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We discuss how the application of real analytic bifurcation theory and finite Morse index solutions of partial differential equations can be used to show that the problem - Laplacian u= r exp(u) in D with u=0 on the boundary of D has infinitely many bifurcation points. Here D is a smooth bounded domain in N-dimensional space and N is between 3 and 9 (and the dimension restriction is best possible).
For More Information:Contact Paul Pearce (P.Pearce@ms.unimelb.edu.au) or Paul Norbury (P.Norbury@ms.unimelb.edu.au)
Given a polynomial in two (or more variables), one may study the zero locus from the point of view of different mathematical subjects (number theory, algebraic geometry, ...). I will explain how tropical geometry allows to encode all topological aspects by elementary combinatorial objects called "tropical varieties."
For More Information:contact: Paul Pearce P.Pearce@ms.unimelb.edu.au OR Paul Norbury P.Norbury@ms.unimelb.edu.au
For the past decade, microarrays have been the assays of choice for a wide range of high-throughput biological assays. Recent improvements in the efficiency, quality, and cost of DNA sequencing are prompting biologists rapidly to abandon microarrays in favor of so-called next-generation sequencers, e.g., Applied Biosystems' SOLiD, Helicos BioSciences' HeliScope, Illumina's Genome Analyzer, and Roche's 454 Life Sciences sequencing systems. These high-throughput sequencing technologies have already been applied to studying genome-wide transcription levels (mRNA-Seq), transcription factor binding sites (ChIP-Seq), chromatin structure, DNA methylation status, copy number and several other tasks. Such studies have overcome many longstanding limitations of microarray-based studies, but of course new technologies raise familiar as well as novel statistical and computational challenges. In this talk I'll outline some of the topics that are currently interesting me in this area.
Anyone who has watched a fly make a flawless landing on the rim of a teacup, or marvelled at a honeybee speeding home after collecting nectar from a flower patch several kilometres away, would know that insects possess visual systems that are fast, reliable and accurate. Insects cope remarkably well with their world, despite possessing a brain that carries fewer than 0.01% as many neurons as ours does. This talk will describe research aimed at understanding the mechanisms underlying visual perception, navigation, learning, memory and “cognition” in honeybees. It will also describe opportunities for incorporating insect-inspired principles into the design of autonomously navigating robots.
Mathematics and Sex? That’s right, the two are very much intertwined. Come and enjoy a dalliance with the equations explaining love, marital bliss, and the number of partners you should have before you stop playing the field to see for yourself. And be prepared for some of the latest mathematical research, as the field goes much, much, further than numbers and probabilities. Mathematics is the study of patterns: their discovery, their interconnections and their implications. And, in the context of human behaviour, patterns abound, and mathematics provides many unique and exciting insights. You’ll leave this presentation with some great relationships tips but dare I say, more importantly, completely reaffirmed in your belief of how creative, evolving, relevant, and downright sexy mathematics is.
Design principles of Mathematica, such as algorithmic automation, integrated symbolic computation, and its all-in-one architecture, will be discussed, with examples to illustrate Mathematica's benefits. This talk will introduce the Mathematica technical computing system and explain the design principles that have guided its development for over 20 years. Using simple examples from a range of science and engineering disciplines, the talk will also explain how these principles have accelerated the development of the system in recent years. Examples will include visualization, data analysis, symbolic computation, image processing, and application development. As well as showing Mathematica's use for free-form problem solving, the talk will discuss scalable deployment capabilities that enable the Wolfram Demonstrations Project and large applications, such as Wolfram|Alpha. http://www.wolfram.com/services/seminars/australia2009/index.html
Recently in collaboration with O.Foda and M. Wheeler, a series of papers were released which explored the correspondence between the quantum six vertex lattice model and the classical KP hierarchy of exactly solvable non linear partial differential equations. In the present talk I shall introduce another quantum lattice model - classical hierarchy correspondence; that between the phase model and the 2-Toda hierarchy.
Despite the generation of vast amounts of genetic data in public genomics projects such as the Human Genome Project the function of many genes remains a mystery. Families which are carrying a genetic disease can have their faulty gene identified through a combination of statistical analysis and benchwork. This not only provides a genetic test for the family but can also give valuable genetic information about the function of the gene. Statistical analysis is used to identify a genomic segment that is likely to harbour the mutated gene. I will illustrate how and when this approach works and compare it to the current, in vogue, approach of genome wide association analysis using case control studies. I will also discuss the way the genetic data is measured (SNP chips) and how one can use the same data for the interpretation of DNA copy number which are deviations to the usual 2 copies of DNA that we all have.
Suppose G is a real reductive Lie group. In its purest form, abstract harmonic analysis asks for a description of the unitary dual of G, the set of equivalence classes of its irreducible unitary representations. For example, if G is the circle group, the unitary dual amounts to the sines and cosines underlying Fourier series. On the other hand, if G is the additive group of real numbers, the unitary dual amounts to the exponential functions appearing in the theory of the Fourier transform. One of the outstanding problems in the subject has been to describe the unitary dual of G in general. Recently a new set of ideas has emerged which appears powerful enough to give such a description under rather mild assumptions. This talk will give a history of the problem, and will report on recent progress (joint with Adams, van Leeuwen, and Vogan).
Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this talk, individual-based models describing cell movement and domain growth are outlined, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-basedmodels are formulated in terms of randomwalkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs.