Pacific Institute for the Mathematical Sciences - PIMS

Mathematics: An experimental Science

We'll discuss some recent results that were conjectured via computer experiments and then proved. The examples will be drawn from number theory, Young tableaux, hypergeometric determinant evaluation, theory of matrices of 0s and 1s, etc. Several u   more››

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Invasion percolation on regular trees

Invasion percolation is a dynamic process closely linked to critical percolation, but without an external parameter. In joint work with Omer Angel, Frank den Hollander and Gord Slade we showed that the cluster of inva   more››

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Markov Chains and Web Ranking: a Multilevel Adaptive Aggregation Method

Google's PageRank method for ranking web pages models how a `random surfer' follows links between web pages in a random fashion. The stationary probability vector of the resulting Markov chain provides a ranking of all the pages in the network. Ca   more››

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Monetary utilities and BDEs

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Oscillating random walk on Z driven by its occupation time at zero

I will consider a nearest-neighbor random walk X_n on integers with the drift at time n (conditional on the past expectation of $X_{n+1}-X_n$) equals to $-sign(X_n)f^{-1}(eta_n),$ where eta_n is the number of visits to zero by time n and f(n) is a   more››

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Differences of semigroups, eigenvalue bounds, and their application to Schroedinger operators

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Boundary Value Problems for Elliptic PDEs and the Morse Index

A symplectic view of linearized elliptic problems leads naturally to the formulation of an appropriate Maslov Index. This is well-known in one space dimension and is generalized here to domains in higher space dimensions. The result is a Morse Ind   more››

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Gamma-Convergence and Saddle Points

We prove a theorem asserting, roughly speaking, that if a sequence of functionals converges to a limiting functional (in the sense of Gamma-convergence, a natural and widely-used notion in the calculus of variations), and if the limiting functiona   more››

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Frailty Correlated Default

We analyze portfolio credit risk in light of dynamic "frailty," in the form of incompletely observed covariates. Common dependence by firms on unobservable time-varying default covariates is estimated to cause large changes in conditiona   more››

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Orbifold String Topology II

In these two lectures I will survey my recent work with Lupercio and Xicotencatl on the properties of the homology of the free loop space of an orbifold. In the first lecture I will review the work of Chas and Sullivan on string topology and I wil   more››

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Spanning Trees, Random Graphs, and Random Walks

In the usual Erdõs-Rényi model of random graphs, each pair of n vertices is connected by an edge independently with probability c/n for some constant c. When c > 1, it has a unique 'giant' component. How quickly does the number of spannin   more››

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Orbifold String Topology I

In these two lectures I will survey my recent work with Lupercio and Xicotencatl on the properties of the homology of the free loop space of an orbifold. In the first lecture I will review the work of Chas and Sullivan on string topology and I wil   more››

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Modular tensor categories from semisimple algebras

Modular tensor categories are finitely semisimple Abelian categories with some additional structure that allows us to obtain combinatorial invariants of oriented framed tangles. Reshetikhin and Turaev invented them in order to construct their 3-ma   more››

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Evolutionary and Population Dynamics of Bacteria and Phage

Bacterial viruses, aka bacteriophage or phage, are ubiquitous in nature, yet many central aspects of host-phage biology have not been integrated into mathematical models. In this talk I present a series of theoretical efforts to understand the div   more››

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Prandtl Conjecture and von Neumann Paradox for Shock Reflections

We plan to discuss the two fundamental problems, the Prandtl Conjecture and von Neumann Paradox, in the theory of supersonic flows with shock waves. These problems were raised in the first half of the 20th century and clearly formulated in the cla   more››

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Mathematical analysis of models for chemotaxis: parabolic, hyperbolic, kinetic

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40th Cascade Topology Seminar

The 40th Cascade Topology Seminar is a bi-annual gathering of topologists from the US Pacific Northwest and Southwestern Canada. All talks will be at the Pacific Institute for the Mathematical Sciences in Room 110 West Mall A   more››

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Braid groups of the projective plane and the splitting problem

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Random walk on the incipient infinite cluster for oriented percolation

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation in d spatial dimensions and one time dimension. For d>6, we obtain bounds on exit times, transition probabilities, and the range o   more››

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Homogenization results for dislocation dynamics

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