Researcher

Associate Professor Catherine Greenhill

Field of Research (FoR)

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Biography

ABOUT ME

 

Biography

My research interests lie at the intersection of combinatorics, probability and theoretical computer science. I study fundamental questions in asymptotic, probabilistic and algorithmic combinatorics. My work has been applied by researchers in many areas such as physics, computer science, statistics and cryptography. I was awarded the 2015 Christopher Heyde Medal for Pure Mathematics by the Australian Academy of Science,...view more

ABOUT ME

 

Biography

My research interests lie at the intersection of combinatorics, probability and theoretical computer science. I study fundamental questions in asymptotic, probabilistic and algorithmic combinatorics. My work has been applied by researchers in many areas such as physics, computer science, statistics and cryptography. I was awarded the 2015 Christopher Heyde Medal for Pure Mathematics by the Australian Academy of Science, jointly with A/Prof Scott Morrison of ANU.

Education

  • D.Phil., University of Oxford, 1996.
  • M.Sc. (Research) in Combinatorics, University of Queensland, 1992.
  • B.Sc. (Hons) in Pure Mathematics, University of Queensland, 1991.

RESEARCH

 Research Goals

  • Design and analysis of Markov chains for sampling from combinatorial sets.
  • Study of random combinatorial structures.
  • Asymptotic enumeration of fundamental combinatorial objects.

Research in Detail

The study of random combinatorial objects, such as random graphs, is an attempt to discover what a "typical" graph with certain properties might look like (for instance, a given number of vertices and edges, or a given number of vertices and every vertex having a fixed degree). This work involves a mixture of combinatorial and probabilistic arguments, and the results obtained are asymptotic, meaning that they hold in the limit as the number of vertices of the graph tends to infinity.

A related area is asymptotic enumeration of various kinds of graphs. This means finding a formula for the number of graphs of interest which is closer and closer to the truth as the number of vertices tends to infinity. For sparse graphs the methods are combinatorial and often involve a probabilistic approach using switchings. For dense graphs the approach is quite different: we want to to know a particular coefficient of the generating function and proceed using complex analysis, evaluating integrals using the saddle-point method.

I am also interested in the design and analysis of randomized algorithms for graphs and other combinatorial structures, particularly Markov chain Monte Carlo algorithms for approximate sampling and/or counting. Here a Markov chain is designed with a given stationary distribution, for instance the uniform distribution on the set of all graphs with a given degree sequence. The aim is to prove that the Markov chain converges rapidly (i.e. in polynomial time) to this distribution.

Research Grants

  • ARC Discovery Grant 2014 – 2016, C.S. Greenhill and B.D. McKay (ANU), A new model for random discrete structures: distributions, sampling and counting.
  • ARC Discovery Grant 2012 – 2014, I.M. Wanless (Monash), C.S. Greenhill and R. Aharoni (Technion), Extremal problems in hypergraph matchings.
  • ARC Discovery Grant 2012 – 2013, D.A. Bright, C.S. Greenhill, A. Ritter and C. Morselli (Montreal), Illicit drug trafficking: the structure of illicit networks and implications for resilience and vulnerability.

Current Student Projects (PhD and Honours)

Topics in the analysis of Markov chains.

Topics in asymptotic analysis of combinatorial objects.

Topics in the study of random combinatorial objects.

Advice for prospective students

If you are interested in working with me, please send me an email. Include some information about the areas that you are interested in, and your background in discrete mathematics, probability or theoretical computer science. If you have a particular topic in mind, please tell me about it.

TEACHING & OUTREACH

Courses I teach

MATH5425 Graph Theory

MATH2601 Higher Linear Algebra

MATH1081 Discrete Mathematics

MATH1241 Higher Mathematics IB

... and whatever else they ask me to teach!

Professional affiliations and service positions

I am an Editor-in-Chief of the Electronic Journal of Combinatorics.

I am on the University Promotions Committee for promotions to Associate Professor.

I am on the Executive Committee and the Equity and Opportunity Committee of the School of Mathematics and Statistics.

AWARDS & ACHIEVEMENTS 

I was awarded the 2015 Christopher Heyde Medal in Pure Mathematics by the Australian Academy of Science.

In 2013 I held the June Griffith Fellowship for Academic Women in Leadership, awarded by the Faculty of Science, UNSW Australia.

I was promoted to Associate Professor in 2013.

I was awarded the 2010 Hall Medal by the Institute of Combinatorics and its Applications. This is an international mid-career medal for research in combinatorics.


My Expertise

Combinatorics, graph theory.

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Location

School of Mathematics and Statistics
UNSW Sydney
NSW 2052, AUSTRALIA
The Red Centre
Room 5105

Contact

9385 7105
57123

Publications

by Associate Professor Catherine Greenhill

Videos

Advanced Mathematics Program