Past Seminars

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Weights on Walls & Combinatorial Calculations with Orthogonal Polynomials
Judy-anne Osborn
The University of Melbourne
Start time: 3:15 pm
Date: Friday 26 May 2006
Location: Theatre 1, Ground Floor, 111 Barry Street, Carlton

Recently an open problem from the 1970's, that of enumerating directed lattice paths in a slit, each wall of which has an independent weight associated with it, was solved using the Constant Term Method. In this talk I will describe that solution, as well as presenting its generalization to the case where each wall has any finite number of independent weights layered against it. The method utilizes rational functions which come from orthogonal polynomials subject to a geometrically apt variable change. I will also present a combinatorial method to directly calculate these transformed orthogonal polynomials.

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PDE Approach to valuation and hedging of credit derivatives
Marek Rutkowski
University of New South Wales,
Start time: 3:15 pm
Date: Friday 12 May 2006
Location: Theatre 2, Ground Floor, 111 Barry Street, Carlton.

Our aim is to examine the PDE approach to the valuation and hedging of a defaultable claim in various settings; this allows us to emphasize the importance of the choice of the traded assets. We start with a general model for the dynamics of the traded primary assets. Subsequently, we further specify our model and we deal with particular defaultable claims such as, for instance, survival claims. For the sake of notational simplicity, we deal throughout with three primary traded assets only. A generalization to the case of any number of primary assets is straightforward, however. First, we examine the no-arbitrage property of a model in terms of a martingale measure. Next, we study the PDE approach to valuation of defaultable claims and we derive the formulae for hedging strategies of a defaultable claim under the assumption that prices of all primary assets are strictly positive. Finally, we show how to adapt the valuation PDEs if some primary assets are defaultable securities with zero recovery, so that their prices vanish after default.

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Long cycles in random graphs
Nick Wormald
University of Waterloo, Canada
Start time: 3:15 pm
Date: Friday 5 May 2006
Location: Theatre 2, Ground Floor, 111 Barry Street, Carlton.

When a random graph on n vertices has enough edges (lines joining the vertices), i.e. about (1/2) n log n edges, it almost surely has a cycle containing all of its vertices, and the circumference (length of longest cycle) is n. What happens in the sparser case? When the graph has much less than n/2 edges, so the average degree of vertices (number of incident edges) is less than 1, there are few cycles and the answer is well understood. I will discuss old results on circumference as well as a new lower bound applying to the intermediate range, where the average degree of a vertex is at least 1 but can be regarded as bounded. (Joint work with Jeong Han Kim of Microsoft Research.)

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Complexity of a System as a Key to Its Optimisation
Galina Korotkikh and Victor Korotkikh
Central Queensland University
Start time: 3:15 pm
Date: Friday 28 April 2006
Location: Theatre 2, Ground Floor, 111 Barry Street, Carlton.

In the talk we discuss results of computational experiments offering the possibility of a general optimality condition of complex systems. “A complex system demonstrates the optimal performance for a problem, when the structural complexity of the system is in a certain relationship with the structural complexity of the problem.” The optimality condition presents the structural complexity of a system [2] as a key to its optimisation. Indeed, according to the optimality condition the optimal result can be obtained as long as the structural complexity of the system is properly related with the structural complexity of the problem. The computational results also indicate that the performance of a complex system may behave as a concave function of the structural complexity and thus reduce the optimisation of a complex system to a one-dimensional concave optimisation. This would become possible irrespective of the number of variables involved in the description of a complex system, because the structural complexity of the system, considered as one variable, could control the performance in such a remarkable way.

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Computational Methods for Analyzing Phylogenetic Trees
Katherine St. John
City University of New York
Start time: 3:15 pm
Date: Friday 21 April 2006
Location: Theatre 3, 1st Floor, 111 Barry Street, Carlton.

Evolutionary histories, or phylogenies, form an integral part of much work in biology. In addition to the intrinsic interest in the interrelationships between species, phylogenies are used for drug design, multiple sequence alignment, and even as evidence in a recent criminal trial. Much work has been done on designing algorithms that build phylogenetic trees given representative sequences of their DNA. The optimization criteria preferred by biologists for building trees is NP-hard. So, heuristics are often used that return many possible trees, instead of single optimal tree. This talk concentrates on the heuristics used for tree reconstruction, as well as how to summarize, analyze, and visualize these sets of trees. In particular, we will focus on fast reconstruction methods with provably nice properties, calculating biologically meaningful distances between trees quickly, and visualizing large sets of trees using the treecomp package designed by our group, as a module for the Mesquite system (developed by Wayne and David Maddison). (This work is joint with Nina Amenta, David Hillis, Tamara Munzner, and Tandy Warnow and is supported by grants from the National Science Foundation.)

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