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International Review of Mathematics
Probability in Cambridge, a Summary
James Norris: 1. The main project I have been working on is a derivation of Smoluchowski's coagulation equation from a system of (Stokes-Einstein-)Brownian particles subject to coagulation on collision. This work is essentially complete but might count as a future plan. There are many (phyically motivated) variations on the problem, for example the case of Ornstein-Uhlenbeck particles, which appear susceptible to the same general strategy. Ines Armedariz is working on this. We are also working in collaboration with Chemical Engineers who are developing stochastic numerical schemes for coagulation processes. Earlier work on mean-field type models was published in
Cluster coagulation. Comm. Math. Phys. 209 (2000), no. 2, 407--435.
2. A second project, in collaboration with Richard Darling, concerns the structure of large random combinatorial structures. We are able to work with a rather general hypergraph model, which appears to offer scope for genuine mathematical modelling. The key idea here is the use of stochastic process methods such as fluid limits and Gaussian approximations to pass to a simplifying limit before doing any calculations. This is the context of Christina Goldschmidt's PhD, which established in particular some new scaling properties of near-critical random hypergraphs. The basic paper is math.PR/0109020.
3. Going back a few years, the paper
Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179 (1997), no. 1, 79--103.
answered a long-standing open problem about the short-time behaviour of heat flow. Using similar methods, the paper
Long-time behaviour of heat flow: global estimates and exact asymptotics. Arch. Rational Mech. Anal. 140 (1997), no. 2, 161--195.
made novel connections between homogenization methods and heat kernel estimates, and gave new variational characterizations of effective constants.
4. A little bit further back,
Twisted sheets. J. Funct. Anal. 132 (1995), no. 2, 273--334
is a geometric theory of two-parameter hyperbolic SPDE. This is a powerful tool for the path-space analysis of Malliavin and others.
Yuri Suhov:
1.
M. Kontsevich and Y. Suhov. Statistics of Klein polyhedra and multi-dimensional continued fractions (with M. Kontsevich). In: {\it Pseudoperiodic Topology} (eds V. Arnol'd, M. Kontsevich and A. Zorich). Amer. Math. Transl. Ser. 2, vol. {\bf 197}. Providence, R.I.: American Mathematical Society, 1999, pp. 9--28
The standard continuous fraction representation $$x=\frac{1}{a_0+\frac{1}{a_1+\frac{1}{a_2+\ldots}}}$$ of a number $x\in (0,1)$ generates a dynamical system (a shift $(a_0,a_1,a_2,\ldots )\mapsto (a_1,a_2,\ldots )$) with an invariant probability measure known as a Gauus, or Gauss-Kuzmin measure (unrelated to the famous Gaussian distribution). Statistical properties of the sequence $(a_0,a_1,a_2,\ldots )$ realtive to the Gauss-Kuzmin distrubution are well-known and play an important role in several applications, including rational approximations of a given real number. Multi-dimensional generalisations of continued fractions lead to reach and intereseting geometric structures and are important, in particular, in rational approximations of a given real vector. The question of studying statistical properties of multi-dimensional continued fractions was posed, by, among others, V. Arnol'd. He stressed importance of studying `typicality' of geometric patterns relative to a `natural' measure replacing the Gauss-Kuzmin measure in multi-dimenmsional situation. This patterns arise when one deals with a random simplicial cone and a lattice inside this cone, with respect to the above measure.
The story goes a follows, as presented in
Arnold, V. I. Higher-dimensional continued fractions. J. Moser at 70. {\it Regul. Chaotic Dyn.} {\bf 3} (1998), no. 3, 10--17.
In 1997 I have proposed the problem of calculation of these statistics to Yu.\,Suhov (Cambridge). It resulted in a remarkable work by Suhov and M.\,Kontsevich [20]. Among other things they proved the following
{\bf Theorem.} {\sl All statistical asymptotic characteristics mentioned above (as well as many others) do exist for almost every simplicial cone and are universal (independent of the cone).}
Unfortunatly, these distributions are still not calculated (while their calculation is reduced to the summation of some series in polylogarithms). For instance, it is unknown, whether the distribution of the integral lenghts of the edges of the sails is shifted toward the longer or toward the shorter lenghts (with respect to the distribution of the integral lenghts of the random integer segments and with respect to the elements of the continued fraction of a random real number).
The main new idea of Kontsevich and Suhov consists in the following inversion of the viewpoint. I suggested to fix the lattice and to move the simplical cone. They are fixing the simplicial cone (i.~e. the ``Cartan'' commutative group $H$ of matrices from $SL(n,\mathbb R)$ diagonal in the same basis), moving the lattice.
This work was highly praised by other sources (see, e.g., Math Reviews MR1733869 (2001h:11101)).
2. My result with J. Martin:
J.B. Martin and Y.M. Suhov. Fast Jackson networks (with J.B. Martin). {\it Annals of Applied Probability}, {\bf 9}, No 4 (1999), 854--870.
3. My result with I. Kurkova
I. Kurkova and Y.M. Suhov. Malyshev's theory and JS-queues (with I. Kurkova). (to appear in {\it Annals of Applied Probability} (2003/4)
I consider these paper as serious achievement in the theory of queueing networks. The paper with Kurkova is still on my website.
4. As many of us, I prefer to talk about new things. In a paper with Chulaevski (in progress) we extend the Anderson localisation theory to the case of several particles, which is the first rigorous result in this direction in the world literature.
Financial Mathematics: In recent years, Professor Rogers and his coworkers have contributed a number of important papers to the development of mathematical finance; among the more significant of these are: `Monte Carlo valuation of American options.' Mathematical Finance 12, 2002, 271--286 `Duality in constrained optimal investment and consumption problems: a synthesis'. CIRANO Workshop on Mathematical Finance and Econometrics, June 2001. `Robust hedging of barrier options' (with H. M. Brown and D. G. Hobson). Mathematical Finance 11, 2001, 285--314. `Markov chains and the potential approach to modelling interest rates and exchange rates' (with F. A. Yousaf). Mathematical Finance - Bachelier Congress 2000, ed. H. Geman, D. Madan, S. R. Pliska, \& T. Vorst, Springer 2002.
The first of these presents a completely novel approach to the pricing of American options, allowing upper bounds on prices to be established; this has had considerable impact on industry practice as well as academic research. The second is a unification and simplification of a substantial recent literature on constrained optimal investment and consumption problems of various types. The third can be seen as a first attempt to understand the use of `quasi-static' hedging techniques, and raises a number of intriguing questions. The final paper develops an approach pioneered by Professor Rogers in the mid-90s to modelling in fixed income and FX markets; it shows that the approach is viable in practice, and subsequent unpublished developments are even more promising.
Copies of these and other papers can be downloaded from http://www.statslab.cam.ac.uk/~chris/papers.html
Rchard Weber:
1. Asymptotics for Provisioning Problems of Peering Wireless LANs with a Large Number of Participants Courcoubetis, C. and Weber, R.R
Statistical Laboratory Research Report, http://www.statslab.cam.ac.uk/cgi-bin/resreps.pl?term=2003-14&field=number
Current research on peer-to-peer networks. We consider a problem in which the owners of wireless LANs have decided to peer with one another so that they can roam in locations other than their own. Each WLAN owner is contribute part of the cost of providing the WLAN resources in his home location so that the qualities of services that are achieved in the locations and a measure of expected welfare is maximized, subject to constraints of incentive compatibility, rationality and feasibility. We prove that as the number of participant becomes large the solution to a limiting problem takes a simple form: it is near optimal to set a fixed fee and allow a participant to use the WLANs in other locations if he is willing to pay this fee. The advantage of this is that the provisioning policy and fee structure can be easily communicated to the participants. We are actively pursuing generalizations of this idea and numerical illustrations.
2. Perfect Packing Theorems and the Average Case Behavior of Optimal and Online Bin Packing, E. G. Coffman, Jr., C. Courcoubetis, M. R. Garey, D. S. Johnson, P. W. Shor, R. R. Weber, and M. Yannakakis, SIAM Review 44 (2002), 95-108. [
This paper was published in SIAM Review 2002 as a "best paper of the year". It uses novel ideas regarding postive recurrence of multidimensional Markov chains, computer proof and strong law of large numbers to evaluate the performances of several bin packing algorithms.
2. Pricing Communication Networks : Economics, Technology and Modelling Costas Courcoubetis, Richard Weber ISBN: 0-470-85130-9 Hardcover 378 pages March 2003.
This new book arises from the authors' extensive interest in modelling of communications systems, and especially pricing. We provide a framework of mathematical models for pricing contracts for network services, and includes background in network services and contracts, network technonology, basic economics, and pricing theory.
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