Pure maths seminars
Seminars will be held in Room 15.206 at 12:30pm unless otherwise noted. Seminars will be 45 minutes long, followed by discussion. The material should be accessible to an audience with an advanced background in Pure Mathematics.
Tuesday, 1 July 2008: Prof Neil Trudinger, Australian National University
On nonlinear PDE with supplementary ellipticity
2.30pm in Rm 15.113
Abstract: [not available] Nonlinear PDE with supplementary ellipticity include some with applications to geometric optics, conformal geometry and optimal transportation.
Thursday, 23 May 2008: Prof. Mark Tomforde of the University of Houston
Vector spaces with an order unit
10.30am in Rm 15.113
Abstract: Motivated by operator systems, we discuss ordered vector spaces with order units. We will first describe what happens in the real setting, and talk about the difficulty involved in forming quotients of Archimedean ordered spaces. We will then talk about complex vector spaces and discuss how a theory of ordered *-vector spaces may be developed. In the complex setting, we will see that the situation is more ramified, and in particular a number of different norms must be considered to understand the theory.
Thursday, 27 March 2008: Glen Wheeler, UoW
Higher Order Geometric Evolution Equations
11.30am in Rm 15.107
Abstract: [not provided]
Friday, 7 March 2008: Prof Michael Cwikel, Technion-Israel Institute of Technology
Interpolation theory in Banach spaces
10.30am in Rm 15.113
Abstract: [in Tex] The problem that we are struggling with goes back to Alberto Calerón's celebrated and beautiful work on his complex interpolation spaces $\left[ A_{0}, A_{1} \right]_{\theta}$, and is now about 44 years old: Suppose that $T$ is a linear operator such that $T: A_{0} \rightarrow B_{0}$ is compact and $T: A_{1} \rightarrow B_{1}$ is bounded, or even compact. Despite many partial results, including some quite recent ones, we still do not know whether, in general, $T: \left[ A_{0}, A_{1} \right]_{\theta} \rightarrow \left[ B_{0}, B_{1} \right]_{\theta}$ is compact.
Unless the audience prefers otherwise, I will briefly recall some of the history and applications of interpolation theory and the definitions and some of the basic relevant facts about Calderón spaces. I will also indicate various connections with Fourier series. Indeed Fourier series are apparently a significant part of the arsenal we have for attacking this problem. (Svante Janson has characterised Calderón's spaces via sequence spaces of Fourier coefficients, and Fedor Nazarov has used Fourier series to give an ``almost counterexample'' to a closely related question.)
I will also briefly mention an application of a partial solution of Calderón's problem to Miscible Oil Recovery.
Some background about these things can be found at
\url{http://www.math.technion.ac.il/~mcwikel/compact}
Thursday, 14 February 2008: Rachael Bunder, UoW
Space curves and calculus of variations
11.30am in Rm 15.113
Abstract: In classical mechanics one often has a known energy functional along a curve which one seeks to extremise. In this talk we consider more generally the problem of minimising a general energy functional defined on an arbitrary space curve. We outline the relevant theory from geometry and calculus of variations, indicate how the relevant Euler-Lagrange equations are derived and briefly provide some classes of energies for which circular helices appear as extremal curves.
Thursday, 20 December 2007: Ben Chad, Oxford University
Consistency results concerning two-point sets
11.30am Rm 15.113
Abstract: A subset of the plane is said to be a two-point set iff it meets every line in exactly two points. We will discuss recent work which studies the isometry groups of two-point sets, and discuss two (relative) consistency results which have arisen. In particular, we will show that it is consistent with ZFC that there exists a two-point
set contained in a countable union of circles.
Thursday 20 September 2007: Dr James Parkinson, University of Sydney
Path combinatorics and retractions in the affine flag variety
11:00am (50 minute talk) Rm 15.113
Abstract: In this talk we demonstrate how the set of \textit{labeled Littelmann paths} occurs naturally as an index set for the points of the \textit{affine flag variety} $G/I$, where $G=G(k((t)))$ is a loop group, and $I$ is an Iwahori subgroup of $G$. This gives a refinement of the Iwahori decomposition of $G$, breaking down a double coset $IwI$ into cells indexed by the possible \textit{shapes} taken by paths of \textit{type} $w$. There are applications to \textit{Mirkovi\'{c}-Vilonen cycles}, the realisation of crystals of representations, presentations of the \textit{affine Hecke algebra} (Iwahori and Bernstein presentations), and computing retractions in the \textit{affine building}. In the talk we will discuss this last application, and possibly more if time permits.
Thursday 13 September 2007: Dr Florica Cirstea, ANU
Large solutions of elliptic equations with a weakly superlinear nonlinearity
11:30am (50 minute talk) Rm 15.113
Abstract: We study the asymptotic behavior near the boundary for large solutions of the semilinear equation $\Delta u+au=b(x)f(u)$ in a smooth bounded domain $\Omega$ of $\RR^N$ with $N\geq 2$, where $a$ is a real parameter and $b$ is a non-negative smooth function on $\overline{\Omega}$. We assume that $f(u)$ behaves like $u(\ln u)^\alpha$ as $u\to \infty$, for some $\alpha>2$. It turns out that this case is more difficult to handle than those where $f(u)$ grows like $u^p \; (p>1)$ or faster at infinity. Under suitable conditions on the weight function $b(x)$, which may vanish on $\partial\Omega$, we obtain the first order expansion of the large solutions near the boundary. Some uniqueness results are also provided. This is joint work with Y. Du.
Thursday 16 August 2007: Professor Ian Doust, UNSW
Conditional and unconditional decompositions of noncommutative $L^p$ spaces
11:30am (50 minute talk) Rm 15.113
Abstract: In a number of problems in different areas of analysis it has been shown that the condition that one needs is not that a certain set be bounded, but rather that it be ‘$R$ bounded'. This concept has been used, for example, in recent important work of Kalton and Weis and of Le~Merdy, concerning $H^\infty$ functional calculus and maximal regularity, and of Celement, de~Pagter, Sukochev and Witvliet on decompositions of Banach spaces.
In this talk I will show how this concept is used in to proving some theorems that can be thought of, either as concerning particular types of decomposition of Banach spaces, or else concerning particular types of functional calculus. As particular applications of these theorems we get analogues of classical multiplier theorems of harmonic analysis, but now acting on the von Neumann-Schatten spaces $S_p$ rather than on the Lebesgue spaces $L^p$. This is joint work with T.A. Gillespie.
Thursday 9 August 2007: Dr Frithjof Dau, UoW
A Diagrammatic Approach to Formal Logic with Peirce's Existential Graphs
11:30am (50 minute talk) Rm 15.113
Abstract: In the last 20 years interest in visual representations of logic has emerged in different fields (Mathematics, Computer Science, Linguistics and psychology). In knowledge representation there are increasing efforts to elaborate visual representations of mathematical logic. In this talk the system of Existential Graphs (EGs) by Charles Sanders Peirce (1839-1914) is presented. Peirce's original EGs were not developed in a manner suited to contemporary mathematical standards in formal logic. Peirce's system can be understood to be the first elaboration of diagrammatic first order logic, including a model-theoretic semantics and a sound and complete diagrammatic calculus (two properties that did not hold for Frege's Begriffsschrift developed two decades earlier).
In this talk, an introduction into Peirce's original system is given, including Peirce's inference rules which are very different to contemporary calculi for first order logic. The speaker will then show how Peirce's graphs can be elaborated in a precise manner. If the remaining time some proof-theoretic properties of Peirce's rules will be presented.
Monday 5 August 2007: Prof John Quigg, Arizona State University
Coactions: a curious category (Operator Algebra Seminar)
2.30pm (50 minute talk) Rm 15.113
Abstract: Within the category of coactions of a locally compact group G on C*-algebras, there are two subcategories which have recently become important: the maximal coactions and the normal coactions, which may be regarded as generalizations of the full and the reduced group C*-algebras. Using some general abstract nonsense, and some standard techniques in noncommutative dynamical systems, we show that these two subcategories are equivalent.
Enquiries to Iain Raeburn (Room 207, phone 4046, email raeburn@uow.edu.au)
Thursday 2 August 2007: Dr Murray Elder, Stevens Inst Tech
Random subgroups of Thompson's group F
11.00am (50 minute talk) Rm 15.113
Abstract: In this talk I will discuss the (new) ideas of random subgroups of groups, where the ideas come from and apply to (cryptography). We place some kind of probability measure on subsets of elements of a group, and compute the relative chance of randomly selecting different kinds of subgroups. With what probability would you get - the trivial group, a free group, the whole group etc? We show that for Thompson's group F some interesting things happen (in contrast to other groups like Braid groups where subgroups are free with probability 1, allegedly).
I will start by defining this group, which has many nice combinatorial incarnations. This talk is joint work with Sean Cleary, Andrew Rechnitzer and Jennifer Taback
Thursday 2 August 2007: A/Prof Jacqui Ramagge, UoW
A totally disconnected perspective on groups
1:30pm (50 minute talk) Rm 15.113
Abstract: This will be an introductory talk on the theory of totally disconnected locally compact groups. Such groups have significant applications within group theory, in the theory of random walks and in computer science because of the connection to automorphism groups of graphs.
The talk will cover recent developments in the theory, mainly due to George Willis and various collaborators. His fundamental observation is that automorphisms of totally disconnected locally compact groups can be equipped with data analogous to eigenvalues and eigenspaces even though there is no vector space structure. My contributions may appear at the end but are more likely to be the topic of a follow-up talk.
Thursdays 7, 14 and 21 June 2007: Dr Nirmalendu Chardhuri, UoW
Calculus of Variations and Microstructures (series of three talks)
9:30am on 7th, others at 11.30am (50 minute talk) Rm 15.107
Abstract: The study of minimum energy solutions and their smoothness is at the heart of Calculus of Variations. The scalar case, (the dimension of the domain or the range of the deformations under consideration is one) is well understood, thanks to Hilbert, Tonelli, De Giorgi, and many other great modern mathematicians. In his landmark 1952 paper, C. B.
Morrey introduced the so-called "quasiconvex nonlinearity" as a necessary and sufficient condition for the existence of minimum-energy solutions in multi-dimensional variational models. Understanding the notion of quasiconvexity is still an outstanding open problem in mathematical analysis. In this series of three lectures, I wish to discuss the basic issues and latest developments in direct and indirect methods of multi-dimensional calculus of variations.
I will also discuss the wildly oscillating solutions and microstructures via Young measures and regularity of system of nonlinear partial differential equations. The first lecture will be of introductory nature, the second lecture will on quasiconvexity and Young measures, and the final lectures will be on geometric rigidity and microstructures.
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Thursday 8 May 2007: Dr Aidan Sims, UoW
Higher-rank graphs and their C*-algebras
2:30pm (50 minute talk) Rm 15.113
Abstract: In two previous seminars, Iain Raeburn and David Pask have introduced graph C*-algebras and discussed some of their important properties. In particular David mentioned a dichotomy which holds amongst simple graph algebras: they are either AF or purely infinite.
I will begin my presentation by summarising briefly some key points of Iain's and David's talks. I will then introduce the notion of a higher-rank graph and define the associated C*-algebra. Higher-rank graphs are analogues of directed graphs in which the
notion of the length of a path is replaced by a higher-dimensional shape or degree, and were introduced by David in joint work with Alex Kumjian in 2000. The associated C*-algebras are now receiving significant international attention.
I will spend the bulk of my talk discussing a particular example of a graph of rank two, which appeared in joint work with David, Iain and Mikael R{\o}rdam. This example, denoted $\textrm{BD}(2^\infty)$, shows amongst other things that the dichotomy mentioned above does not hold in the realm of higher-rank graphs and their C*-algebras.
Time permitting, I will discuss a number of ways of thinking of $\textrm{BD}(2^\infty)$ and indicate how each of these has led to new classes of examples of higher-rank graphs. These examples are giving us new insight into just how much broader is the class of C*-algebras associated to higher-rank graphs than the one associated to their 1-dimensional cousins.
Thursday 24 April 2007: A/Prof David Pask, UoW
Looking for simplicity in graph C*-algebras
2:30pm (50 minute talk) Rm 15.113
Abstract: Since some of my audience may not have had the benefit of Iain's first lecture, I shall begin my presentation by giving the definition of a Cuntz-Krieger E-family of operators associated to a directed graph E. The C*-algebra generated by a “universal” Cuntz-Krieger E-family is denoted C* (E), and called a graph C*-algebra. I will describe a couple of informative examples of graph C*-algebras.
A C*-algebra is simple if it has no closed 2-sided ideals. Simple C*-algebras form the basic building blocks in operator algebra theory and so it is important to be able tell whether a given C*-algebra is simple.
For a graph C*-algebra C*(E), the conditions for simplicity can be stated in terms of the properties of the graph E. The huge benefit of this result is that in most cases it is relatively straightforward to see if a graph C*-algebra is simple or not.
It also turns out that a simple graph C*-algebra must belong to one of two important categories of C*-algebras. If time permits I shall explain this dichotomy.
Tuesday 27 March 2007: Professor Iain Raeburn, UoW
Linear operators and the geometry of Hilbert space
2:30pm (50 minute talk) Rm 15.113
Abstract: Several of our group's current research projects involve graph algebras: C*-algebras associated to directed graphs and other combinatorial structures. These algebras have an elegant structure theory which relates properties of the algebra to the behaviour of paths on the underlying graph. In this series of talks, Iain, David and Aidan will give a general introduction to this subject and topics of current interest, starting from what they hope is common ground in elementary functional analysis.
In this first talk, Iain will review the algebraic structure of the set of bounded linear operators on Hilbert space, and will show how geometric properties of operators and subspaces can be encoded algebraically. We will then discuss the Cuntz-Krieger families of operators associated to a directed graph, examine the structure of the algebras they generate, and give examples to show they are likely to be of interest in a variety of areas.
Thursday 22 Feb 2007: Professor Reinhard Wolf, University of Salzburg, Austria
Distance Geometry in Quasihypermetric Spaces
3:30pm (50 minute talk) Rm 15.113
Abstract: Let $(X, d)$ be a compact metric space. Further, define
\[
M(X) = \sup_\mu \int\!\!\int d(x, y) \, d\mu(x) d\mu(y),
\]
where $\mu$ ranges over all signed Borel measures on~$X$ of total mass~$1$. We give a discussion of the properties of the constant~$M(X)$ in the case when $X$ enjoys the so-called quasihypermetric property. For example, we analyse the finiteness of~$M(X)$, its special properties in finite metric spaces, and its relation to the geometry of certain measure spaces.
Biography: Prof. Reinhard Wolf works in the Institute of Mathematics in the University of Salzburg, Austria. His areas of research are functional analysis, metric geometry and discrete mathematics.
Thursday 15 February 2007: Dr Min-Chun Hong, University of Queensland
Anti-self-dual connections and their related flow on 4-manifolds
11:30am (50 minute talk) Rm 15.113
Abstract: In this talk, we establish existence of the maximal time of the smooth solution to the anti-self-dual (ASD) flow in vector bundles over a 4-dimensional compact Riemannian manifold M and present a different proof of the Taubes' existence theorem on anti-self-dual connections on 4-manifolds.
Monday 5 February 2007: Professor Neil Trudinger, Centre for Mathematics and its Applications, Australian National University
On the regularity of optimal transportation mappings
2:30pm (50 minute talk) Rm 15.113
Wednesday, 31 January 2007: Sarah Neville, UoW
A fourth order Euler-Lagrange equation arising in biology
11:30am (30 minute talk) Rm 15.113
Abstract:The unusual biconcave shape of the Red Blood Cell has been researched in both Biological and Mathematical fields. We have approached the problem using the Calculus of Variations. The focus is on minimising the bending energy of the cell's membrane (according to certain constraints), which involves deriving the Euler-Lagrange equation for the problem.
Thursday 7 December 2006: Graham Williams, UoW
The Minimal Surface Equation
11:30am (50 minute talk) Rm 15.113
Abstract: The Minimal Surface Equation is a nonlinear partial differential equation satisfied by functions whose graphs have the least area in some class. For example in the Dirichlet Problem we consider functions which take on prescribed values on the boundary of a domain. In some cases it can be shown that this Dirichlet problem has a solution but in other cases it can be shown that no solution exists. Of course the existence of solutions depends on the nature of the prescribed boundary values but more critical is the geometry of the domain. In this talk we will survey some of the results and the tools used in the proofs.
Friday 10 November 2006: Glen Wheeler, UoW
Existence of solutions to the Dirichlet problem for second order elliptic partial differential equations
11:30am (50 minute talk) Rm 15.113
Abstract:Although applications of partial differential equations originate from the problems of physics, modern times have seen their uses expand. These include solving long-standing problems in other areas, facilitating new fields of mathematics, and imaginative practical applications. As a subset to this, the theory of second order elliptic partial
differential equations is primarily concerned with questions of existence, uniqueness, and
properties of solutions. In this talk I will cover the issue of existence and uniqueness of solutions to the Dirichlet problem (a boundary value problem) for a reasonably general class of linear elliptic differential equations. Emphasis will be placed on understanding the larger picture rather than rigorous attention to technical detail.
Thursday 31 October 2006: Ben Chad
Symmetries of two-point sets
11:00am, Rm 15.113
Abstract: A subset of the plane is said to be a two-point set iff it meets every planar line in exactly two points. The motivating question for this talk is ``What are the topological symmetries of a two-point set?". Our main results assert the existence of rigid two-point sets, and the existence of rotationally invariant two-point sets.
Thursday 19 October 2006: Dongvu Tonien (SITACS), UoW
A mathematical excursion on Diophantine equations
11:30am, Rm 15.113
Abstract: I will talk about Diophantine equations, such as, why the
equations a^2+b^2+c^2+d^2=15^n or a^3+b^3+c^3-3abc=2006^n always have
solutions for any value of n, and how to find all solutions to the
equation a^3+2b^3+4c^3-6abc=1 by the method of descent. Pell's equations, quaternions, norm and trace, etc., will also be mentioned. The talk is intended for general audience.
Thursday 12 October 2006: Professor Martin Bunder, UoW
On binary reflected Gray codes and functions
11:30am (30 minute talk) Rm 15.108
Abstract: A Binary Gray code of length n is a sequence of 2^n distinct strings of 0s and 1s, with the property that each string differs from its successor in only one digit. The sequence codes the integers 0 to 2^n - 1. The binary reflexive Gray code is generated by coding 0 by 00....0 and in a table listing 0 to 2^k - 1 and their codes, reflecting all the codes obtained so far, about a line below 2^k -1 and adding a 1 to the kth place.
The binary reflected Gray code function b is defined by: b(m) is the integer obtained when the initial zeros are omitted from the code of length n for the integer m, for any n>log_2m. This paper examines this function and its inverse and gives new simple algorithms to evaluate both. It also simplifies an expression of Conder for the jth letter of the code of length n for k.
21 September 2006: Paul-James White, UoW
Partial differential equations
11:30am (50 minute talk) Room: 15.108
Abstract: [results of research project supervised by James McCoy and Graham Williams.] I shall be discussing the concept of a weak solution to a PDE and the corresponding method of solution, this shall then be compared to the classical method. In particular the differing approaches shall be illustrated via the solving of Laplace's equation using the weak method.
14 September 2006: Dr James McCoy, UoW
Bernstein properties of solutions to some fourth order partial differential equations
11.30am, Rm15.108
Abstract: Recently we proved a Bernstein property of solutions to a class of prescribed affine mean curvature equations arising in affine differential geometry. In this talk we explore further applications of similar techniques to other fourth order PDEs.
29 June 2006: Dr Murray Elder, UoW
Grigorchuk's group (Joint work with Mauricio Gutierrez and Zoran Sunik)
11.30am, Rm15.108
Abstract: Carrying on from last time, we consider the growth properties of a different example, called the Baumslag-Solitar group (and you thought that stood for something else). This group has a rational generating function for its growth, but John Groves proved that no set of words that gives this growth forms a regular language. So we get into generating functions of formal languages. I proved that it does have a context-free set of words (geodesics). So I'll attempt to describe some of this.
We might also discuss the "Lamplighter groups", yet another cool family (they are wreath products of cyclic groups, but called lamplighter for a reason).
22 June 2006: Dr Murray Elder, UoW
A Bernstein property of solutions to a class of prescribed affine mean curvature equations
11.30am, Rm15.108
Abstract: In this talk I'll define the "growth" of a group, which is a function that describes how many elements the group has. We might be interested in this as a series of numbers, or if they are too hard to nail down, we might just ask are they growing polynomially or exponentially, or otherwise. And if you are in MATH 222, you might want to find a "generating function" for it, and describe that (in terms of rational/algebraic, or otherwise).
After exploring some easyish examples, we'll introduce a great construction of Grigorchuk, of a group described by actions of infinite strings and automata, which turns out to have crazy growth properties. My recent work with Mauricio and Zoran attempts to nail down some specifics about words in this group.
The talk should be accessible to students and staff, and only a vague idea of what a group is is required.
8 June 2006: Dr James McCoy, UoW
A Bernstein property of solutions to a class of prescribed affine mean curvature equations
11.30am, Rm15.108
Abstract: Inspired by recent work of Lin and Jia on a Bernstein property of affine maximal hypersurfaces of dimension n=2 or n=3, we generalise their result to complete n-dimensional hypersurfaces satisfying a related fourth order Monge Ampere-type fully nonlinear PDE. We first illustrate the key techniques with a new generalisation of the classical result of Yau that an entire harmonic function on a complete Riemannian manifold of nonnegative Ricci curvature must be identically constant. We then discuss the extra ingredients for the fourth order case.
21 April 2006, 2.00pm in Rm15.206: Prof Rudolf Wille (Darmstadt)
Does a continuum consist of points?
Abstract: The famous mathematician Herman Weyl claimed that a continuum does not consistent of points. For him points are only "swimming in a space sauce". Aristotle had already argued that the time continuum does not consist of time points. In this lecture it will be shown using method of formal concept analysis that linear continua without points can be mathematically modelled and completed by points as bounds of sub continua.
About the speaker: Rudolf Wille is Professor Emeritus in Mathematics from The Technical University of Darmstadt in Germany. From 1970 until 2003 Wille was Professor of Mathematics in Darmstadt. Author of more than 200 articles and books, Prof. Wille has supervised more than 50 PhDs to completion and created the field of "formal concept
analysis", a branch of applied lattice and order theory.
11 April 2006, 2.30pm in Rm15.113: Dr Nirmalendu Chaudhuri (ANU)
Generalized Liouville Theorem and its application to thin martensitic films
Abstract: The classical Liouville theorem asserts that if the gradient of a vector field V is close to the rotation group SO(n), V is close to an affine map. In this lecture we will prove that if the gradient of a deformation is close to two incompatible copies (does not differ by a matrix of rank-one) of SO(n), then the deformation is close to a rigid motion. Such a result can be viewed as the nonlinear version of the well-known Korn's inequality in linear elasticity. As an application we will also discuss the scaling behaviour of materials undergoing phase transformations. Our proof is mainly based on the regularity of system of elliptic PDEs together with fine covering arguments. This is a joint work with Stefan Mueller.
2 March 2006, 12.30 pm in Rm3.121: Dr Murray Elder (SMAS)
An introduction to pattern-avoiding permutations
Abstract: In "combinatorics" we start with some set of objects, like paths on a
grid of length n, or permutations of length n, or graphs with n edges
or vertices - with some (natural) restrictions - and we count the
number of such things for each n. Then we look at the sequence of
integers we get, see how they relate to others, and try to find
compact formulas or descriptions for them.
For example, count the number of arrangements of n ('s and n )'s, so
that the brackets are "balanced". So for length 2 we get (()) and () (). For length 3 we get: ((())), ()()(), (())(), ()(()), (()()).
In the past couple of years there has been a lot of interest in
counting sets of permutations with some restrictions that I will
describe in the talk (they avoid some patterns). Two years ago Marcus
and Tardos resolved a longstanding conjecture of Stanley and Wilf,
and last year Albert, me, Rechnitzer, Westcott and Zabrocki knocked
over a conjecture of Arratia. I will try to explain these new theorems.
My most recent work involves permutations that are generated by
passing an ordered sequence through some series of stacks. Knuth
looked at this for one stack and came up with a nice answer. More
than one stack and things get interesting.
No specific background will be assumed, and computer scientists,
applied mathematicians and everyone else is most welcome to attend.
15 February 2006, 11.00am in Rm15.113: Dr Nirmalendu Chaudhuri (ANU)
On twice differentiability of certain non-convex functions
Abstract: A classical theorem of Alexsandrov asserts that convex functions in R^n
are twice differentiable almost everywhere. We will discuss the twice
differentiability property for a larger class of non-convex functions,
namely the subharmonic functions with respect to the Pucci minimal
operator. As a consequence, we will see that the so-called k-convex
functions are twice differentiable almost everywhere for k>n/2.
15 December 2005: Amos Koeller, from Frei Universitat Berlin
Consequences of the approximate j-dimensionality of minimal surface singularity sets
11am in Rm15.113
5 December 2005: Dr James McCoy, UoW
A Bernstein property of affine maximal hypersurfaces
11am in Rm15.108
19 October 2005, 12.30pm in Rm15.206: Professor Bruce Berndt, Department of Mathematics, University of Illinois
Ramanujan's Forty Identities for the Rogers-Ramanujan Functions
Abstract: The Rogers-Ramanujan identities are among the most famous identities in combinatorics. Late in his stay in England, Ramanujan derived forty further identities relating the two Rogers-Ramanujan functions at different arguments. Although almost all of the forty identities have now been proved, principally by L. J. Rogers, G. N. Watson, D. Bressoud, and A. J. F. Biagioli, an impenetrable fog still lies over the ideas which led Ramanujan to derive these identities. In the past four years, the speaker and several of his former doctoral students have been attempting to find proofs in the spirit of Ramanujan's mathematics. At this moment, we have proofs of thirty-five of the identities that could have been given by Ramanujan. In this mostly expository talk, the various methods that have been used to prove the forty identities are discussed. It is possible that Ramanujan used asymptotic analysis of the Rogers-Ramanujan functions to discover, but not to prove, the identities. We also describe the ideas behind this approach.
15 September 2005, 12.30pm in Rm15.111: Dr James McCoy, UoW
Mixed Volume Preserving Curvature Flows
The talk should be of interest to members of the School at large, and all are welcome.
Abstract:
We consider a class of fully nonlinear, parabolic evolution equations for compact, strictly convex hypersurfaces in Euclidean space. The speed of the evolving surfaces consists of a function which is positive, monotone, homogeneous of degree one in the principal curvatures, balanced by a global term which fixes any particular mixed volume under the flow. We show that the evolution has a smooth solution for all time which converges
exponentially to a sphere. This work generalises the author's earlier results for mixed volume preserving mean curvature flows. Special cases of the flow can be used to re-prove the Minkowski inequalities of convex geometry.
4 August 2005: Prof. Reinhard Wolf, Institute of Mathematics, University of Salzburg, Austria
Average Distances in Compact Metric Spaces
We will start by giving a survey of some known results on certain constants relating to average distances amongst sets of points in compact metric spaces. In the main part of the talk we will give an introduction to some new results in this area which have been developed with Peter Nickolas since his visit to Salzburg last year.
7 July 2005: Frank Prokop
The Role of Multifunctions in the Theory of Relations
A multifunction is essentially a relation with fixed "domain". Since multifunction can be represented as the union of functions, it will be shown that there is a class of simple multifunctions which behave in many respects like functions. Further, I will attempt to show that multifunctions are a useful tool in providing an alternate view of topics that vary from simple things like the Implicit Function Theorem and implicit differentiation to topological continuity. If things go 'well' I would hope to get to an introduction to a simple ' work which was at one time in progress' approach to differentiable multifunctions.
23 June 2005: Rod Nillsen
The Fourier transform and the expression of functions as sums of finite differences
Let $L^2(\Real)$ denote the space of square integrable functions on $\Real$ and let $\widehat f$ denote the Fourier transform of $f\in L^2(\Real)$. Let $\delta_x\ast f$ denote the translation of $f$ by $x$ (that is, the graph of $f$ is moved through a distance $x$). Then a function $f$ in $L^2(\Real)$ is called a {\it first order difference} if there are $x \in \Real$ and $g \in L^2(\Real)$ such that $$ f= g-\delta_x\ast g.$$ Then, for $f\in L^2(\Real)$, $$\int_{-\infty}^{\infty}{|{\widehat f}(x)|^2\over |x|^2}\ dx<\infty$$ if and only if $f$ is a finite sum of first order differences, and this characterizes when a function is the derivative of a function in the Sobolev space of order $1$. In this case, $f$ is a sum of $3$ first order differences and this estimate $3$ is sharp. In general, for $f\in L^2(\Real)$ $$\int_{-\infty}^{\infty}{|{\widehat f}(x)|^2\over |x|^{2s}}\ dx<\infty$$ if and only if $f$ is a finite sum of $2s+1$ ``differences of order $s$". In this talk, the background of this area will be addressed, with the main problem to be discussed being when the estimate of $2s+1$ is sharp.
The abstract once processed by LateX (in .pdf format)
9 June 2005: Pam Davy
Harmonic Hypercubes
Finite binary sequences are of fundamental interest in many areas of mathematics, statistics and computer science. Such sequences can be interpreted in many ways, such as vertices of a hypercube, nodes of a tree or points of a lattice. Two-dimensional visualisation of hypercubes of dimension n > 3 is not a natural task for the inhabitants of a 3-dimensional world. This talk will present some unusual graphical representations of binary sequences based on harmonic numbers. The resulting graphs raise some interesting problems in number theory.
26 May 2005: Keith Tognetti
The Search for the Golden Sequence
When some patterns of points are variously stretched , folded and interlaced, the resultant metamorphoses can display most interesting self similarity properties . This is an attempt to describe the journey I have taken in analysing such processes in a search for the blueprint for the formation of discrete shapes in nature such as with trees , leaf arrangements and seed packing . The hope is that one will stumble onto the discrete analogue of the logarithmic spiral which is the blueprint for constructing continuous natural structures such as sea shells.
My journey has taken me off the main path into various byways( a thicket of broken sticks dearly loved by economists- heaps of folded paper ) but mainly it has been associated with the arrangement of points around a circle so that they are spread out evenly( which does not mean that they are necessarly uniformly distributed) . The latter part of the seminar will attempt to describe the self similarity of integer part sequences and their relationship to Moire patterns . A claim will be made that these self matching patterns represent the simplest discrete analogue to the stretching and folding within chaos .
12 May 2005: Martin Bunder
An Introduction to Nonclassical, and in particular, Rough Consequence Logics
Many will be familiar with classical propositional logic, as based on truth tables. There are many restrictions and extensions of this logic that can't be based on truth tables and have to be based on axioms and rules of inference. The talk will briefly discuss a number of such logics (intuitionistic logic,paraconsistent logics, relevant logics, BCK, BCI, BCIW logics, modal logics, multivalued logics, predicate logics) and their applications in mathematics and computer science. We will look in more detail at rough consequence logic. We look at variants of the rule:
If A and B->C hold and A and B are roughly equivalent, then C,
and how the logics that stem from these are related.
28 April 2005: Peter Nickolas
Average Distances in Euclidean and Other Spaces, Part 2
\bigskip Let $(X, d)$ be a compact metric space. If $x_1, \ldots, x_n \in X$ and $w_1, \ldots, w_n$ are non-negative real weights adding to~$1$, then $\sum_{i,j} w_i w_j d(x_i, x_j)$ is the corresponding \emph{weighted double average} of the mutual distances amongst the~$\{x_i\}$. We define $M(X)$ to be the supremum of all such double averages. Further, we define $\overline{M}(X)$ identically, except that the weights, though still adding to~$1$, may now be arbitrary real values. The properties of $M(X)$ are rather good (though its value is usually hard to calculate), but the properties of $\overline{M}(X)$ in contrast are quite pathological. This contrast, and other fascinating properties of $M(X)$ and $\overline{M}(X)$, can be explained fairly well by reference to the geometric properties of~$X$ and topological properties of certain subspaces of the space of finite Borel measures on~$X$. The talk centres around these ideas. Few proofs will be given; the emphasis will be on the overall shape of the theory. This talk is billed as Part~2 of a pair, of which the first part was given a fortnight ago by Ben Chad. In fact, however, this talk will be largely independent of Ben's. This means that you should not feel that you must have attended Ben's talk to attend this one; it will be comprehensible or otherwise in its own right.
14 April 2005: Ben Chad
Average Distances in Euclidean and Other Spaces, Part 1
Let $X$ denote the circle of diameter 1. Given any finite collection of points $x_i \in X$, it can be shown that there exists $y \in X$ such that the average distance from $y$ to the $x_i$ is $2/\pi$. Further, $2/\pi$ is the only ``magic number" (particular to X) for which this works! If we consider the mutual average distances amongst finite sets of points in $X$, it can be shown that the supremum of such averages is again $2/\pi$. By generalising the space $X$, and using different notions of the term ``average distance", we will discuss such properties of metric spaces within a measure theoretic framework.
17 March 2005: Eugenio Fedriani Martel (Visiting fellow at UOW, Associate Professor at Pablo de Olavide University, Sevilla)
A First Approach to Topological Graph Theory
We begin with a brief summary of the most basic definitions in Graph Theory directed to the introduction of the notion of a graph drawing, or embedding, which implies that its vertices are represented by points in the Euclidean plane, its edges are represented by curves between these points, and different curves meet only in common endpoints. We also explain the concept of outerplanarity and its generalizations, giving some recent results about these concepts.
3 March 2005: Adam Piggott
Todd--Coxeter Using Graphs
The Todd--Coxeter Coset Enumeration Procedure is a cornerstone of computational group theory. We discuss an illuminating geometric interpretation and some applications.
Abstract: The subject of optimal transportation goes back over two hundred years to the Monge problem of moving soil from one place to another with the least amount of work. In recent years, its theory has blossomed in the wake of a diverse range of applications to areas such as astronomy, economics, image processing and meteorology, with new applications looming in biology and computer networks. In this talk we briefly survey the area and present sharp conditions on cost functions and domains so that the resultant optimal mappings are smooth.
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