Critical phenomena in lattice polymers
My most reacent research focused on the model of lattice self-avoiding trails. Self-avoiding trails are paths on a lattice (like the square lattice picture in the figure on the right) which can visit the same vertex twice but cannot cross the same edge more than once.
My reseach studied the critical properties of these geometrical objects.
This research has produced a number of unexpected results. I have shown that the collapse of trails in three dimension exhibits a non-standard scaling for which there is currently no theoretical prediction available. I also explored how the phase diagram changes when we introduce a second parameter such as stiffness or nearest-neighbour interactions.
Lattice trails are not only an important problem in the study of polymeric systems. The geometry of intersecting paths lies at the base of our theoretical understanding of many completely different phenomena, for example it is connected to the role of disorder in the Anderson metal-insulator transitions in two dimensions. The problem of characterising the collapse transition of trails is considered an open problem in the field, with contrasting predictions, both theoretical and numerical.
Exact calculation of graph polynomials
I developed an algorithm to compute the Tutte polynomial of a graph, an object of central importance in both mathematics and physics. My algorithm extends the traditional approach based on the transfer matrix with the computer-science concept of tree-decomposition. The result is two-fold: a software package to compute the Tutte polynomial of a given graph faster than any other, and a very general framework that can be easily adapted to compute many other kinds of graph polynomials.
This work allowed many researchers working on the chromatic polynomial to state conjectures and to test hypotheses in a fraction of the time required before.
Using my algorithm I investigated the distribution of the roots of the chromatic polynomial, over the ensemble of planar graphs. This is an area of very active research given the numerous conjectures on the possible loci of such roots that get stated and often disproven. The results of this study have been published in Journal of Physics A (DOI 10.1088/1751-8113/43/38/385001).
Super-symmetric methods in combinatorial problems
My PhD research was devoted to the study of combinatorial problems through the use of a formalism based on the Grassmann calculus. This is called a “fermionic” formalism in reference to the Pauli principle, which states that two half-spin particles (fermions) of the same kind cannot occupy the same physical state. This principle is reflected in the nilpotency of algebraic variables. Using this formalism I obtained formulas for enumerating the forest configurations (subgraphs without cycles) one can draw on a graph. My method is able to deal with forests on a more general structure, a hypergraph, extending the results previously only available for ordinary graphs. I proved that the asymptotic behaviour of the number of forests on a hypergraph depends on the ratio between the number of connected components in the forest (trees, in this case) and the number of vertices in the hypergraph. This fact gives rise to a phase transition in the canonical ensemble, where a fugacity parameter controls the number of trees. Furthermore I proved that this phase transition is consequence of the spontaneous breaking of a hidden supersymmetry of the model and is associated with the appearance of a “giant component”.
During my time at The University of Melbourne I taught the following subjects.
2014 Experimental Mathematics (MAST90053)
Modern computers have developed far beyond being great devices for numerical simulations or tedious but straightforward algebra. Experimental Mathematics covers some of the great advances made in using computers to algorithmically discover and prove nontrivial mathematical theorems.
I was lecturer in charge for this subject. I produced a revised set of lecture notes (available on GitHub), Mathematica notebooks, assignments and exam questions.
2012 and 2013 Calculus 2 (MAST10006)
Calculus 2 extends knowledge of calculus from school. I was part of a team of four lecturers. I delivered lectures and practice classes. I produced and marked assignment and exam questions.
- L Morawska, P K Thai, X Liu, A Asumadu-Sakyi, G Ayoko, A Bartonova, A Bedini, F Chai, B Christensen, M Dunbabin, J Gao, G S W Hagler, R Jayaratne, P Kumar, A K H Lau, P K K Louie, M Mazaheri, Z Ning, N Motta, B Mullins, M M Rahman, Z Ristovski, M Shafiei, D Tjondronegoro, D Westerdahl, and R Williams. Applications of low-cost sensing technologies for air quality monitoring and exposure assessment: How far have they gone? Environment International, 116 (2018).
- G Menconi, A Bedini, R Barale, and I Sbrana. Global Mapping of DNA Conformational Flexibility on Saccharomyces cerevisiae. PLOS Computational Biology, 11 (2015).
- A Bedini, A L Owczarek, and T Prellberg. Lattice polymers with two competing collapse interactions. J. Phys. A: Math. Theor., 47 (2014).
- A Bedini, A L Owczarek, and T Prellberg. Numerical simulation of a lattice polymer model at its integrable point. J. Phys. A: Math. Theor., 46 (2013).
- A Bedini, A L Owczarek, and T Prellberg. Semi-flexible interacting self-avoiding trails on the square lattice. Physica A: Statistical Mechanics and its Applications, 392 (2013).
- A Bedini, A L Owczarek, and T Prellberg. Self-avoiding trails with nearest-neighbour interactions on the square lattice. J. Phys. A: Math. Theor., 46 (2013).
- A Bedini, A Owczarek, and T Prellberg. Weighting of topologically different interactions in a model of two-dimensional polymer collapse. Phys. Rev. E, 87 (2013).
- A Bedini, A L Owczarek, and T Prellberg. Anomalous critical behavior in the polymer collapse transition of three-dimensional lattice trails. Phys. Rev. E, 86 (2012).
- A Bedini, and J L Jacobsen. A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings. J. Phys. A: Math. Theor., 43 (2010).
- A Puliti, C Rizzato, V Conti, A Bedini, G Gimelli, R Barale, and I Sbrana. Low-copy repeats on chromosome 22q11.2 show replication timing switches, DNA flexibility peaks and stress inducible asynchrony, sharing instability features with fragile sites. Mutation Research - Fundamental and Molecular Mechanisms of Mutagenesis, 686 (2010).
- A Bedini, S Caracciolo, and A Sportiello. Phase transition in the spanning-hyperforest model on complete hypergraphs. Nuclear Physics B, 822 (2009).
- A Bedini, S Caracciolo, and A Sportiello. Hyperforests on the complete hypergraph by Grassmann integral representation. J. Phys. A: Math. Theor., 41 (2008).