Plenary speakers
Plenary Speakers
|
Abstracts |
|
| Clay Lecturer Mohammed Abouzaid (MIT) | Functoriality in Homological Mirror Symmetry |
| Werner Ballmann (Bonn/MPI Bonn) | Index theory on non-compact manifolds |
| Clay Lecturer Danny Calegari (Caltech) | |
| Michael Eastwood (ANU) | Re-inventing the wheel: differential operators on the sphere |
| Dennis Gaitsgory (Harvard) | Quantum Geometric Langland Program |
| Ezra Getzler (Northwestern) | n-groups |
| Kerry Landman (Melbourne) | Simulating a developmental cell invasion process |
| DSTO Lecturer Bill Moran (Melbourne/NICTA) | Golay-Rudin-Shapiro: A rose by any other name! |
| Jacqui Ramagge (Wollongong) | |
| Clay-Mahler Lecturer Terence Tao (UCLA) | The proof of the Poincare conjecture |
| ANZIAM Lecturer Peter Taylor (Melbourne) | Some interesting problems in modelling ad hoc mobile networks |
| Early Career Lecturer Akshay Venkatesh (Stanford) | The Cohen-Lenstra heuristics over global fields |
Plenary Abstracts
Clay Lecturer Mohammed Abouzaid (MIT)
Mohammed Abouzaid received a B.S. in 2002 from the University of Richmond, and a Ph.D. in 2007 from the University of Chicago under the supervision of Paul Seidel. He is interested in symplectic topology and its interactions with algebraic geometry and differential topology, in particular the homological mirror symmetry conjecture. Since 2007 he has been a postdoctoral fellow at MIT, and a Clay Mathematics Institute Research Fellow.
Functoriality in Homological Mirror Symmetry
Kontsevich's original version of the Homological Mirror
Symmetry Conjecture was a statement about pairs of Calabi-Yau manifolds,
with no indication of any connection between mirrors of varieties which are
related to each other. I will describe recent progress which reveals
situations in which Homological mirror symmetry exhibits more "functorial"
properties. This conjectural functoriality is clearest for the case of the
inclusion of an anticanonical divisor in a Fano variety.
The talk will focus on examples starting in dimension 1, and on explaining
the geometric source of these phenomena.
Werner Ballmann (Bonn/MPI Bonn)
Werner Ballmann studied of mathematics with Diplom, 1976 and Promotion,
1979 at the University of Bonn. Habilitation, University of Bonn 1984.
Scientific Assistant, University of Bonn 1979-1985. Research Scholarship (DFG),
University of Pennsylvania 1980-1981. Associate Professor, University of
Maryland 1984-1986. Associate Professor, University of Bonn 1986-1987. Full
Professor, University of Zürich 1987-1989. Full Professor, University of
Bonn since 1989. Scientific Member and Director, MPI for Mathematics since
2007.
Index theory on non-compact manifolds
Elliptic differential operators on vector bundles over compact manifolds are a central object of classical global analysis, as for example in Hodge theory and in the original Atiyah-Singer index theorem. In their seminal work on boundary value problems for Dirac operators, Atiyah, Patodi, and Singer also consider manifolds with cylindrical ends. Their work marks the beginning of a whole series of studies on relations between the geometry of ends and properties of geometric operators over non-compact spaces. In the talk, I will explain some recent work on index theory for Dirac operators on manifolds with cusp-like ends.
Clay Lecturer Danny Calegari (Caltech)
Danny Calegari is the Richard Merkin Professor of Mathematics at the California Institute of Technology, and is one of the recipients of the 2009 Clay Research Award for his work in geometric topology and geometric group theory. He received a B.A. in 1994 from the University of Melbourne, and a Ph.D. in 2000 from the University of California, Berkeley under the joint supervision of Andrew Casson and William Thurston. From 2000–2002 he was Benjamin Peirce Assistant Professor at Harvard University, after which he joined the Caltech faculty; he became Richard Merkin Professor in 2007.
Faces of the scl norm ball
It often happens that a solution of an extremal problem in geometry has more regularity and nicer features than one has an a priori right to expect. I will show how a simple topological problem - when does an immersed curve on a surface bound an immersed surface? - is unexpectedly related to linear programming in nonseparable Banach spaces, and gives rise to geometric and dynamical rigidity and discreteness of symplectic representations.
Michael Eastwood (ANU)
Michael Eastwood received a BA in Mathematics from the University of Oxford in 1973 and a PhD from Princeton University, as a Fulbright Scholar, in 1976. He is currently an Australian Research Council Federation Fellow at the Australian National University and prior to that an Australian Research Council Professorial Fellow at the University of Adelaide. His primary research interests are in differential geometry, integral geometry, complex geometry, and the representation theory of Lie groups.
Re-inventing the wheel: differential operators on the sphere
The usual mathematical `wheel' is the circle acted upon by the rotation group SO(2). There are plenty of natural differential operators on this usual circle, manufactured from the basic d/dθ. But the circle is also acted upon by SL(2,R) and this larger symmetry group cuts down the list of natural differential operators to something smaller but more interesting! I shall explain what happens and how this phenomenon generalises to spheres under various symmetry groups. Some familiar differential operators emerge, such as `div,' `grad,' and `curl' but also some less familiar ones such as the mapping `strain→stress' in linearised elasticity. These constructions are part of a general theory (of the `Bernstein-Gelfand-Gelfand resolution') but they have numerous unexpected applications, for example in suggesting a new stable nite-element scheme in linearised elasticity (due to Arnold, Falk, and Winther).
Dennis Gaitsgory (Harvard)
Dennis Gaitsgory is a Professor of Mathematics at Harvard University known for his research on the geometric Langlands program. He studied Mathematics at Tel Aviv University under Joseph Bernstein. He received his doctorate in 1997 for a thesis entitled "Automorphic Sheaves and Eisenstein Series". Gaitsgory has been awarded a Harvard Junior Fellowship, a Clay Research Fellowship, and the prize of the European Mathematical Society. Prior to his 2005 appointment at Harvard he was an Associate Professor at the University of Chicago from 2001-2005.
Quantum Geometric Langland Program
Let be a reductive group, and let Ğ denote its Langlands dual. The usual (function-theoretic) Langlands program aims to establish a correspondence (ideally, but not really, a bijection) between representations of the Galois group into Ğ of a global field and irreducible automorphic representations of G.
A geometrization suggested by Drinfeld replaces the space of automorphic functions by the category of automorphic sheaves, which are, by defnition, D-modules on the moduli space BunG(X) of principal G-bundles on a given algebraic curve X.
This paves a way to formulating the ambitious "Classical Geometric Langlands Conjecture" that there is an equivalences between the derived categories of D-modules on BunG(X) and that of quasi-coherent shaves on the moduli space of local systems on X.
The Quantum Geometric Langlands Conjecture is a 1-parameter deformation of the classical one. A remarkable feature is that away from the zero value of the parameter, the statement of the conjecture becomes symmetric in G and Ğ: both D-mod(BunG(X)) and QCoh(LocSystG(X)) become replaced by appropriately defined categories of twisted D-modules.
Ezra Getzler (Northwestern)
Ezra Getzler was an undergraduate at ANU, where he studied Pure Mathematics and Theoretical Physics. Since completing his doctorate at Harvard, he has taught in the mathematics departments at MIT and Northwestern University, where he is currently a professor. He has held visiting professorships at Kyoto University and Imperial College, and a membership of the IAS. His interests include topological field theory, Gromov-Witten theory, and operads.
n-groups
In this talk, we give a brief introduction to a natural generalization of groups, called n-groups.
Just as discrete groups represent the homotopy types of acyclic spaces, n-groups realize homotopy types of connected topological spaces X such that πi(X)=0 for i>n. In this talk, we adopt the formalism of simplicial sets, and define n-groups as simplicial sets satisfying certain filling conditions (introduced by Duskin).
In the first part of the talk, we explain what 2-groups look like: this material is contained in any textbook on simplicial sets. We indicate how 2-groups arise in topological quantum field theory.
In the second part of the talk, we explain a generalization of Lie theory to n-groups, in which the role of Lie algebras is taken by differential graded Lie algebras, and the role of the ordinary differential equations underlying Lie theory is taken by the Maurer-Cartan equation for flat superconnections on simplices.
Kerry Landman (Melbourne)
Professor Kerry Landman obtained her PhD in mathematics from The University of Melbourne, followed by six years working as an applied mathematician in USA. She returned to Melbourne to join Siromath, a mathematical sciences consulting firm, before joining the Department of Mathematics and Statistics at the University of Melbourne. Her research interests arise from real-world problems. She is currently collaborating with experimentalists in the areas of developmental biology and tissue engineering.
Simulating a developmental cell invasion process
During the development of the gastrointestinal nervous system, neural crest cells must first invade and colonise the entire gut from stomach to anal end. These cells form the enteric nervous system, which gives rise to normal gut function and peristaltic contraction. Failure of the neural crest cells to invade the whole gut results in a lack of neurons in a length of the terminal intestine. This is a potentially fatal condition.
Here we discuss the development of mathematical models of neural crest cell invasion to replicate and connect the current population-level and cell-level experimental data, as well as generate experimentally testable predictions.
DSTO Lecturer Bill Moran (Melbourne/NICTA)
Bill Moran is the Research Director of Melbourne Systems Laboratory (MSL)
in the Department of Electrical and Electronic Engineering at the University
of Melbourne, where he has been a Professor of Electrical Engineering since
2001. Previously he was Professor of Mathematics, Head of the Department of
Pure Mathematics, Dean of Mathematical and Computer Sciences at the
University of Adelaide, and Head of the Mathematics Discipline at the
Flinders University of South Australia. He was the Head of the Medical
Signal Processing Program in the Cooperative Research Centre for Sensor
Signal and Information Processing. He was elected to the Fellowship of the
Australian Academy of Science in 1984. He holds a Ph.D. in Pure Mathematics
from the University of Sheffield, UK (1968), and a First Class Honours B.Sc.
in Mathematics from the University of Birmingham (1965).
His main areas of research interest are in signal processing both
theoretically and in applications to radar, waveform design and radar
theory, sensor networks, and sensor management. He also works in various
areas of mathematics including harmonic analysis, representation theory, and
number theory.
Golay-Rudin-Shapiro: A rose by any other name!
In the first half of 1951 a paper in infra-red spectroscopy, and a mathematical Master’s thesis appeared. The first was by an electrical engineer, Marcel Golay, at the US Army Signal Corps Laboratories in New Jersey, the second was by a mathematician, Harold Shapiro, at MIT. Remarkably, despite the divergence of research areas, the papers independently described the same sequences of $\pm 1$. A few years later, Rudin described these sequences again, in the context of a problem of harmonic analysis.
Much controversy has been generated over time about the contributions of the people whose names are attached by the various communities to the sequences, known to engineers as Golay Sequences, and to mathematicians as Rudin-Shapiro Sequences, or more commonly in terms of the (trigonometric) polynomials of which the sequences are the coefficients: Rudin-Shapiro polynomials. Whatever the name, these sequences continue to provide unsolved problems to interest mathematicians, and issues of application to interest engineers, in communications, in navigation, and in sensing.
I will present some of the ideas, results, and unsolved problems associated with these sequences, as well as some new ways in which they are being considered and generalized for applications in radar. New ways of thinking about these sequences involve the representation theory of the discrete Heisenberg-Weyl group. Surprisingly, in the radar application, another sequence of $\pm 1$ s, also with a triple-barrelled name, is needed to make the GRS sequences work well — the so-called Prouhet-Thue-Morse sequence.
Jacqui Ramagge (Wollongong)
Jacqui Ramagge is an Associate Professor at the University of Wollongong. She graduated with a BA from the University of Warwick in 1988 and liked it so much that she stayed and in 1993 graduated with a PhD on Kac-Moody groups supervised by Roger Carter. She won the BH Neumann Prize at the AustMS conference in Perth in 1991 and accepted a Level A position at Newcastle (Australia) just before she graduated. She has since worked on operator algebras arising from group actions on buildings, Hecke algebras, and topological groups. She moved to the University of Wollongong in 2007.
Totally disconnected, locally compact groups
Each locally compact group is an extension of a connected group by a totally disconnected group. Connected, locally compact groups can be approximated by Lie groups and we therefore know a lot about them. In contrast, little was known about general totally disconnected, locally compact groups until recently. In 1994 George Willis introduced highly innovative tools which have revolutionized subject. He essentially provided analogues of linear-algebraic concepts such as eigenvalues and eigenspaces in a context that is group-theoretic rather than linear in nature. There is now an international team of researchers working on exploiting the tools to their full potential.
I will give an introduction to Willis' key ideas, give an overview of what has been achieved so far, and identify directions for future research.
Clay Mahler Lecturer Terance Tao (UCLA)
Terence Tao was born in Adelaide, Australia, in 1975. He has been a professor of mathematics at UCLA since 1999, having completed his PhD under Elias Stein at Princeton in 1996. Tao's areas of research include harmonic analysis, PDE, combinatorics, and number theory. He has received a number of awards, including the Salem Prize in 2000, the Bocher Prize in 2002, the Fields Medal and SASTRA Ramanujan Prize in 2006, and the MacArthur Fellowship and Ostrowski Prize in 2007. Terence Tao also currently holds the James and Carol Collins chair in mathematics at UCLA, and is a Fellow of the Royal Society and the Australian Academy of Sciences (Corresponding Member).
The proof of the Poincare conjecture
In a series of three papers from 2002-2003, Grigori Perelman gave a spectacular proof of the Poincare conjecture (every smooth compact simply connected three-dimensional manifold is topologically isomorphic to a sphere), one of the most famous open problems in mathematics, by developing several new groundbreaking advances in Hamilton's theory of Ricci flow on manifolds. In this talk I describe in broad detail how the proof proceeds, and briefly discuss some of the key turning points in the argument.
ANZIAM Lecturer Peter Taylor (Melbourne)
Peter Taylor received a B Sc(Hons) and a PhD in Applied Mathematics from the University of Adelaide in 1980 and 1987 respectively. After periods at the Universities of Western Australia and Adelaide, he moved at the beginning of 2002 to the University of Melbourne. In January 2003, he took up a position as the inaugural Professor of Operations Research and became Head of the Department of Mathematics and Statistics in 2005.
Peter's research interests lie in the fields of stochastic processes and applied probability, with particular emphasis on applications in telecommunications. Recently he has become interested in the interaction of stochastic modelling with optimisation and optimal control.
Some interesting problems in modelling ad hoc mobile networks
In conventional mobile telephone networks, users communicate directly with a base station, via which their call is transferred to the recipient. In an ad hoc mobile network, there is no base-station infrastructure and users need to communicate between themselves, either directly if they are close enough or via transit nodes if they are not.
Two interesting questions immediately arise in the modelling of ad hoc mobile networks. The first concerns the `amount of resource' that a network needs in order to be able to operate with a reasonable quality of service. The second addresses what needs to be done to prevent selfish users from making their handset unavailable as transit nodes. There are many ways to approach these questions from a mathematical modelling point of view. In this talk I shall discuss some of these.
Early Career Lecturer Akshay Venkatesh (Stanford)
Akshay Venkatesh commenced his PhD at Princeton University in 1998 under Peter Sarnak, which he completed in 2002, producing the thesis Limiting forms of the trace formula. He was supported by the Hackett Fellowship for postgraduate study. He was awarded the Salem Prize and the Packard Fellowship in 2007. He also won the 2008 SASTRA Ramanujan Prize. Venkatesh held a Clay Research Fellowship from the Clay Mathematics Institute from 2004 to 2006, and was an associate professor at the Courant Institute of Mathematical Sciences. Since September 2008, he is a professor at Stanford University.
The Cohen-Lenstra heuristics over global fields
A "class group," in number theory, is a finite abelian group measuring the failure of unique factorization in a ring of integers. The study of class groups began with Gauss, who noted, for example, that the class group of Z(\sqrt{d}) is "often" trivial when d is positive. In these cases, Z(\sqrt{d}) has unique factorization.
What does "often" mean? A conjectural answer to this -- and many other related questions -- was given by Cohen and Lenstra. These conjectures are called the "Cohen-Lenstra heuristics." I will begin by reviewing their work; no background in number theory will be assumed.
I will then describe joint work with Jordan Ellenberg (U. Wisconsin) and Craig Westerland (U. Melbourne) where we obtain partial results towards the Cohen-Lenstra heuristics over "function fields." The heuristics in that context have interesting relations with topology and combinatorial group theory.