News:
I have accepted a 3-year postdoctoral position as an Emory Mathematical Fellow at Emory University . I will be moving to Atlanta and starting as of Fall '09.Research:
My research interests center in two areas; spectral graph theory and probabalistic combinatorics. I am especially enamored with problems that somehow combine these topics, such as the following: Suppose we take some random subgraph (in some sense) of a some 'real', not necessarily regular, given host graph, where we have some sort of spectral control of the host graph. What kind of things can we say about the random subgraph? This can be thought of as generalizing some natural questions about G(n,p); where in G(n,p) the host graph is the complete graph. My advisor is Fan Chung Graham .Previously, at RPI, I did some work under the guidance of Prof. Mark Goldberg relating to algorithmically finding (many) dense subgraphs of a graph.
Papers:
The spectal gap of a random subgraph of a graph , to appear, Internet Mathematics (joint with F. Chung)Teaching:We establish a bound on the spectral gap of a random subgraph of graph generated by percolating edges with some probability p. In particular if pd_min is large enough, we show that the spectral gap of the percolated subgraph is close to the spectral gap of the original graph.Intersecting Domino Tilings , preprint, (joint with S. Butler, and E. Tressler)
We prove an analogue to the celebrated Erdos-Ko-Rado theorem on the maximum size of an interesecting family for certain types of domino tilings.Diameter of random spanning trees in a given graph, preprint, (joint with F. Chung and L. Lu)
We establish bounds on the diameter of a random spanning tree in a graph with arbitrary degree sequence with some spectral control. The lower bound improves an earlier result of Aldous on the regular case.Dynamic models of online social networks, (joint with A. Bonato, N. Hadi, P. Pralat, and C. Wang)
Properties of a cloning-type model for geneterating large graphs is investigated. Among other properties, spectral properties of both the adjacency and normalized Laplacian, are studied.The giant component in a random subgraph of a given graph , preprint, (joint with F. Chung and L. Lu)We show that, under some conditions on the degree sequence of G and some spectral conditions on G, that the sharp threshold for the emergence of the giant component is 1/\tilde{d}, where \tilde{d} is the second order average degree of G. Our conditions on the degree sequence are somewhat more flexible than those admitted by other approaches.Independent dominating sets in graphs with girth 5 , preprint, (joint with A. Harutyunyan and J. Verstraete)
It is well known that d-regular graphs contain dominating sets of size asymptotic to n log(d)/d . Here we show that if the graph has girth 5, such a dominating set can be taken to be independent. Furthermore, we show that if G is not regular, the corresponding statement with d replaced by the minimum degree does not hold.An Alon-Boppana type result for the normalized Laplacian, in preparation.
We prove a version of the Alon-Boppana result for graphs with a general degree sequence and for the normalized Laplacian using a different notion of average degree. This result also confirms the optimality, up to a constant factor, of the result of Chung, Lu and Vu on the spectrum of graphs with a given expected degree sequence.From my undergrad years:
Statistical Modeling of Social Groups on Communication Networks, Proceedings of NAACSOS, (with M. Goldberg, M. Magdon-Ismail, W. Wallace, J. Riposo, D. Siebecker, and B. Yener).
This quarter (Spring '09) I am TAing for Math 163 .Last fall, I taught Math 10B. Information is here. Last summer I taught Math 20F, information is archived here
In previous quarters, I have TAed for 4C, 10B, 20ABF, 180ABC, and 183. In Spring 2008, I assisted with Fan Chung's 261 course.
Schedule:
This quarter I am travelling a fair amount. The week of 1/4-1/9 I will be at the AMS meetings in DC. From 1/25-1/31 I will be visiting the Fields Institute in Toronto. I will be in Barcelona 2/10-2/15 attending WAW 2009.
Classes:
In Fall '05, I took a course (Math 262) on the regularity lemma, and the combinatorics of patterns in subsets and graphs from Prof. Fan Chung. A complete set of notes are typed and are maintained by Steve. I also maintain the notes for the Fall 2005 on Prof. Fan Chung's course on random walks on directed graphs, Math 261a. The notes are here.
Personal:
I play guitar, albeit badly. Some of my (admittedly poorly recorded, written and played) songs are online. I also enjoy hiking, backpacking, and national parks. For Halloween, my roommate and I carved some mathematical pumpkins.