Mathematics

Risk Sharing in Over the Counter Markets 3

Speaker: 
Darrel Duffie
Date: 
Fri, Jul 25, 2014
Location: 
PIMS, University of British Columbia
Conference: 
The Economics and Mathematics of Systemic Risk and Financial Networks
Abstract: 
I will begin with an overview of the purpose and structure of OTC markets, and how they can be a source of systemic risk.
This will be followed by a brief review of search-based theories of trade and information sharing in OTC markets. Then I will turn to theories and evidence regarding the use of collateral, the role of central clearing, and failure management. The failure management topic will finish with a model of the efficient application of legal stays that could be imposed on OTC contracts at the point of bankruptcy or administrative failure resolution. These stays can yield effective payment or settlement priority to OTC contracts. Stays can be efficient, or not efficient, depending on the setting. The affected OTC contracts include derivatives, repurchase agreements, securities lending agreements, and clearing agreements. I assume a basic knowledge of game theory and of measure-theoretic probability theory, particularly counting processes with an intensity.
Class: 
Scientific
Subject: 
Mathematics
Econometrics
Applied Mathematics

Channels of Contagion in Financial Systems 3

Speaker: 
Rama Cont
Date: 
Fri, Jul 25, 2014
Location: 
PIMS, University of British Columbia
Conference: 
The Economics and Mathematics of Systemic Risk and Financial Networks
Abstract: 

Understanding the mechanisms underlying systemic risk requires to change the traditional focus of risk modelling and examine the link between the structure of the financial system and its stability, with a focus on contagion mechanisms which may lead to large scale instabilities in the financial system. Some channels of contagion which have played an important role in past crises are: insolvency contagion through counterparty exposures, withdrawal of liquidity in funding channels and price-mediated contagion through fire sales of assets.

We review some recent work on the mechanisms underlying these channels of contagion, with a focus on the nature of the 'network' underlying each contagion mechanism and the implications of these results for the monitoring and regulation of systemic risk. In particular, we will attempt to illustrate the importance of the ineraction between these various channels and how this interaction may undermine regulatory efforts focussed only on a single mechanism.

Class: 
Scientific
Subject: 
Mathematics
Econometrics
Applied Mathematics

Channels of Contagion in Financial Systems

Speaker: 
Rama Cont
Date: 
Thu, Jul 24, 2014
Location: 
PIMS, University of British Columbia
Conference: 
The Economics and Mathematics of Systemic Risk and Financial Networks
Abstract: 

Understanding the mechanisms underlying systemic risk requires to change the traditional focus of risk modelling and examine the link between the structure of the financial system and its stability, with a focus on contagion mechanisms which may lead to large scale instabilities in the financial system. Some channels of contagion which have played an important role in past crises are: insolvency contagion through counterparty exposures, withdrawal of liquidity in funding channels and price-mediated contagion through fire sales of assets.

We review some recent work on the mechanisms underlying these channels of contagion, with a focus on the nature of the 'network' underlying each contagion mechanism and the implications of these results for the monitoring and regulation of systemic risk. In particular, we will attempt to illustrate the importance of the ineraction between these various channels and how this interaction may undermine regulatory efforts focussed only on a single mechanism.

Class: 
Scientific
Subject: 
Mathematics
Econometrics
Applied Mathematics

Risk Sharing in Over the Counter Markets 2

Speaker: 
Darrel Duffie
Date: 
Wed, Jul 23, 2014
Location: 
PIMS, University of British Columbia
Conference: 
The Economics and Mathematics of Systemic Risk and Financial Networks
Abstract: 

These lecture notes are part of a series on "Risk Sharing in Over-the-Counter Markets"

Class: 
Scientific
Subject: 
Mathematics
Econometrics
Applied Mathematics

Undecidability in Number Theory

Speaker: 
Bjorn Poonen
Date: 
Mon, May 26, 2014
Location: 
PIMS, University of British Columbia
Conference: 
2014 Niven Lecture
Abstract: 

Hilbert’s Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers.  Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists.  However, the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be.

Class: 
Scientific
Subject: 
Mathematics
Number Theory

High dimensional expanders and Ramanujan complexes

Speaker: 
Alexander Lubotzky
Date: 
Fri, Sep 19, 2014
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 

Expander graphs have played, in the last few decades, an important role in computer science, and  in the last decade, also in pure mathematics.  In recent years a theory of "high-dimensional expanders" is starting to emerge - i.e., simplical complexes which generalize various properties of expander graphs. This has some geometric motivations (led by Gromov) and combinatorial ones (started by Linial and Meshulam).  The talk will survey the various directions of research and their applications, as well as potential applications in math and CS.  Some of these lead to questions about buildings and representation theory of p-adic groups.

 

We will survey the work of a number of people. The works of the speaker in this direction are with various subsets of  { S. Evra, K. Golubev,  T. Kaufman,  D. Kazhdan , R. Meshulam, S. Mozes }

Class: 
Scientific
Subject: 
Mathematics
Computer Science

The rank of elliptic curves

Speaker: 
Benedict Gross
Date: 
Fri, Oct 10, 2014
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 

After quadratic equations in two variables come cubic equations, or elliptic curves. The set of rational points on an elliptic curve has the structure of a finitely generated abelian group. I will recall the basic theory of elliptic curves, then discuss the conjecture of Birch and Swinnerton-Dyer, which attempts to predict the rank of the group of rational points from the number of solutions (mod p) for all primes p. I will also discuss some recent results on the average rank, due to Manjul Bhargava and his collaborators. (PIMS-UBC Distinguished Colloquium)

Class: 
Scientific
Subject: 
Mathematics
Number Theory

Reconstructing carbon dioxide for the last 2000 years: a hierarchical success story

Speaker: 
Doug Nychka,
Date: 
Thu, Oct 16, 2014
Location: 
PIMS, University of British Columbia
Conference: 
SCAIM Seminar
Abstract: 

Knowledge of atmospheric carbon dioxide (CO2) concentrations in the past are important to provide an understanding of how the Earth's carbon cycle varies over time. This project combines ice core CO2 concentrations, from Law Dome, Antarctica and a physically based forward model to infer CO2 concentrations on an annual basis. Here the forward model connects concentrations at given time to their depth in the ice core sample and an interesting feature of this analysis is a more complete characterization of the uncertainty in "inverting" this relationship. In particular, Monte Carlo based ensembles are particularly useful for assessing the size of the decrease in CO2 around 1600 AD. This reconstruction problem, also known as an inverse problem, is used to illustrate a general statistical approach where observational information is limited and characterizing the uncertainty in the results is important. These methods, known as Bayesian hierarchical models, have become a mainstay of data analysis for complex problems and have wide application in the geosciences. This work is in collaboration with Eugene Wahl (NOAA), David Anderson (NOAA) and Catherine Truding.

Class: 
Scientific
Subject: 
Mathematics
Atmospheric and Oceanic Physics: Climate Modelling
Applied Mathematics

Imaging with Waves in Complex Environments

Speaker: 
Liliana Borcea
Date: 
Fri, Oct 31, 2014
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC/IAM Distinguished Colloquium
Abstract: 

The talk is concerned with the application of sensor array imaging in complex environments. The goal of imaging is to estimate the support of remote sources or strong reflectors using time resolved measurements of waves at a collection of sensors (the array). This is a challenging problem when the imaging environment is complex, due to numerous small scale inhomogeneities and/or rough boundaries that scatter the waves. Mathematically we model such complexity (which is necessarily uncertain in applications) using random processes, and thus study imaging in random media. I will focus attention on the application of imaging in random waveguides, which exhibits all the challenges of imaging in random media. I will present a quantitative study of cumulative scattering effects in such waveguides and then explain how we can use such a study to design high fidelity imaging methods.

Class: 
Scientific
Subject: 
Mathematics
Applied Mathematics

Blowup or no blowup? The interplay between theory and computation in the study of 3D Euler equations

Speaker: 
Thomas Hou
Date: 
Fri, Feb 27, 2015
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities.  To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes with a maximum effective resolution of order $10^12$ in each direction. A careful local analysis also suggests that the solution develops a highly anisotropic self-similar profile which is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. Very recently we prove rigorously that the 1D model develops finite time singularity.This is a joint work of Prof. Guo Luo.

Class: 
Scientific
Subject: 
Mathematics
Partial Differential Equations

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