Past Probability and Statistics:

**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

Matthew JungeDuke 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

**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 24

**Speaker**institution

**Abstract**: TBA

October 31

**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

**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

**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.

November 21

**Speaker**institution

**Abstract**: TBA

November 28

**Speaker**institution

**Abstract**: TBA

December 5

**Speaker**institution

**Abstract**: TBA

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu