes to give an explicit sequence of elementary collapses that collapses the orbit space of Robinson's subcomplex to the vertex corresponding to the conjugacy class of Sylow p-subgroups.
In the 1960s Higman was able to characterize the finitely generated subgroups of finitely presented groups, that is, groups defined using a finite set of generators and finite set of defining relations. His result, which is called the Higman Embedding Theorem, is a key result in combinatorial group theory which makes precise the connection between group presentations and logic. In this talk I will present a result of a similar flavour, proved in recent joint work with Mark Kambites (Manchester), in which we characterise the groups of units of inverse monoids defined by presentation where all the defining relators are of the form w=1. I will explain what an inverse monoid is, the motivation for studying this class of inverse monoids, and also outline some of the geometric ideas that we developed in order to prove our results.
The theory of condensed sets, developed by Dustin Clausen and Peter Scholze, provides a framework well-suited to study algebraic objects that carry a topology. In my talk, I will discuss the basic properties of the cohomology of condensed groups and its relation to continuous group cohomology. Johannes Anschütz and Arthur-César le Bras showed that for locally profinite groups and solid (e.g. discrete) coefficients, condensed group cohomology agrees with continuous group cohomology. On the other hand, if G is a locally compact and locally contractible topological group (e.g., a Lie group), and M is a discrete group with trivial G-action, then the condensed group cohomology of G, M (the sheaves of continuous functions into G and M) is isomorphic to the singular cohomology of the classifying space of G with coefficients in M, whereas the continuous group cohomology of G with coefficients in M is isomorphic to the singular cohomology of the classifying space of π0(G) with coefficients in M.
Generalizing results of Johannes Anschütz and Arthur-César le Bras on locally profinite groups, I will explain that continuous group cohomology with solid coefficients can be described as a cohomological δ-functor in the condensed setting for a large class of topological groups.
Recently, Nies, Segal and Tent started an investigation of finite axiomatizability in the realm of profinite groups. Among the classes of profinite groups under their consideration is the class of p-adic analytic pro-p groups. In joint work with Benjamin Klopsch, we consider two key invariants of these groups, namely rank and dimension, and show that they can be characterized by a single first-order sentence. Before discussing these results I will introduce the relevant background. If time permits, I will also present some natural generalisations.
Given a group G and two vectors v and w in a representation V, one can ask: do v and w lie in the same orbit? This is what we call the orbit problem. For the action of GLn on m-tuples of n × n matrices, the orbit problem translates to the module isomorphism problem for which there is a known polynomial time algorithm. In joint work with Visu Makam, we also gave a polynomial time algorithm for deciding whether the orbit closures of v and w intersect. We also have similar results for the left-right action of SLn × SLn on the space of m-tuples of matrices. These results are related to interesting problems in algebraic complexity theory, such as non-commutative rational identity testing. The graph isomorphism problem can also formulated as an orbit problem. No polynomial time algorithm for the graph isomorphism problem is known. I will discuss an algorithm for the graph isomorphism problem that is more powerful than the higher-dimensional Weisfeiler-Leman algorithm when it comes to distinguishing pairs of non-isomorphic graphs in polynomial time. (This algorithm could potentially be polynomial time, but we will not make such a bold claim.)
We show that the celebrated Friedlander-Suslin theorem - on finite generation of cohomology of a finite group scheme G over a field - remains valid for a finite flat group scheme G over a commutative noetherian ring. In view of earlier work it suffices to put a uniform bound, depending on G only, on torsion in cohomology of G-modules.
The intermediate symplectic characters, introduced by R. Proctor, interpolate between Schur functions and symplectic characters. They arise as the characters of indecomposable representations of the intermediate symplectic group, which is defined as the group of linear transformations fixing a (not necessarily non-degenerate) skew-symmetric bilinear form. In this talk, we present Jacobi-Trudi-type determinant formulas and bialternant formulas for intermediate symplectic characters. By using the bialternant formula, we can derive factorization formulas for sums of intermediate symplectic characters, which allow us to give a proof and variations of Hopkins' conjecture on the number of shifted plane partitions of double-staircase shape with bounded entries.
Upper triangular matrices with ones on the diagonal and entries which are integers (or algebraic integers) arise in many contexts, e.g. as Stokes matrices in the theory of meromorphic connections with irregular poles, in many situations in algebraic geometry (often related to Stokes matrices), especially in quantum cohomology and the theory of isolated hypersurface singularities, but also in the theory of Coxeter groups.
Concepts from singularity theory like vanishing cycles, monodromy groups, Seifert forms, tuples of (pseudo-)reflections and distinguished bases can be derived from upper triangular
matrices in cases beyond singularity theory and are interesting to study.
Additionally, always braid group actions on the matrices and on the distinguished bases are in the background. They give rise to certain covering spaces of the classifying space of the braid group. These are interesting natural global manifolds. Some are well known, others are new.
The talk presents concepts and old and new results. It puts emphasis on some cases from singularity theory and some 3x3 cases.
We prove a variant of Quillen's stratification theorem in equivariant homotopy theory for a finite group working with arbitrary commutative equivariant ring spectra as coefficients, and suitably categorifying it. We then apply our methods to the case of Borel-equivariant Lubin-Tate E-theory. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing tensor-ideals of the category of equivariant modules over Lubin-Tate theory, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.
