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Statement of Research Interests

Christopher Williamson2013

My primary research interests are in the mathematics that form the foundations oftheoretical computer science (TCS) a large part of which is in the realms of abstract

algebra, combinatorics, and graph theory. Below, I describe the connections between myinterests, and their potential for future research at Brown.

Group theory expander graphs

We begin by exploring the relationship between groups (more specifically, their Cayleygraphs) and expander graphs. Cayley graphs are defined in terms of some group and achoice of a generating set for that group; expanders are a type of graph that have fewedges, yet despite this, are surprisingly well connected (perhaps in the sense that thediameter of the graph is small). A vast amount of research has been conducted to studywhich groups have a generating set that yields an expanding Cayley graph. An example

of this research is the work of Kassabov [1] in which it was proved that every symmetricgroup has a generating set yielding a Cayley graph that is an expander.

Expanders Ramanujan graphs

The consideration of expander graphs leads naturally to the study of Ramanujangraphs. In order to understand the definition of Ramanujan graphs, first note that oneway of quantifying how well a k-regular graph holds the expansion property is by look-ing at the second-largest (by absolute value) eigenvalue, 2, of its adjacency matrix. Thisconstant, which here I call the expansion constant, is one convenient way of characterizing

the expansion property, as graphs with a large spectral gap (typically defined as k 2)can be shown to have the property that the probability distribution of the location of arandom walker on the graph tends rapidly towards uniform. In fact, the L2 norm of thedifference between the vector representing the probability distribution over the verticesafter a t step walk and the uniform distribution vector, (1/n, ..., 1/n), is bounded above

by2k

t. Having small 2 is a sign that the graph is well connected imagine that in

the limiting case, we have a complete graph and the location of the walker after even justa single random step is completely unpredictable (but of course, a complete graph is apoor expander due to its large number of edges). This expansion constant is fundamentalto the study of spectral graph theory. Many fundamental properies of graphs are relatedto the expansion constant. For example, the diameter of a connected n-vertex, k-regular

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graph is bounded above by 1 + log(n)log(k)log(2)

.

This brings us finally back to Ramanujan graphs, which are optimal expanders in thattheir spectral gap is as large as possible. Specifically, we know that ifGn,k is a family ofconnectedk-regular,n-vertex graphs, then for fixed k as n tends to infinity,

2(Gn,k) 2

k 1 o(1)where 2(Gn,k) is the expansion constant. This motivates the definition of Ramanujangraphs as those with expansion constant less than 2

k 1. A current area of research

that I find interesting is the quest to find k-regular families of Ramanujan graphs for newvalues ofk. Currently, the best result is that of [2, 3], where explicit families of degree

pn + 1 are constructed (forp prime andn Z). Whether or not families of other degreesexist is unknown, a fact made more enticing by the fact that, in a certain sense, almostall graphs are almost Ramanujan [4], meaning that for all > 0, a random k-regular

graph on n vertices satisfies 2 2k 1 + with probability 1 o(1). An excitingnew development from this year has demonstrated the existence of bipartite Ramanujangraphs of all degrees greater than two [5].

Expanders extractors

Algebra, combinatorics, and graph theory are critical for the creation of extractors,functions commonly studied in theoretical computer science that turn input bits sampledfrom a weakly-random source (typically quantified by min-entropy), and sometimes arandom seed that is as short as possible, into output bits that are very nearly completely

random. Demonstrating another surprising connection between pure mathematics andtheoretical computer science, the combinatorial study of line-point incidence theorems hasproven useful in the creation of extractors. For example, the Szemeredi-Trotter theoremstates that:

|I(P, L)| =O

(|L||P|) 23 + |L| + |P|

,

whereP andL are finite sets of points (resp. lines) in R2 andI(P, L) = {(p, l) PL |pl}. The finite-field version of this theorem was critical in the creation of Bourgainsextractor (see [6, 7]).

Graph theory also takes part in the creation of extractors, some of which are constructed

by performing random walks on expander graphs. The conceptual reason for this is easyto see: one can associate with each vertex of an expander a string of bits (used as output)and use the input string of bits as defining a walk on the graph. If one then outputs thestring of bits associated with the last vertex on the walk, then the fast mixing rate of theexpander graph implies that these bits are highly unpredictable after a relatively shortwalk.

Extractors pseudorandom generators

Extractors are frequently used in the creation of pseudorandom generators (PSGs).An example of this application of extractors is in [8], where PSGs for regular branching

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programs are created using the expander random walk extractors created by Goldreichand Wigderson in [9]. Pseudorandom generators are functions that turn a small numberof truly random bits into a longer string of bits that cannot be easily distinguished froma random string of that length by some model of computation. Thus, a PSG is a functionG: {0, 1}s {0, 1}n such that for some class of functions f,

|ExUs [f(G(x))] ExUn [f(x)]| < .The endeavor to improve existing PSGs (by reducing the random seed lengths) and tocreate PSGs for other models of computation, especially certain classes of circuits, is anactive area that I would like to study further.

Extractors graph theory

By considering a weakened version of an extractor, we will return to graph theory

and find one final avenue of potential research. A disperser is a weakened extractor inthe sense that, rather than requiring the output bits be close to uniform, we only requirethat the size of the support of the output distribution is large. It is an elementary factthat (and we will refrain from making all necessary definitions for the sake of brevity) ifone has a 2-source disperser for weakly-random sources of min-entropy k, then one canuse that to efficiently construct a bipartite, 2k-Ramsey graph (a j-Ramsey graph is agraph with no cliques or independent sets of size j). Current constructions of Ramseygraphs vary widely in their complexity. At one end are the highly intricate constructionsof Barak, Rao, Shaltiel, and Wigderson [10], and at the other end are the simple Frankl-Wilson constructions [11], which can be described in a few sentences (the complexity of[10] is justified in that the graphs constructed there are j-Ramsey for lower j ). An opentask is to find more explicit and easily understood Ramsey graph constructions that alsooutperform the Frankl-Wilson construction.

References

[1] M. Kassabov, Symmetric groups and expander graphs, Invent. Math. 170 (2007), no.2, 327-354.

[2] M. Morgenstern, Existence and explicit constructions of q+1 regular Ramanujangraphs for every prime power q, J. Combin. Theory Ser. B 62 (1994), 44-62.

[3] A. Lubotzky, B. Samuels and U. Vishne, Explicit constructions of Ramanujan com-plexes of type Ad, European J. Combin. 26 (2005), no. 6, 965-993.

[4] J. Friedman, A proof of Alons second eigenvalue conjecture and related problems,CoRR (2004).

[5] A. Marcus, D. Spielman, N. Srivastava, Interlacing Families I: Bipartite RamanujanGraphs of All Degrees, eprint arXiv:1304.4132 (2013).

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[6] J. Bourgain. More on the sum-product phenomenon in prime fields and its applica-tions. International Journal of Number Theory, 1:1 - 32, 2005.

[7] A. Rao. An Exposition of Bourgains 2-Source Extractor. ECCC (2007).

[8] M. Braverman, A. Rao, R. Raz, A. Yehudayoff. Pseudorandom Generators for Reg-ular Branching Programs. FOCS 10 Proceedings of the 2010 IEEE 51st AnnualSymposium on Foundations of Computer Science (40-47).

[9] Oded Goldreich and Avi Wigderson. Tiny families of functions with random proper-ties: A quality-size trade-off for hashing.Random Struct. Algorithms, 11(4): 315-343,1997.

[10] B. Barak, A. Rao, R. Shaltiel, A. Wigderson.2-Source Dispersers for Sub-PolynomialEntropy and Ramsey Graphs Beating the Frankl-Wilson Construction. STOC (2006).

[11] P. Frankl and R. M. Wilson. Intersection theorems with geometric consequences.Combinatorica, 1(4):357-368, 1981.

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