Prime graphs of solvable groups

This project is at the intersection of finite group theory and graph theory and will be supervised by Thomas Keller. There are several graphs associated with finite groups which are currently of interest in the field. The subject of this project are graphs related to the element orders of a finite group. The best known such graph is the prime graph Γ(G) of the group G and is defined as follows. The vertices are the prime numbers dividing the order of the group, and two such primes are linked by an edge if and only if their product divides the order of an element of the group. For solvable groups it is known that Γ(G) has at most two connected components, and the structure of the group in case of two components is well understood. Moreover, for solvable groups a purely graph theoretical description of the prime graphs is known. This has recently been extended to larger classes of groups. There are two main types of questions which are commonly studied in this context. First, how do restrictions on the graph influence the group structure? For example, groups whose prime graph is a tree have been classified. Second, how do restrictions on the group influence the prime graph? In this project, Keller will lead students to study some interesting questions of this type of various levels of difficulty. Of particular interest would be to obtain classifications of the prime graphs of some more classes of groups.

Depending on the ability and interest of the students, they can choose whether they prefer to work mostly on the group theoretical end or on the graph theory side of things. It is assumed that students have had at least one semester of Modern Algebra or Abstract Algebra. Explanations and background of additional topics, such as solvable groups and Frobenius groups, will be provided. The students will also have an opportunity to learn the computer algebra system GAP, which is useful in examining specific examples of groups that may be useful in making or confirming conjectures.

On Gluck’s Conjecture

Motivated by a theorem of Jordan, one can reasonably ask whether there is an abelian subgroup of index bounded in terms of the largest degree of the complex irreducible characters. An affirmative answer to this question for nilpotent groups was given many years ago by Isaacs and Passman. Let G be a finite group, and denote by b(G) the largest degree of an irreducible character of G. Isaacs and Passman showed that if G is nilpotent, then G has an abelian subgroup of index at most b(G)^4. Later, Gluck proved that in all finite groups, the index of the Fitting subgroup F(G) in G is bounded by a polynomial function of b(G). For solvable groups, Gluck further showed that |G : F(G)| <= b(G)^(13/2) and conjectured that |G : F(G)| <= b(G)^2. Many research has been done by studying the orbit structure of linear group actions. In this project, we plan to push the limit of the current technique.

Hypothesis testing based on deep ReLU networks

Deep ReLU Neural networks (DNN) play a central role in modern Artificial intelligence (AI) and have been successfully used in areas such as natural language processing and image recognition. While most applications focus on prediction and classification, there is an increasing interest in studying statistical inference of neural networks. The study of DNN statistical inference can enhance our understanding of DNN. Moreover, it can facilitate the DNN-based hypothesis testing so that DNNs can be applied to hypothesis-driven clinical and biomedical research. Two main regimes will be considered to construct test statistics based on DNNs. One is the under-parametrized regime, where the number of parameters in a DNN is smaller than the sample size, but they both increase to infinity. The other is the modern over-parametrized regime, where the number of parameters in a DNN is much larger than the sample size. In the past year, the mentor has been working with an undergraduate research assistant and has figured out a sufficient condition for a DNN to achieve weak consistency in the under-parametrized regime. This result will serve as the theoretical background in this project to construct a goodness-of-fit test statistic and a quasi-likelihood ratio test statistic, as well as establishing their asymptotic distributions in the under-parametrized regime. Depending on the ability and interest of the students, the mentor will provide opportunities for students to explore conditions so that similar test statistics can be constructed in the over-parametrized regime.

Chip firing on signed graphs and beyond

Chip-firing is a game played on a graph G: a number of chips are placed at each vertex of G, and when a vertex “fires” it gives one chip to each of its neighbors. This simple game has surprising depth and has become an important area of study in combinatorics and related fields. It has beautiful connections to algebraic geometry, group theory, representation theory, and various other fields of mathematics, as well as to physics and biology. Our aim in this project is to investigate and generalize the known constructions and results for chip-firing on graphs to a larger class of objects. In 2022 we investigated chip-firing on “signed graphs” using a template called chip-firing pairs provided by Klivans-Guzman. In 2023 we studied chip-firing on “hypergraphs”, utilizing the fact that such objects can be encoded as bipartite vertex-edge incident graphs. In 2024 we were able to extend the duality between critical and superstable configurations to integral chip-firing pairs. We plan to continue this investigation, starting with extending the duality to non-integral M-matrices and again use this to extend to non-integral chip-firing pairs. We are also going to investigate other related questions, such as within the signed graph model, whether we can describe the biggest common subgroup among all critical groups that can appear given an underlying graph.