Joel Friedman


Research Classification

Research Interests

Algebraic Graph Theory
Computer Science Theory

Relevant Thesis-Based Degree Programs

Affiliations to Research Centres, Institutes & Clusters



Master's students
Doctoral students
Postdoctoral Fellows
Any time / year round

Computer Science Theory, Graph Theory, Combinatorics, Algebraic Topology applied to Combinatorics

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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Riemann functions, their weights, and modeling Riemann-Roch formulas as Euler characteristics (2022)

In this thesis, we discuss modelling with (virtual) k-diagrams a class of functions from ℤⁿ to ℤ, which we call Riemann functions, that generalize the graph Riemann-Roch rank functions of Baker and Norine. The graph Riemann-Roch theorem has seen significant activity in recent years, however, as of yet, there is not a satisfactory homological interpretation of this theorem. This may be viewed as disappointing given the rich homological theory contained in the classical Riemann-Roch theorem. For a graph Riemann-Roch function, we will be able to express the associated graph Riemann-Roch formula as a (virtual) Euler characteristic of the modelling (virtual) k-diagram. From the development of this approach, we obtain a new formula for computing the graph Riemann-Roch rank of the complete graph Kn.

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Abelian Girth and Gapped Sheaves (2016)

The girth of a graph is the length of the shortest cycle in a graph, and the abelian girth of a graph is the girth of the graph's universal abelian covering graph. We denote the abelian girth of a graph G as Abl(G) and show that for d-regular graphs on n vertices with d constant and n growing we have Abl(G) ≤ 6 log_{d-1} n plus a vanishing term. This can be seen as a version of the Moore bound for abelian girth. We also prove Girth(G) ≤ Abl(G)/3, which implies that any multiplicative improvement to the abelian girth Moore bound would also improve the standard Moore bound. Sheaves on graphs and two of their homological invariants, the maximum excess and the first twisted Betti number, were used in the proof of the Hanna Neumann Conjecture from algebra and may be of use in proving several related unresolved conjectures. These conjectures can be proven if certain sheaves called ρ-kernels have vanishing maximum excess. Ungapped sheaves have maximum excess equal to the first twisted Betti number, and it is easy to compute the maximum excess of a given sheaf in the case that the sheaf is not gapped. For general sheaves though, there is no known way of computing the maximum excess in polynomial time. We give several conditions that gapped sheaves must satisfy. These conditions include that a gapped sheaf must have edge dimension at least as large as the abelian girth of the underlying graph. The ρ-kernels are subsheaves of constant sheaves. We prove that gapped subsheaves of constant sheaves exist, implying that finding maximum excess of some ρ-kernels may be computationally difficult.

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Alon's second eigenvalue conjecture: Simplified and generalized (2013)

For a fixed graph, B, we study a probability model of random covering maps of degree n. Specifically, we study spectral properties of new eigenvalues of the adjacency matrix of a random covering, and its Hashimoto matrix (i.e., the adjacency matrix of the associated directed line graph).Our main theorem says that if B is d-regular, then for every positive epsilon, the probability that a random covering has a new adjacency eigenvalue greater than 2(d-1)^(1/2) + epsilon tends to zero as n tends to infinity. This matches the generalized Alon-Boppana lower bound.For general base graphs, B, Friedman conjectured in that the new eigenvalue bound holds with 2(d-1)^(1/2) replaced with the spectral radius of the universal cover of B. We refer to this conjecture as the generalized Alon conjecture; Alon stated this conjecture in the case where B has one vertex, i.e., the case of random d-regular graphs on n vertices. However, for some non-regular base graphs B, we cannot yet prove any non-trivial new eigenvalue upper bound with high probability.We use trace methods, as pioneered by Broder and Shamir for random, d-regular graphs; these methods were eventually refined by Friedman to prove the original Alon conjecture, i.e., in the case where B has one vertex. Our methods involve several significant simplifications of the methods of Friedman.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Variations on sparsifier constructions (2023)

We studied Joshua Batson, Daniel A. Spielman, andNikhil Srivastava's Twice-Ramanujan Sparsifiers. On this basis, we try to generalize their potential function. Inspired by the idea of non-backtracking matrices, we designed the new potential formula that puts all eigenvalues on the complex plane. Following the construction of the twice Ramanujan sparsifier, we conclude some obstacles must be encountered to improve their bound.

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Linear information theory and its application to the coded caching problem (2022)

We investigate the peculiar properties of information theory when all random variables involved are linear functions of a given source endowed with the structure of a Z/2Z-vector space.We describe the coded caching problem and review the progress made in particular cases of the problem.We devise a new caching scheme for a special instance of the problem and extend some approaches in the coded caching literature to the linear case such that we can use our results in linear algebra to drive new lower bounds.We show one example within coded caching, with the linearity assumption, where our results in information theory lead to stronger bounds.Our results infer certain "qualitative" aspects of caching schemes, using the information they must contain, rather than giving a complete analysis of such schemes.

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