Algebra section
Invited speakers:
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Contributed talks:
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Programme (all talks are in Abacws/3.02):
Mon 30th March, 2026
| 3:00pm – 4:00pm | Anja Meyer (Manchester) |
| “Cohomology and invariant theory of finite matrix groups” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 4:45pm | Charlotte Llewellyn (Glasgow) |
| “The combinatorics of maximal modifying modules” (abstract) | |
| 4:45pm – 5:00pm | Chun-Yu Bai (Edinburgh) |
| “Derived skein modules” (abstract) | |
| 5:00pm – 6:00pm | Grigory Garkusha (Swansea) |
| “Projective schemes for non-commutative graded algebras with extra symmetries” (abstract) |
Tue 31st March, 2026
| 3:00pm – 4:00pm | Jan Spakula (Southampton) |
| “Shalom's Conjecture and boundary representations” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 5:00pm – 6:00pm | Natasha Blitvic (Queen Mary) |
| “Permutation patterns” (abstract) |
Wed 1st April, 2026
| 3:00pm – 4:00pm | Christian Walmsley Hagendorf (UC Louvain) |
| “Spin chains and totally symmetric alternating sign matrices” (abstract) | |
| 4:00pm – 4:30pm | Coffee break |
| 4:30pm – 4:45pm | Alexander Jackson (Durham) |
| “Polynomial results for \(\mathrm{GL}_n(o)\)” (abstract) | |
| 4:45pm – 5:00pm | Galina Kaleeva |
| “Universal equivalence of linear groups over rings” (abstract) | |
| 5:00pm – 6:00pm | Nelly Villamizar (Swansea) |
| “Koszul homology and the dimension of spline spaces” (abstract) |
Algebra section organisers:
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Abstracts:
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Chun-Yu Bai (Edinburgh): Derived skein modules
Skein modules are central to the study of quantum topology. It's important to study a derived version, which will contain more interesting information. In this talk, we study an axiomatic model for derived skein modules. After explaining motivations, we compute derived skein modules for the groups \(\mathrm{SL}_2\) and \(\mathrm{GL}_1\). Then we give some computable theorems, and finiteness theorem. Finally, we talk about some open questions and conjectures.
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Natasha Blitvic (Queen Mary): Permutation patterns
In the area of combinatorics known as ‘permutation patterns’, seemingly innocuous questions can conceal a surprising degree of difficulty, giving rise to combinatorial problems that range from trivial to unsolved (despite decades of work). In this talk, we will explore several ways in which permutation pattern questions interface with probability – sometimes giving rise to probabilistic processes, sometimes illuminated by probabilistic reasoning. Some of these will be generalizations of known processes, others new, and still others conjectural. The proof techniques we employ build on known connections between consecutive permutation patterns and posets. In particular, we examine posets arising from packings of consecutive permutation patterns and show that they lead to natural classification problems while also yielding new enumerative results. These results appear in several forms, including explicit integer sequences, generating functions, integrals and matrix products. This is based on forthcoming joint work with Slim Kammoun and Einar Steingrimsson.
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Grigory Garkusha (Swansea): Projective schemes for non-commutative graded algebras with extra symmetries
One of the key problems in constructing schemes for non-commutative rings is that often in non-commutative ring theory there basically are not enough prime ideals. However, the situation changes for non-commutative graded algebras with extra symmetries like, for example, the tensor algebra of a vector space. We introduce and study symmetric projective schemes associated to such algebras as well as their categories of symmetric quasi-coherent sheaves. The latter category is defined in terms of graded representations of the symmetric groups. Symmetric projective schemes have the same topological properties as classical ones and their categories of symmetric quasi-coherent sheaves are closed symmetric monoidal Grothendieck with invertible generators. We also prove that classical projective schemes (resp. the classical category of quasi-coherent sheaves on a projective scheme) are recovered out of symmetric projective schemes (resp. out of symmetric quasi-coherent sheaves). It is worth mentioning that main ideas and methods here originate in stable homotopy theory.
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Christian Walmsley Hagendorf (UC Louvain): Spin chains and totally symmetric alternating sign matrices
Alternating sign matrices are central objects in enumerative combinatorics and closely related to the six-vertex model of statistical mechanics. Totally symmetric alternating sign matrices (TSASMs) form a highly constrained symmetry class whose enumeration is only partially understood. In this talk, I describe a connection between TSASM enumeration and a distinguished eigenvector of an integrable lattice model called the open XXZ spin chain. This eigenvector arises from a Laurent-polynomial solution of the boundary quantum Knizhnik-Zamolodchikov (bqKZ) equations. It leads to a multivariate Laurent-polynomial generalisation of the sum of the eigenvector's entries, which is uniquely characterised by a set of algebraic properties. We show that a suitable partition function of a six-vertex model associated with TSASMs satisfies the same properties. By unique characterisation, this partition function coincides with the generalised sum of the eigenvector's entries. As a consequence, TSASM enumeration can be studied using the bqKZ solution. In particular, this connection yields a multiple contour-integral formula for the number of TSASMs of arbitrary order.
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Alexander Jackson (Durham): Polynomial results for \(\mathrm{GL}_n(o)\)
The smooth representations of \(\mathrm{GL}_n(o)\), where \(o\) is the ring of integers of a non-Archimedean local field, have been studied extensively in connection with the local Langlands correspondence for \(\mathrm{GL}_n\). In relation to these groups, Onn (2008) studied automorphism groups of finite \(o\)-modules. For this class of groups, Onn conjectured that the dimensions of the representations, and the number of representations of each dimension, are given by finitely many polynomials in the residue cardinality \(|o/p|\). I prove this conjecture in a special case by direct computation. Using tools from algebraic geometry, I also prove an alternative version of the polynomial property for \(G(o)\), where \(G\) is a smooth affine group scheme over \(\mathbb{Z}\).
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Galina Kaleeva: Universal equivalence of linear groups over rings
Two algebraic structures are said to be universally equivalent if they have the same universal theory (the set of all universal sentences, i.e., sentences whose prenex normal form contains only universal quantifiers). A classical question about linear groups is whether two groups \(\mathrm{GL}_n(K)\) and \(\mathrm{GL}_m(L)\) are isomorphic (or, more generally, equivalent in some sense) if and only if \(n=m\) and the rings \(K\) and \(L\) are isomorphic (equivalent in the same sense). In general, the answer is no, even for isomorphism. In this talk, we discuss the analogous question for universal equivalence of general linear groups over (possibly noncommutative) local rings.
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Charlotte Llewellyn (Glasgow): The combinatorics of maximal modifying modules
Maximal modification algebras (MMAs) serve as natural analogues of minimal models in algebraic geometry. To study the combinatorics of these structures, we examine their associated maximal modifying (MM) modules, whose interactions are governed by a type of categorical mutation. In this talk, we give an overview of this mutation and highlight some applications of this technology, including the first construction of perverse schobers associated to \(cA_2\) singularities.
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Anja Meyer (Manchester): Cohomology and invariant theory of finite matrix groups
The invariant theory and the cohomology of a group are closely linked. For a finite group \(G\), \(H^0(G,k)\) with \(k\) a field is the invariant module \(k^G\). This fact helps with the computation of group cohomology via the Lyndon-Hochschild-Serre spectral sequence. Furthermore, invariant theory plays an important role in group cohomology computation via the stable elements method. In this talk, we look at both cases and focus on \(H^*(\mathrm{SL}_2(\mathbb{Z}/3^n),\mathbb{F}_3)\) for \(n>1\) as a worked example. All concepts, such as spectral sequences and stable elements, will be introduced in the talk.
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Jan Spakula (Southampton): Shalom's Conjecture and boundary representations
Shalom's Conjecture asserts that any Gromov hyperbolic group admits a uniformly bounded representation with a proper cocycle. This is one of the more relaxed versions of Gromov's a-T-menability (a.k.a. the Haagerup property), which for a given group, asks for an existence of a proper action by isometries on a Hilbert space. I will explain the statements, context, and current status. This is based on joint work with K. Boucher.
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Nelly Villamizar (Swansea): Koszul homology and the dimension of spline spaces
Splines are piecewise polynomial functions defined over a polyhedral complex satisfying prescribed smoothness conditions across codimension-one faces. Their study leads naturally to problems in commutative and homological algebra, and computing the Hilbert function of spline spaces has been a central challenge since the foundational work of Billera and Schenck–Stillman. In this talk we focus on spline spaces associated to fans arising from central hyperplane arrangements in three-dimensional space, where the smoothness is constant along hyperplanes. Our main result establishes that, for generic hyperplane arrangements (no three hyperplanes meeting along a common line), the dimension of the spline space in each degree can be expressed in terms of the dimensions of the zeroth and first Koszul homology modules of the sequence of powers of linear forms defining the arrangement, where the exponents are determined by the smoothness orders. This connection unlocks several new results. Using bounds on the Castelnuovo–Mumford regularity of the ideal generated by these powers of linear forms, we show that for sufficiently large degree—and for any number of hyperplanes—the dimension of the spline space is given by an explicit combinatorial formula depending only on the number of hyperplanes and the smoothness orders. For arrangements of up to five hyperplanes, we obtain a complete formula valid in all degrees, which is moreover a combinatorial invariant. In contrast, we show that for six or more hyperplanes, the Hilbert function of the spline space genuinely depends on the geometry of the arrangement. This is joint work with Carles Checa, Michael DiPasquale, Pablo Mazón, Thái Thành Nguyen, Liana Sega, Prajwal Udanshive, and Adam Van Tuyl.
- BehrendR at cardiff dot ac dot uk
Contact:
All further questions and inquiries can be addressed to Roger Behrend at: