Tag - Group theory

Ian Tan: Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

30th September, 2024

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(ℂ) × SL2(ℂ) → SO4(ℂ), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

Hadi Salmasian: Lie Groups and Quantization

4th September, 2024

This is a 23-lecture course, with each lecture being around 80 minutes long, given by Hadi Salmasian.

The goal of the course is to first cover the foundational theory of Lie groups and then move on to more advanced topics that expose the audience to areas of active research. The following is the list of topics that are intended to be covered:

  • Foundational theory of Lie groups: Lie groups, the exponential map, Lie correspondence. Homomorphisms and coverings. Closed subgroups. Classical groups: Cartan subgroups, fundamental groups. Manifolds. Homogeneous spaces. General Lie groups.
  • Introduction to quantization: Symplectic manifolds, pre-quantization, the orbit method. Poisson manifolds, Manin triples. Universal enveloping algebras, quantum sl(2) and its representations, quantum symmetric spaces.

Jonas Deré: Simply transitive NIL-affine actions of soluble Lie groups

26th August, 2024

Although not every 1-connected soluble Lie group G admits a simply transitive action via affine maps on ℝn, it is known that such an action exists if one replaces ℝn by a suitable nilpotent Lie group H, depending on G. However, not much is known about which pairs of Lie groups (G,H) admit such an action, where ideally you only need information about the Lie algebras corresponding to G and H. In recent work with Marcos Origlia, we show that every simply transitive action induces a post-Lie algebra structure on the corresponding Lie algebras. Moreover, if H has nilpotency class 2 we characterize the post-Lie algebra structures coming from such an action by giving a new definition of completeness, extending the known cases where G is nilpotent or H is abelian.

Susan Hermiller: Subgroups of the group of dyadic piecewise linear homeomorphisms of the real line

3rd May, 2024

The group of dyadic orientation-preserving piecewise linear (PL) homeomorphisms of the unit interval is called Thompson's group F, and the question of which groups are - or cannot be - subgroups of F has yielded many interesting results. In this talk I'll discuss the question of what groups can or cannot be subgroups of Aut(F) (the automorphism group of F), and more particularly subgroups of an index 2 subgroup of Aut(F) that is isomorphic to a group of dyadic PL homeomorphisms of the real line.

Stuart White: Simple amenable C*-algebras

27th March, 2024

I'll give an overview of recent progress in the structure and classification of simple amenable C*-algebras, making parallels to the Connes-Haagerup classification of amenable von Neumann algebras and drawing examples from group actions.

Artem Chernikov: Recognizing groups in Erdős geometry and model theory

15th March, 2024

Erdős-style geometry is concerned with difficult questions about simple geometric objects, such as counting incidences between finite sets of points, lines, etc. These questions can be viewed as asking for the possible number of intersections of a given algebraic variety with large finite grids of points. An influential theorem of Elekes and Szabó indicates that such intersections have maximal size only for varieties that are closely connected to algebraic groups. Techniques from model theory - variants of Hrushovski’s group configuration and of Zilber’s trichotomy principle - are very useful in recognizing these groups, and led to far reaching generalizations of Elekes-Szabó in the last decade. I will overview some of the recent developments in this area, in particular explaining how all of this is not just about polynomials and works for definable sets in o-minimal structures.

Laura Ciobanu: Group equations, constraints and decidability

8th March, 2024

In this talk I will discuss group equations with non-rational constraints, a topic inspired by the long line of work on word equations with length constraints. Deciding algorithmically whether a word equation has solutions satisfying linear length constraints is a major open question, with deep theoretical and practical implications. I will introduce equations in groups and several kinds of constraints, and show that equations with length, abelian or context-free constraints are decidable in virtually abelian groups (joint with Alex Evetts and Alex Levine). This contrasts the fact that solving equations with abelian constraints is undecidable for non-abelian right-angled Artin groups and hyperbolic groups with ‘large’ abelianisation (joint work with Albert Garreta).

Antonio Viruel: Permutation representations of finite groups via evolution algebras

18th December, 2023

In the wake of the influential work by Elduque-Labra, it is known that every finite-dimensional evolution K-algebra X such that X2 = X, namely X is idempotent, has a finite group of automorphisms. Building on this foundation, works of Costoya et al. show that given any finite group G, there exists an idempotent finite-dimensional evolution algebra X such that Aut(X) ≅ G. Moreover, when the base field is sufficiently large in comparison to the group G, such an X can be selected to be simple. As a result, Sriwongsa-Zou propose that idempotent finite-dimensional evolution algebras can be classified based on the isomorphism type of their group of automorphisms and dimension. Within this context, we establish that the natural representation of highly transitive groups cannot be realized as the complete group of automorphisms of an idempotent finite-dimensional evolution algebra. For instance, for any sufficiently large integer n, there exists no evolution algebra X such that X2 = X, dim X = n, and Aut(X) is isomorphic to the alternating group An. However, we demonstrate that for any (not necessarily faithful) permutation representation ρ : G → Sn and any field K, there exists a finite-dimensional evolution K-algebra X such that X2 = X, Aut(X) ≅ G and the induced representation given by the Aut(X)-action on the natural idempotents of X is ρ.

Markus Lohrey: Streaming word problems

8th December, 2023

We are interested in highly efficient algorithms for word problems of groups: the algorithm should read the input word once from left to right symbol by symbol (such algorithms are known as streaming algorithms), spending ideally only constant time for each input letter. Moreover, the space used by the algorithm should be small, e.g. O(log n) if n is the length of the input word. To achieve these goals we need randomization: the algorithm is allowed to make random guesses and at the end it gives a correct answer (is the input word trivial in the underlying group?) with high probability. We show that for a large class of groups such algorithms exist, where in particular the space complexity is bounded by O(log n). These groups are obtained by starting with finitely generated linear groups and closing up under the following operations: finite extensions, graph products, and wreath products where the left factor is f.g. abelian. We also contrast this result with lower bounds. For instance, for Thompson’s group F every randomized streaming algorithm for the word problem for F has space complexity Ω(n) (n is again the length of the input word).