Seminars 2008 Arun Ram

I have an interest in combining the many strengths in Melbourne (and Australian) mathematics and the stream of seminars will be important to this goal. The following is something of a seminar blog. This page is the archive from 2008.

• Algebra-Geometry-Topology seminar page 2009
• Algebra-Geometry-Topology seminar page 2008
• Department seminar page

December 2008 Seminar Blog

• 16 December 2008 13:00 Room 107
  
  Matthew Zuparic (University of Melbourne) An introduction to the tau functions of the KP hierarchy
  
  Abstract: In the this talk we shall offer an introduction to the construction of the KP tau function using free fermion calculus. We shall then attempt to show a fundamental result of this construction: the bilinear identity.
  
  
• 15 December 2008 13:00 Room 213
  
  Jorgen Rasmussen (University of Melbourne) W-extended fusion algebras of logarithmic minimal models
  
  Abstract: We consider the continuum scaling limit of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p') as `rational' logarithmic conformal field theories with extended W symmetry. Critical dense polymers and critical percolation correspond to LM(1,2) and LM(2,3), respectively, and are used as illustrations. The W symmetry allows the countably infinite number of indecomposable Virasoro representations to be reorganized into a finite number of W-indecomposable representations. We classify these representations, which can have rank 1, 2 or 3, and discuss their characters. Using a lattice implementation of fusion on a strip, we determine the fusion rules for the W-indecomposable representations and find that they generate a closed fusion algebra, albeit one without identity for p>1.
  
  
• 9 December 2008 13:00 Room 213
  
  Anita Ponsiang (University of Melbourne) A solution of the q-deformed Knizhnik-Zamolodchikov equation
  
  Abstract: The two-boundary bond percolation model, may be described by the Temperley-Lieb algebra and a space of link patterns. The transfer matrix of this system has a groundstate eigenvector which also satisfies the two-boundary q-deformed Knizhnik-Zamolodchikov equation. I will describe the relationship this equation has to the system and present a solution for general system sizes.
  
  
• 8 December 2008 13:00 Room 213
  
  Andrew Rechnitzer (University of Melbourne) Counting elements of Thompson's group F
  
  Abstract: Richard Thompson's group F is a widely studied group which has provided examples of and counter-examples to a variety of conjectures in group theory. It is also an extremely combinatorially appealing object which has a beautiful description in terms of binary trees. In this talk I will give a description of some of the combinatorics of the group and mostly talk about some enumeration questions associated with F. This is work together with Sean Cleary, Murray Elder, Eric Fusy, Buks van Rensburg and Jennifer Taback.
  
  
• 2 December 2008 14:15 Room 213
  
  Nora Ganter (University of Melbourne) Introduction to K-theory IV
  
  

November 2008 Seminar Blog

• 27 November 2008 15:15 Room 213
  
  Stephan Tillmann (University of Melbourne) Applications of tropical geometry to manifolds
  
  
• 25 November 2008 13:00 Room 107
  
  Peter Tingley (University of Melbourne) Three combinatorial models for affine sl(n) crystals continued
  
  Abstract: This is a continuation of a talk given Nov 4. We will continue discussing combinatorial models for affine sl(n) crystals, focusing this time on the appearance of cylindric plane partitions. We will also see how level-rank duality appears.
  
  
• 25 November 2008 14:15 Room 213
  
  Nora Ganter (University of Melbourne) Introduction to K-theory III
  
  
• 20 November 2008 15:15 Room 213
  
  John Groves (University of Melbourne) Applications of tropical geometry to group theory
  
  
• 18 November 2008 14:15 Room 213
  
  Nora Ganter (University of Melbourne) Introduction to K-theory II
  
  
• 18 November 2008 13:00 Room 107
  
  Steve McAteer (University of Melbourne) An aspect of the large n behaviour of the partitions of n
  
  Abstract: A probability distribution, which is related to hook length, is defined for the set of partitions of a given number n. We then ask the questions about the expected shape of a partition for large n. It turns out that the distribution concentrates on a partition whose Young diagram has a boundary of a particular simple shape.
  
  
• 18 November 2008 12:00 Russell Love
  
  Thomas Handscomb (University of Melbourne) Surface bundles and Lefschetz fibrations
  
  Abstract: Surface bundles and Lefschetz fibrations appear in many diverse branches of mathematics. The rich structure of these objects illuminates powerful interactions between areas of complex analysis, algebraic geometry and symplectic topology. The colloquium will provide an easy going and pleasant introduction to surface bundles and will describe an accessible overview of their natural relationship with the moduli space. We will emphasize intuition rather than definition and will illustrate concepts with simple pictures. Indeed we will see how the edgewise gluing of triangles, and a little imagination, can uncover deep connections between the Fibonacci numbers and the existence of certain 4-manifolds.
  
  
• 17 November 2008 13:00 Room 213
  
  Richard Brak (University of Melbourne) Combinatorics of the Rothe diagram
  
  Abstract: The Rothe diagram is one of the earliest combinatorial models and appeared in a memoir by Rothe in 1800 concerning simultaneous linear equations. I will discuss a range of combinatorial results that can be obtained from the diagram. These range from the elementary to the not-so-elementary: inversion codes, Coxter words for S_n, multiset bijections and a (newish) RSK tableaux correspondence method, the coplactic monoid and, time permitting, crystals bases. I will also discuss a new bijection, using half of the Rothe diagram, between partially directed walks in a wedge and involutions with no fixed points, conjectured by Rechnitzer, Prellberg and van Rensberg at FPSAC 07.
  
  
• 13 November 2008 15:15 Room 213
  
  Peter Bouwknegt (Australian National University) T-duality: An overview
  
  Abstract: T-duality is a symmetry that relates string theories on two distinct manifolds. It has important ramifications in different areas of mathematics, such as algebra, geometry and topology, where it is related to constructions like Takai duality, mirror symmetry, Fourier-Mukai transforms, the Connes-Thom isomorphism, etc. In this talk I will give an overview of some of the recent developments in the field.
  
  
• 12 November 2008 13:15 Geoff Opat Seminar room (360), School of Physics
  
  Peter Bouwknegt (Australian National University) Gauge Theory and Langlands Duality
  
  Abstract: In this talk I will explain how electric-magnetic duality (S-duality) of N=4 supersymmetric Yang-Mills theory in 4 dimensions is related to the Geometric Langlands program. We will see how important physical concepts, such as Wilson and 't Hooft operators, branes, mirror symmetry of sigma models, and others, when applied to N=4 SYM, give rise to the ingredients and statements of the Geometric Langlands conjecture. At the same time, this talk will serve as an introduction to the Geometric Langlands program.
  
  
• 11 November 2008 15:15 Room 213
  
  Nora Ganter (University of Melbourne) Introduction to K-theory I
  
  
• 11 November 2008 13:00 Room 107
  
  Emily Peters (University of California, Berkeley) Planar algebras and knot invariants
  
  Abstract: A planar algebra is a family of algebras whose structures are tied tightly together, by an action of the 'planar operad.' (Formal sums of) link diagrams are probably the most natural example of a planar algebra, and considering planar algebra homomorphisms from link diagrams to other planar algebras is a good way to construct knot invariants. For example, the Jones polynomial and colored Jones polynomial can be constructed by mapping link diagrams to the Temperley-Lieb algebra. After introducing planar algebras, I will discuss these constructions, and also mention the D_2n planar algebras and the link invariants they give rise to (this is joint work with Scott Morrison and Noah Snyder).
  
  
• 11 November 2008 12:00 Russell Love
  
  Alana Moore (University of Melbourne) Estimating Probabilities for Managing Biological Populations
  
  Abstract: The work in my thesis was motivated by a desire to understand the value of information, and hence the role of monitoring, for managing biological populations. The value of information for making a one-off decision is reasonably well understood, however, understanding the value of information when making a sequence of decisions is a much more difficult problem. In the first half of my talk, I will consider the problem of controlling an invasive pest species when only partial observations are available at each time step. In the model considered, the monitoring data available are binomial observations of a probability which is an index of the population size. In the second half of my talk I will discuss a different problem, concerned with the benefit of sacrificing short-term rewards in order to improve estimates of a static probability which may improve long-term profits. When addressing economic problems which span a number of years, it is typical to discount future rewards. Naturally, this has an impact on how much we are willing to invest now in order to increase future returns. However, it has recently been argued that traditional geometric discounting is not appropriate for making decisions of social concern as it does not address issues of intergenerational equity and sustainability. Consequently several alternative discounting schemes have been proposed. I use a model of a theoretically harvested population with an unknown parameter to compare how different types of discounting influence the value of learning.
  
  
• 10 November 2008 13:0 Room 213
  
  Ian Enting (MASCOS) Some Statistics from Lattice Statistical Mechanics
  
  Abstract: Lattice statistical mechanics is based on the Gibbs specification for probability exp[-E(x1, ..xN)/kT]/Z for microstate x1,...xN. This talk looks at applications in which this form of probability distribution is applied in contexts that have no relation to actual temperature. Applications of Ising-like models in image processing are noted briefly. A special class of random fields, introduced by Pickard, is discussed as giving more tractable statistical models. The connections to statistical mechanics are discussed and conjectures about higher-order generalisations proposed.
  
  
• 5 November 2008 13:00 Room 107
  
  Peter Tingley (University of Melbourne) Three combinatorial models for affine sl(n) crystals
  
  Abstract: We present some combinatorial descriptions of the crystals associated to integrable highest weight representations of affine sl(n). The underlying sets of these crystals can be parameterized by partitions, by "configurations on an abacus" or by cylindric plane partition. Similar models have been studied before, but the exact forms we use have some advantages; we see that the generating function of cylindric plane particular is a specialization of the Weyl character formula, and also observe some well known forms of level-rank duality.
  
  
• 04 November 2008 12:00 Russell Love
  
  Ashish Gupta (University of Melbourne) Quantum polynomials
  
  Abstract: Quantum polynomials (in two variables X and Y) differ from ordinary polynomials in that XY = YX is no longer valid, but is replaced by XY = cYX, where c may be a complex number. Quantum polynomials were already mentioned in H. Weyl's book: The theory of groups and quantum mechanics. More recently quantum polynomials have gained importance for their role in Mathematical physics and non-commutative geometry. One of the concepts worth studying is that of modules over these polynomials which are quite analogous to the familiar vector spaces(the module elements may be multiplied by polynomials as well as scalars). As modules over quantum polynomials are mostly infinite dimensional as vector spaces, the suitable concept of dimension here is the "growth" of the module which is recorded in the degree a special polynomial - the Hilbert-Samuel polynomial. This new type of dimension is known as the Gelfand-Kirillov(GK) dimension. Allowing negative integers for the exponents of the variables of quantum polynomials, we get what are known as quantum Laurent polynomials(QLPs). QLPs are multiplicative analogues of Weyl algebras. The latter had attracted a great deal of interest in the recent decades. Some questions that were asked for the Weyl algebras then appeared for the multiplicative Weyl algebras or QLPs. We shall discuss one such question: Do QLPs have non-holonomic simple modules? This was first considered in 1987 by McConnell and Pettit and has not been fully answered till now. The motivation behind it is that holonomic modules over Weyl algebras have very nice properties(they are the central theme of Bjork's book: Rings of differential operators). If time permits we shall also discuss some fascinating conjectures on the simple modules of QLPs satisfying a certain condition.
  
  

October 2008 Seminar Blog

• 28 October 2008 13:00 Room 107
  
  Jan de Gier (University of Melbourne) qKZ equation for the two-boundary Temperley-Lieb equation Part III
  
  
• 28 October 2008 12:00 Russell Love
  
  Tony Guttmann (University of Melbourne) The Ising model --- a paradigm of co-operative behaviour
  
  Abstract: The Ising model is the most celebrated model of phase transitions, with a 90 year history. The model has an enormous range of applications. The biggest breakthroughs in solving the problem in two dimensions were due to Onsager, who in 1944 found the free energy (the zeroth field derivative), and to C.N. Yang, who found the magnetisation (the first field derivative) in 1952. For 50 years people tried and failed to find the susceptibility (the second field derivative). In the last decade great progress in determining the susceptibility has been made, which has in turn illuminated other problems. Both the history and recent progress will be discussed.
  
  
• 27 October 2008 13:00 Old Geology 2
  
  Tony Guttmann (University of Melbourne) Calabi-Yau equations, Lattice Green's functions and related problems
  
  Abstract: This will be a "work in progress" discussion, that originated in some earlier work I did with Larry Glasser on the 4d hypercubic lattice Green's function. We found a Fuchsian ODE, but were unable to totally solve it. In recent years a family of 4th order Fuchsian ODEs with special properties have been called Calabi-Yau equations, and hundreds of them have been solved. Among the solutions is the 4d hypercubic lattice Green's function. This raises the question of the solution of other 4d Green's functions. In a rambling, disconnected and ill-prepared seminar I propose to enlarge on the above.
  
  
• 27 October 2008 10:00 Alice Hoy 330
  
  Paul Norbury (University of Melbourne) Modular forms in geometry
  
  Abstract: I will give a geometric view of Alex Ghitza's talk.
  
  
• 21 October 2008 13:00 Room 107
  
  Jan de Gier (University of Melbourne) qKZ equation for the two-boundary Temperley-Lieb equation Part II
  
  
• 20 October 2008 10:00 Alice Hoy 330
  
  Alex Ghitza (University of Melbourne) Modular forms in arithmetic geometry
  
  Abstract: The purpose of this talk is to introduce various notions of modular forms from the point of view of arithmetic geometry. I will define these objects, explain why one might be interested in them, and describe some of the aspects that I have been working on.
  
  
• 14 October 2008 12:00 Russell Love
  
  Harold Kusner (Brown University) Stochastic Approximation and the Analysis of Proportional Fair-Sharing Algorithms in Communications Theory
  
  Abstract: Recursive stochastic algorithms that take the form of discrete-tim stochastic dynamical systems are known as stochastic approximations, and have an enormous variety of applications. A brief description of the basic algorithms and techniques of proof via weak convergence methods will be given.
  
  
• 14 October 2008 13:00 Room 107
  
  Michael Couch (University of Melbourne) Moment Maps: projection of Hamiltonian systems to coadjoint orbits
  
  Abstract: In Hamiltonian mechanics the time evolution of a system is governed by a Hamiltonian, or 'energy' function, or more precisely by set of first order equations dependent upon the Hamiltonian. One is often interested in systems in which the equations of motion admit a symmetry - invariance under rotations about an axis, for example. I argue that for symmetries of an appropriate form, one is able to project the dynamics forward to a system on which the equations of motion take on the well known Lax pair form.
  
  
• 13 October 2008 13:00 Old Geology 2
  
  Gerasim Iliev (University of Melbourne) Directed walk models of semiflexible polymers
  
  Abstract: The use of directed walk models to study polymeric systems in a variety of environments has provided much insight into the full self-avoiding walk model of such problems. Many of the previous studies have focused on the behaviour of long, flexible polymers, however on sufficiently small length scales the polymer essentially behaves as a rigid rod. Modifying the usual directed walk models (Dyck paths, Motzkin paths and partially-directed walks) we can obtain models which have a tunable persistence length. These models are subsequently used to study the phenomenon of polymer adsorption at an impenetrable surface in the presence and absence of an elongational force. In such studies, we obtain the phase diagram for each model as a function of the interaction parameter with the adsorbing surface and a stiffness parameter. In all models studied, the adsorbed/desorbed phase transition in the absence of a tensile force is of second order for all finite values of the stiffness. When an elongational force is introduced, several regimes with respect to the stiffness parameter become apparent as we observe the critical value of the desorbing force. Work in collaboration with: E. Orlandini and S. Whittington.
  
  
• 13 October 2008 10:00 Alice Hoy 330
  
  Omar Foda (University of Melbourne) 196883 + 0.5 = 196884 - 0.5
  
  Abstract: The purpose of the talk will be to introduce the conformal field theory, vertex operator algebras and so forth (hence the 196883.5) that interpolate the monster on the one hand and modular functions on the other.
  
  
• 7 October 2008 13:00 Room 107
  
  Laura Hattam (University of Melbourne) Orthogonal polynomials as a mechanism to find solutions to the KP hierarchy
  
  
• 7 October 2008 12:00 Russell Love
  
  Michael Eastwood (University of Adelaide) Div, grad, curl and all that
  
  Abstract: These well-known differential operators are, of course, important in applied mathematics. This is just the tip of an iceberg. I shall indicate some of what lies beneath the surface. There are links with topology, physics, symmetry groups, finite element schemes, and more besides. This talk will touch on these different topics by means of examples. Little prior knowledge will be assumed beyond the equality of mixed partial derivatives.
  
  
• 6 October 2008 10:00 Alice Hoy 330
  
  Michael Eastwood (University of Adelaide) Conformal invariants and semiconformal mappings
  
  Abstract: A submersion between two Riemannian manifolds is said to be semiconformal if it is conformal orthogonal to the fibres. Joint work with Paul Baird produces many examples semiconformal mappings from Euclidean 3-space to Euclidean 2-space and, under mild non-degeneracy assumptions, gives necessary and sufficient conditions in order that a function on 3-space be one of the components of such a mapping. These conditions are in the form of non-linear partial differential equations. They may also be regarded as conformal invariants of a smooth function. As such, some of these invariants have interesting geometric significance. Nothing much will be assumed of the audience and all terminology will be explained from scratch.
  
  
• 2-3 October 2008 9:00 RMIT Building 8, Level 9 Room 66
  
  Victorian Algebra Conference, RMIT University
  
  
  • Talks will be in the Access Grid Room 8.9.66 (Building 8, Level 9, Room 66). Building 8 is accessible from Swanston St, between Latrobe St and Franklin St.
  • Registration and coffee will be in the Mathematics Staff Room 8.9.03 (Building 8, Level 9, Room 3).

September 2008 Seminar Blog

• 30 September 2008 12:00 Russell Love
  
  Jon Borwein (Dalhousie and Newcastle) Computer assisted discovery and proof
  
  Abstract: I shall describe some of the methods that are now available for computer-assisted discovery and proof of highly non-trivial mathematical formulas and identities.
  
  
• 25 September 2008 13:00 Room 107
  
  Scott Morrison (Microsoft Research) Generators and relations for the representation theory of U_q(sl_n) as a planar algebra
  
  Abstract: I'll begin by explaining what a planar algebra is, and then show you how the representation theory of SU(2) and SU(3) can be given a "finite presentation by generators and relations" as a planar algebra. This may be familiar to some, as the Temperley-Lieb algebra for SU(2) or Kuperberg's spider for SU(3). Next, I'll explain my work on generalising this to all SU(n). We'll start with a category of diagrams, generated by some trivalent vertices, and a surjective map to the representation theory; the difficulty will be understanding the relations amongst these diagrams. The main trick is to remember that SU(n) sits inside SU(n+1), and conversely representations of SU(n+1) break up (or "branch") as representations of SU(n). I'll explain how to understand the combinatorics of branching in terms of my diagrams, and how to use this to "lift" relations for diagrams from one level to the next.
  
  
• 25 September 2008 14:15 Room 213
  
  Ole Warnaar (University of Queensland Interpolation Macdonald polynomials
  
  Abstract: The interpolation Macdonald polynomials are a generalisation of the famous Macdonald polynomials to multivariable polynomials that are neither symmetric nor homogeneous. In this talk I will give an introduction to the interpolation Macdonald polynomials, describing their connection with classic Newton interpolation, the (extended affine) Hecke algebra and multiple hypergeometric series.
  
  
• 19 September 2008 12:00 Old Geology 2
  
  Arun Ram (University of Melbourne) Generalising Pascal's triangle
  
  Abstract: There are generalizations of Pascal's triangle which "control" the symmetric groups, Temperley-Lieb algebras, and many other diagram algebras. I'll give a short tour of some of my favourites.
  
  
• 16 September 2008 13:00 JH Michell Theatre
  
  MUMS University Math Olympics
  
  Details about this awesome event: Date: Tuesday 16th September (last Tuesday before mid semester break!) Time: 1-2pm Location: J.H. Mitchell Theatre, Richard Berry Building Register: Forms outside the MUMS room (G24) or downloadable from http://www.ms.unimelb.edu.au/~mums/olympics/umo.html
  
  
• 16 September 2008 12:00 Room 107
  
  Chunhoa Chen (University of Melbourne) Stochastic Loewner Evloution
  
  
• 15 September 2008 13:00 Old Geology 2
  
  Nick Witte (University of Melbourne) The Bethe ansatz from an analytical perspective
  
  Abstract: Recently the 75th anniversary of Bethe's 1931 solution for the Heisenberg model via his now-famous ansatz - the functional equation for the eigenvalues - was celebrated. Although not wishing to downplay the novelty, let alone the originality of the work, it has since become apparent that such systems of equations have an older genealogy, going back to Heine and Stieltje's work on the functional equations for the zeros of orthogonal polynomials. This was made clear in an observation by Ismail et al in 2004 which demonstrated that the Bethe anstaz equations for a particular open boundary condition case of the generic XXZ model are the functional equations for the roots of a sequence of polynomials satisfying a $q$-analog of a second-order Sturm-Liouville equation. The coefficients of this equation were generated from a weight which is a natural generalisation of the Askey-Wilson weight, and in fact the Askey-Wilson weight is the case of the shortest chain length. However they had no theory to link the polynomial system orthogonal with respect to this weight to such a $q$-Sturm-Liouville equation, and we show how such a theory can be constructed and its implications. This work is part of a joint project with Paul Pearce.
  
  
• 8 September 2008 10:00 Alice Hoy 330
  
  Craig Westerland (University of Melbourne) 196884 -1 = 196883
  
  
• 9 September 2008 13:00 Room 107
  
  Jan de Gier (University of Melbourne) qKZ equation for the two-boundary Temperley-Lieb equation
  
  
• 9 September 2008 12:00 Russell Love
  
  Matt Wand (University of Woolongong) Variational Approximations in Statistics
  
  Abstract: Variational approximations have been used extensively in Statistical Physics and Computer Science as a means of overcoming difficult probability calculus problems, and offer an alternative to (Markov chain) Monte Carlo methods. We describe recent work on the transferral and adaptation of variational approximation methodology to contemporary Statistics settings. Particular attention is paid to generalised linear mixed models and semiparametric regression. This talk represents joint research with Dr John T. Ormerod.
  
  
• 8 September 2008 18:30 Theatre A Elisabeth Murdoch
  
  Ary Hoffmann (University of Melbourne) Biodiversity vs extinction in a stressful world: Maintaining healthy environments at a time of climate change and increasing population pressure
  
  Abstract: Are we entering a period of irreversible mass extinction, when more than 90% of all our animals and plants will disappear? Will our urban and rural environments become biodiversity deserts, dominated by a few common invasive imports and local species? How can we preserve biodiversity and resilience in species and in ecosystems? In his Vice-Chancellor’s Research Lecture, University of Melbourne evolutionary biologist Professor Ary Hoffmann (a Federation Fellow in the Bio21 Institute) will tap into recent research to give some answers.
  
  
• 8 September 2008 10:00 Alice Hoy 330
  
  Nora Ganter (University of Melbourne) 196884 = 196883 + 1
  
  Abstract: will give an introduction to the subject of Moonshine, named by Conway after a seemingly crazy observation by McKay concerning the Monster finite group and modular forms. The original Moonshine conjectures, formulated by Conway and Norton, were proved by Borcherds and won him the Fields medal. However, the geometry underlying the picture is still a bit of a mystery. A generalization of the original conjecture, known as "generalized Moonshine conjecture" (now also mostly proved) sheds some light on what this picture will be. I will explain my interpretation of Norton's generalized Moonshine conjecture and explain why it is natural to use power operations in elliptic cohomology to rephrase some of the basic concepts of Moonshine. Time permitting, I will conclude with some questions.
  
  
• 2 September 2008 12:00 Russell Love
  
  Thomas Forster (University of Cambridge) Quine's New Foundations For Mathematical Logic
  
  Abstract: In 1937 Quine published an article (Reprinted many times in successive editions of *From a Logical Point of View*) in which he presented a system of set theory ("NF") based on Rusell's theory of Types. It looks like a straightforward enhancement but the situation is subtle and the consistency of NF clearly doesn't follow from the consistency of Russell's system. The consistency of NF is probably the oldest open problem in set theory. I shall sketch the history, explain why the problem is still open, and explain some partial positive results.
  
  
• 1 September 2008 13:00 Old Geology 2
  
  Peter Forrester(University of Melbourne) Interlacing variables in random matrix theory
  
  Abstract: I'll review a result of Baryshnikov in queueing theory which relates to the minors of a random Hermitian matrix. I'll then go on to show how the point process defined by other natural sequences of interlaced eigenvalues appears in some statistical mechanical problems relating to tilings.
  
  
• 1 September 2008 10:00 Alice Hoy 330
  
  Peter Milley(University of Melbourne) Mom-technology and small hyperbolic 3-manifolds
  
  Abstract: A "Mom-structure" is a type of cellular complex inside a hyperbolic 3-manifold, with connections to both the concept of spines and the concept of canonical triangulations. In this talk I will define Mom-technology and discuss how it recently has been used to solve two open problems in 3-manifold topology: finding the minimum volume hyperbolic 3-manifold (by Gabai, Meyerhoff, and myself), and finding the maximal number of exceptional Dehn surgeries (by Lackenby and Meyerhoff).
  
  

August 2008 Seminar Blog

• 26 August 2008 10:00 Architecture 103
  
  Roger Behrend (University of Melbourne) Counting alternating sign matrices with a fixed number of osculations
  
  Abstract: Alternating sign matrices of size n correspond to certain configurations of n osculating paths. In this informal talk, I'll show that the number of these with k osculations is a particular polynomial in n of degree 2k.
  
  
• 26 August 2008 10:00 Architecture 103
  
  Murray Elder (University of Queensland) Amenability and percolation for infinite graphs
  
  Abstract: In this talk I will introduce two properties of infinite graphs: being amenable and percolation. I will give lots of examples, and outline some of the major results about them, open problems, and connections to different fields.
  
  
• 25 August 2008 13:00 Old Geology 2
  
  Iwan Jensen (University of Melbourne) Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
  
  Abstract: We calculate very long low- and high-temperature series for the susceptibility $\chi$ of the square lattice Ising model as well as very long series for the five-particle contribution $\chi^{(5)}$ and six-particle contribution $\chi^{(6)}$. The series for $\chi$, $\chi^{(5)}$ and $\chi^{(6)}$ are now extended to 2000 terms. The analysis of these very long series shows an accumulation of singularities in $\chi$ (arising from individual n-particle contributions $\chi^{(n)}$) and lends strong support to the notion that $\chi$ has a natural boundary in the complex plane. We find the presence of singularities at $w=1/2$ for the linear ODE of $\chi^{(5)}$, and $w^2=1/8$ for the ODE of $\chi^{(6)}$, which are not singularities of the ``physical'' $\chi^{(5)}$ and $\chi^{(6)}$, that is to say the series-solutions of the ODEs analytic at $w=0$.
  
  
• 25 August 2008 10:00 Alice Hoy 330
  
  Craig Hodgson (University of Melbourne) Harmonic deformations of hyperbolic 3-manifolds
  
  Abstract: This talk will give an introduction to my work with Steve Kerckhoff on harmonic deformations of hyperbolic 3-manifolds, and describe some topological applications. In particular, this work gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem, including precise estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling.
  
  
• 19 August 2008 12:00 Russell Love
  
  Hyam Rubinstein (University of Melbourne) Representing maximum classes and sample compression.
  
  Abstract: In statistical learning theory, the aim is to predict a concept from amongst an appropriate class, choosing the best fit to a given sample. A classical result is that a concept class is learnable (in the sense of Valiant) if and only if it has finite "VC dimension", which is a measure of complexity. An interesting question is whether learnability is also equivalent to having a finite size compression scheme. Ben and I have recently shown that every maximum concept class can be represented by a simple hyperplane arrangement, answering a conjecture of Kuzmin and Warmuth. This gives a nice geometric way of showing that maximum classes have compression schemes and the background to it will be described in a talk suitable for a general audience, covering ideas from combinatorics, geometry and topology. This is joint work with Ben Rubinstein.
  
  
• 18 August 2008 10:00 Alice Hoy 330
  
  Arun Ram (University of Melbourne) M_g,n and Fock space
  
  Abstract: In his January lectures at MSRI, Okounkov outlined how to use partition combinatorics and Fock space to give formulas for the coefficients of the M_{g,n} volume polynomial. I shall try to explain this combinatorics and summarize the results of Okounkov-Pandharipande.
  
  
• 12 August 2008 12:00 Russell Love
  
  Reinout Quispel (LaTrobe University) Faithful and Discrete: the Geometric Numerical Integration of Differential Equations
  
  Abstract: Geometric integration is the numerical integration of a differential equation, while preserving one or more of its geometric/physical properties exactly, i.e. to within round-off error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase-space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. The field has tantalizing connections to dynamical systems, as well as to Lie groups. In this talk we will present a survey of geometric numerical integration methods for differential equations. Our aim has been to make the review of interest for a broad audience.
  
  
• 11 August 2008 13:00 Old Geology 2
  
  Jan de Gier (University of Melbourne) Slowest relaxation mode of the partially asymmetric simple exclusion process with open boundaries
  
  Abstract: he asymmetric simple exclusion process (ASEP) is one of the most widely studied stochastic processes. Despite its known integrability, due to an underlying Temperley-Lieb algebra, it was only in 2005 that the transition matrix of the process with open boundaries was diagonalised by the Bethe ansatz method. Due to this result, the (asymptotic) dynamics of the ASEP with asymmetric hopping in the bulk and open boundaries came for the first time within reach of analytic treatment. In this talk I will present recent work on the analysis of the Bethe equations and the smallest eigenvalue of the transition matrix which governs the slowest relaxation mode towards the stationary state. The resulting dynamical phase diagram turns out to be surprisingly rich.
  
  
• 11 August 2008 10:00 Alice Hoy 330
  
  Craig Westerland (University of Melbourne) Hurwitz spaces and string topology
  
  Abstract: Continuing in the theme begun in the previous two weeks with talks by Paul Norbury and Arun Ram, I will talk about moduli spaces of curves, and moduli of branched covers of curves (Hurwitz spaces). The goals are twofold: to study these moduli themselves (with an eye towards computing their homology in some sort of limit), as well as elucidating a little-explored connection with string topology. It is unclear whether we will achieve both goals in the time alotted.
  
  
• 5 August 2008 12:00 Russell Love
  
  Ruth Williams (University of California, San Diego) Stochastic Networks with Resource Sharing
  
  Abstract: Stochastic networks are used as models for complex systems involving dynamic interactions subject to uncertainty. Application domains include manufacturing, the service industry, telecommunications, and computer systems. Networks arising in modern applications are often highly complex and heterogeneous, with network features that transcend those of conventional queueing models. The control and analysis of such networks present challenging mathematical problems. In this talk, a concrete application will be used to illustrate a general approach to the study of stochastic networks using more tractable approximate models. Specifically, we consider a data network model that represents the randomly varying number of flows present in a network where bandwidth is shared fairly amongst elastic documents. This model, introduced by Massoulie and Roberts, can be viewed as a stochastic network with simultaneous resource possession. Elegant fluid and diffusion approximations will be used to study the performance of this model. The talk will conclude with a summary of the current status and description of open problems associated with the further development of approximate models for general stochastic networks. This talk is based in part on joint work with W. N. Kang, F. P. Kelly, and N. H. Lee.
  
  
• 5 August 2008 10:00 Architecture 103
  
  Loretta Bartolini (University of Melbourne) One-sided Heegaard splittings of 3-manifolds
  
  
• 4 August 2008 10:00 Alice Hoy 330
  
  Arun Ram (University of Melbourne) M_g,n, counting and recursions
  
  Abstract: This talk will attempt an elementarification/bigpicturification of the topic of Paul Norbury's talk last week.
  
  
• 31 July 2008 15:15 Alice Hoy 330
  
  Discussion group -- Moduli spaces and branched covers
  
  We'll do some parsing of Paul Norbury's Monday talk.
  
  
• 31 July 2008 14:00 Room 107
  
  Richard Brak (University of Melbourne) More on the plactic monoid
  
  

July 2008 Seminar Blog

• 29 July 2008 13:00 Elisabeth Murdoch Theatre
  
  Glyn Davis (University of Melbourne) External and budget challenges the University faces in 2009
  
  Discussion session with Faculty of Science
  
  
• 28 July 2008 13:00 Old Geology 2
  
  Uwe Schwerdtfeger (University of Bielefeld) Exact solution of two classes of prudent polygons
  
  Abstract: A prudent walk is a walk on the square lattice consisting of nearest neighbour steps, such that in the course the walker never steps towards an occupied vertex, no matter at what distance. In particular, such a walk is self-avoiding. Three subclasses of such walks, called one-, two- and three-sided
  prudent walks, have been solved so far. The first class has a rational generating function. The second was shown to have an algebraic generating function by E. Ducci in 2005. Recently, Mireille Bousquet-Melou solved these two classes and the third, whose generating function turns out to be non-holonomic. In this talk we study polygon versions of the two- and three-sided walks, meaning walks ending at a vertex adjacent to their starting point. Similar to the walk cases, we find an algebraic generating function for the former and a non-holonomic one for the latter class.
  
  
• 25 July 2008 11:30 Room 107
  
  Geordie Williamson (University of Freiburg) Knot homology and equivariant homology
  
  Abstract: Khovanov-Rozansky link homology is a categorified knot invariant: starting with a knot or link one obtains triply-graded "homology groups" whose dimensions are invariants of the knot or link. Moreover, taking an Euler characteristic yields the HOMPFLYPT polynomial. I will begin by sketching one way of constructing this invariant due to Khovanov (using ideas of Rouquier). I will then explain ongoing joint work with Ben Webster which offers a geometric interpretation of Khovanov's construction in terms of equivariant cohomology.
  
  
• 22 July 2008 10:00 Room 107
  
  Alexander Kleshchev (University of Oregon) Introduction to cyclotomic Hecke algebras
  
  Cyclotomic Hecke algebras are quotients of affine Hecke algebras.
  
  
• 18 July 2008 13:00 Room 107
  
  Alexander Kleshchev (University of Oregon) Introduction to Whittaker modules
  
  Whittaker modules are natural generalizations of Verma modules.
  
  
• 22 July 2008 12:00 Russell Love
  
  Loretta Bartolini (University of Melbourne) One-sided Heegaard splittings of 3-manifolds
  
  Abstract: Heegaard splittings, a key area of 3-manifold theory, deal with one of the difficulties in the study of 3-manifolds: visualization. This technique gives a convenient way to assimilate a 3-manifold for the human mind, by splitting it into smaller 'visible' pieces. Whilst the traditional field of two-sided Heegaard splittings decomposes a 3-manifold into two pieces, having split along an embedded two-sided surface, the one-sided analogue - a one-sided Heegaard splitting - splits the manifold instead along an embedded non-orientable surface, resulting in a single piece. I will describe this technique through concrete examples, making it accessible to those without a background in the field. In the context of two-sided splittings, there are obvious analogues to be pursued in the one-sided case. I will discuss some of these, illustrating both similarities and differences, as established in my thesis.
  
  
• 18 July 2008 10:30 Room 107
  
  Alexander Kleshchev (University of Oregon) Introduction to W-algebras
  
  Abstract: We will try to motivate the study of finite W-algebras as a natural part of Lie theory connected to many other classical objects, such as Whittaker modules, primitive ideals, canonical bases, polynomial representations of GL(n). We will explain how the category of finite dimensional modules over a W-algebra of type A categorifies polynomial representations of GL(n) and sketch a higher level Schur-Weyl duality between finite W-algebras and (degenerate) cyclotomic Hecke algebras.
  
  W-algebras are generalizations of the Virasoro algebra and are useful in conformal field theory.
  
  Alexander Kleshchev revolutionised the modular representation theory of the symmetric group
  when he proved the rule for restricting a simple S_n module to S_{n-1}. This rule later was connected to the theory of crystals and to Fock representations. He is an expert in the fast moving theory of W-algebras and we can expect to learn a lot from him during his visit.
  
  We will plan to go to Thresherman's for lunch after his Friday talks.
  
  
• 15 July 2008 13:00 Room 107
  
  Richard Brak (University of Melbourne) The plactic monoid.
  
  This talk summarised the chapter of Lascoux in the new edition of Lothaire's book Combinatorics on words. There is some summary of the relationship between this story and crystals in the paper Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux. Omar Foda pointed out that the map that realises the crystal isomorphism between the product B(\mu)\otimes B(\nu) and B(\nu)\otimes B(\mu) is of interest. Peter Tingley is an expert in these crystal commutors.
  
  Section 5 of math.CO/0408114 will be the closest to the point of view presented in the talk, but other references of interest may be
  • arXiv:0707.2248 The crystal commutor and Drinfeld's unitarized R-matrix. Joel Kamnitzer, Peter Tingley. AIM 2007 - 36. math.QA (math.CO).
  • math/0610952 A definition of the crystal commutor using Kashiwara's involution. Joel Kamnitzer, Peter Tingley. AIM 2006-79. math.QA (math.CO).
    
  • math.CO/0408114 The octahedron recurrence and gl(n) crystals. Andre Henriques, Joel Kamnitzer. math.CO (math.QA).
    
  • math.QA/0406478 Crystals and coboundary categories. Andre Henriques, Joel Kamnitzer. math.QA (math.CO math.CT).
    
  
  
• 14 July 2008 13:00 Room 213
  
  Isabel Hubard (University of Auckland) Abstract polytopes and their symmetries
  
  Abstract: Abstract polytopes are combinatorial structures that generalize the classical concept of convex polytopes. Of particular interest to us will be polytopes with a high degree of symmetries. In this talk we shall give an introduction to the theory of abstract polytopes and take a detailed look at the automorphism groups of regular and two-orbit polytopes.
  
  This talk left me thinking that we must try to understand the classification of reflection groups in terms of its maximal reflection subgroups is. This classification should be understood in the context of a cohomology theory or obstruction theory along the lines of the exitence/classification of p-compact groups that one finds in Anderson, Grodal, Moller, et al.