Fall 2018
Time & Location: All talks are on Wednesdays in Gibson Hall 126 at 3:00 PM unless otherwise noted.
Organizer: Scott McKinley and Swati Patel
September 26
Topic: Chase-Escape
Matthew Junge | Duke University
Abstract:
Imagine barnacles and mussels spreading across the surface of a rock. Barnacles move to adjacent unfilled spots. Mussels too, but they can only attach to barnacles. Barnacles with a mussel on top no longer spread. What conditions on the rock geometry (i.e. graph) and spreading rates (i.e. exponential clocks) ensure that barnacles can survive? Chase-escape can be formalized in terms of competing Richardson growth models; one on top of the other. New, tantalizing open problems will be presented. Joint work with Rick Durrett and Si Tang.
October 17
Topic: Estimating evolutionary rates from pattern in the CRISPR defense memory of prokaryotes
Franz Baumdicker | University of Freiburg
Abstract:
Tacteria and archaea are under constant attack by a myriad of viruses. Consequently, many prokaryotes harbor immune systems against such viral attacks. A prominent example is the CRISPR system, that led to the CRISPR-Cas genome editing technology.
The prokaryotic CRISPR defense system includes an array of spacer sequences that encode an inheritable memory of previous infections. These spacer sequences correspond to short sequence samples from past viral attacks and provide a specific immunity against this particular virus.
Notably, new spacer sequences are always inserted at the beginning of the array, while deletion of spacers can occur at any position in the array. In a sample of CRISPR arrays, spacers can thus be present in all or a subset of the sample, but the order of spacer sequences in the array will be conserved across the sample. This order represents the chronological infection history of the host.
In an evolutionary model for spacer acquisition and deletion we derived the distribution of the number of different spacers between spacers that are present in all arrays. In particular, the order of spacers in the arrays can be used to estimate the rate of spacer deletions independently of the spacer acquisition rate. A property that is usually hard to obtain in population genetics. These estimates provide a basis to study the co-evolution of CRISPR possessing prokaryotes and their viruses.
October 31
Topic: Universality for Products of Random Matrices
Phil Kopel | University of Colorado Boulder
Abstract:
Random matrices have played an increasingly significant role in diverse areas of pure and applied mathematics over the last fifty years. In the first half of this talk, I will provide a brief overview of the subject: some of the things we study, some of the methods we use, some major results obtained, and several tantalizing unsolved problems. This will hopefully be accessible and informative for a general mathematical audience -- no prior experience required! In the second half of the talk, I will discuss some recent results obtained (in collaboration with Sean O'Rourke and Van Vu) establishing universality of fluctuations in the spectral bulk for products of independent entry ensembles, and outline the proofs.
November 7
Topic: Iterative projection methods for noisy and corrupted systems of linear equations
Jamie Haddock | UCLA
Abstract:
Iterative linear solvers have gained recent popularity due to their computational efficiency and low memory footprint for large-scale linear systems. We will discuss two classical methods, Motzkin's method (MM) and the Randomized Kaczmarz method (RK). MM is an iterative method that projects the current estimation onto the solution hyperplane corresponding to the most violated constraint. Although this leads to an optimal selection strategy for consistent systems, for noisy least squares problems, the strategy presents a tradeoff between convergence rate and solution accuracy. We analyze this method in the presence of noise. RK is a randomized iterative method that projects the current estimation onto the solution hyperplane corresponding to a randomly chosen constraint. We present RK methods which detect and discard corruptions in systems of linear equations, and present probabilistic guarantees that these methods discard all corruptions.
November 14
Topic: Consistent Inter-Model Specification for Time-Homogeneous SPX Stochastic Volatility and VIX Market Models
Andrew Papanicolaou | institution (Host:Nathan Glatt-Holtz)
Abstract:
This talk is about the recovering of stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a derivative of the S\&P500 (SPX) and it is common practice to price SPX derivatives using an SVM. Hence, a consistent model for both SPX and VIX derivatives would be one where the SVM is obtained by inverting the market model. A function for stochastic volatility function is the solution of an inverse problem, with the inputs given by a VIX futures market model. Several models are analyzed mathematically and explored numerically.
December 5
Topic: The tempered fractional Lévy process
Cooper Boniece | Tulane University
Abstract:
The celebrated family of fractional stochastic processes exhibits a phenomenon called long-range dependence, or slow decay of their covariance as the time separation between two observations increases.
In recent development have been tempered fractional models, which exhibit semi-long range dependence — i.e. their covariance exhibits slow decay over an intermediate time scale, but ultimately transitions to exponential decay. Tempered models have recently found applications in nanobiophysics and finance, where this type of transient behavior is observed in real data.
In this talk, I will introduce a new stochastic process called the tempered fractional Lévy process (TFLP) and discuss some of its probabilistic properties. This is joint work with Farzad Sabzikar (Iowa State) and Gustavo Didier (Tulane).
