Incidence homology for the hyperoctahedral group

The incidence structure of the cross-polytope gives rise to certain modular representations for the hyperoctahedral group. In this thesis we introduce and begin the study of these natural representations. In particular we show that they satisfy a branching rule. This branching rule is used to extract information about the representations and underlying combinatorial objects. Amongst the information extracted is a formula for the dimensions of the representations. This has applications in calculating the p-rank of incidence matrices arising from the cross-polytope. We also construct explicit generators for the representations and identify cases where the representations are irreducible

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Enveloping Algebras and Geometric Representation Theory (hybrid meeting)

The workshop brought together experts investigating algebraic Lie theory from the geometric and categorical viewpoints

Hopf ring structures on the cohomology of certain spaces

[from the introduction]: The structure of this work is as follows. In the first chapter, we recall the preliminary notions and results we will make use of, in particular, some facts regarding Coxeter groups and their classifying spaces, results for the homology of 1-loop spaces and Hopf rings. The second chapter is devoted to the exposition of the results involving the cohomology of the symmetric groups. That chapter basically follows the treatment of two published papers, one by Giusti, Salvatore and Sinha, and the other by the author of this thesis. In the third chapter, we calculate the cohomology of the Coxeter groups of Type Bn and Dn as (almost-)Hopf rings. We also carry on the calculation of the restriction to elementary abelian subgroups and of the Steenrod algebra action. Finally, the fourth chapter deals with some results concerning the mod p cohomology of D(X) and Q(X) for a topological space X

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Uniform Symbolic Topologies in Non-Regular Rings

When does a Noetherian commutative ring R have uniform symbolic topologies (USTP) on primes -- read, when does there exist an integer D>0 such that the symbolic power P^{(Dr)} lies in P^r for all prime ideals P in R and all r >0? Groundbreaking work of Ein -- Lazarsfeld -- Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. Their work shows that there exists a D depending only on the Krull dimension: in other words, the exact same D works for all regular rings as stated of a fixed dimension. Referring to this last observation, we say in the thesis that the class of excellent regular rings enjoys class solidarity relative to the uniform symbolic topology property (USTP class solidarity), a strong form of uniformity. In contrast, this thesis shows that for certain classes of non-regular rings including rational surface singularities and select normal toric rings, a uniform bound D does exist but depends on the ring, not just its dimension. In particular, for rational double point surface singularities over the field C of complex numbers, we show that USTP solidarity is plainly impossible. It is natural to sleuth for analogues of the Improved Ein -- Lazarsfeld -- Smith Theorem where the ring R is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This thesis lies in the overlap of these research directions, working with Noetherian domains.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/149907/1/robmarsw_1.pd

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic