Abstract: In this talk, we will consider the properties of finite groups that are preserved under isomorphism, which we call group invariants. The richness of finite group theory is reflected in the zoo of invariants that have been defined and studied. These invariants typically stem from the abstract group structure or the representation theory. We will discuss how group invariants can shed light on the classification of finite simple groups. This talk aims to be accessible and appealing to a broad mathematical audience.

Abstract: We will discuss some results on pro-$p$ groups, with a focus on cases in which pro-$p$ groups behave like Lie groups in some sense. In particular, we will discuss the notion of an analytic structure on certain $p$-pro groups, and ways that this structure is analogous to the analytic structure on Lie groups. This talk will cover no original work, but rather focus on presenting work of Lazard, Serre, and others.

Abstract: Let $M$ be the Euclidean or hyperbolic $n$-space with a regular tessellation. By a finitary rearrangement of $M$ (with respect to decompositions of $M$ into finitely many pieces of a pre-specified shape) we mean the process of cutting $M$ along tile-boundaries into finitely many tessellated pieces $X_i$ each of which is either a single tile or isometric to one of the specified infinite shapes, and then mapping each piece by a tessellation-respecting isometric embedding $f_i : X_i \to M$, to a new position, with the property that the images $f_i(X_i)$ cover $M$ and have pairwise disjoint interiors. The union of the maps $f_i$ is well defined on all interior points of the tiles and induces a permutation of the set $\Omega$ of all tile-centers which we call a finitary piecewise isometric tile-permutation of $M$.

It is interesting to replace $M$ by the abstract planar tree $T$ with all its vertices of degree $3$, view it as tessellated by its edges, and consider a corresponding construction. For if one restricts attention to finitary rearrangements of $T$ with respect to decompositions of $T$ whose infinite pieces specified to be rooted dyadic trees (with root of degree $2$ and all other vertices of degree $3$), then one recovers Richard Thompson's key tool to the structure of his groups: It exhibits the the elements of Thompson's group – up to finite permutations – as finitary piecewise planar-tree-isometric edge-permutations of $T$.

Moreover, the tree $T$ occurs as the dual of the tessellation of the hyperbolic plane $H^2$ given by an ideal triangle and its iterated reflections over the sides; and the close connection between the isometries of $H^2$ and the planar-tree-isomorphisms of $T$ allows one to show that Thompson's groups can also be interpreted as groups of piecewise hyperbolic-isometric tile-permutations of $H^2$.

The observation that Thompson's groups are part of the game could nurture hopes that other basic regular tessellations might lead to similar gem-stones. The Euclidean space $E^n$ with its standard unit-cube tessellation is certainly the most natural down-to-earth case; and it is also natural to handle it by decomposing $E^n$ into finitely many orthants: i.e. intersections of tessellated half-spaces. In this situation it is convenient to describe the set of all tile centers directly, identifying them with the lattice $\Omega = Z^n$. The decomposition of $E^n$ into orthants thus decomposes $Z^n$ into a finite pairwise disjoint union of single points, discrete rays, discrete quadrants, … , discrete rank-$n$ orthants; and the elements of our piecewise Euclidean-isometric permutation group $G = pei(Z^n)$ have to rearrange them to a new disjoint-union-decomposition of $Z^n$. On the face of it this looks rather complicated, but those who nurture hope for gem-stones should be prepared to dig deep.

However, it turned out to be doable. In my talk I will indicate how a $G$-equivariant germs-of-orthants structure at infinity of $Z^n$ with a corank-$1$ flow helps to get information on the group structure of $G$ to show:

(1) The group of all piecewise Euclidean isometric permutations of $Z^n$ is the fundamental group of a CW-complex with finite $(2n – 1)$-skeleton. This is joint work with Heike Sach, ArXiv:2016. (Recall that Thompson's groups are of type $F_\infty$).

(2) For each $0\leq k\leq n$ the subgroup $G_k$ consisting of all elements of $G$ supported on the $k$-skeleton of $E^n$ is normal in $G$. In my pandemic related sabbatical from BU I added: Every normal subgroup $N$ of rank $k$ in $G_k$ contains the unique subgroup of index $2$ of $G_{k-1}$. See J. London Math. Soc. 2022. (Recall that Thompson's groups $V$ and $T$ are simple and every normal subgroup of $F$ contains $F'$).

Abstract: The solubility graph associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. Properties of this graph can have dramatic consequences on the structure of $G$. For example, it has recently been proved that if some non-identity element has prime degree then $G$ is an abelian simple group. So in this talk, we focus our attention on the set of neighbors of a vertex $x$ which we call it the solubilizer of $x$ in $G$, $Sol_G(x)$. We investigate both arithmetic and structural properties of this set and discuss how restrictions on the structure of this set affect the structure of $G$.

Abstract: This is an expository introduction to fusion algebras for affine Kac-Moody algebras, with major focus on the algorithmic aspects of their computation and the relationship with tensor product decompositions. Explicit examples are included illustrating the rank 2 cases. Previous work of the author and collaborators on a different approach to fusion rules from elementary group theory is also explained.

Abstract: Not much is known about decomposition numbers of spin representations of symmetric groups. For example it is not even known whether in general the decomposition matrices are triangular. I will show how certain parts of the decomposition matrices look like and in particular focus on rows corresponding to spin representations labeled by partitions with at most 2 parts.

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Abstract: We introduce a generalization of $K$-$k$-Schur functions and $k$-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule are described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases promise to be detailed in the future.

Abstract: One method of studying a local commutative ring is to study its category of finitely generated modules. An invariant of particular interest is a module’s Betti sequence. This sequence contains various information about the module, but in particular it measures the growth of the minimal free resolution of the module. Studying such sequences can also provide information about the ring. In particular, it is known that if every finitely generated module has a minimal free resolution that grows on the order of a polynomial, then the ring must be a complete intersection. One can say much more in this case; each Betti sequence will be governed by a quasipolynomial of period 2. We will explore such sequences, and establish a bound on the discrepancy between the polynomials which govern their even and odd terms. The bound is computed using an invariant of the ring called its quadratic codimension. (Joint work with Lucho Avramov and Mark Walker.)