Mathematics Home / Colloquium: Fall 2021

Time & Location: The Tulane Math Colloquium this semester will be via Zoom. All talks are on Thursdays at 3:30 pm unless otherwise noted. In order to have time to talk informally with the speakers, we will schedule a time we call “Tea with the speaker” that everyone is welcome to join.

Organizer: Gustavo Didier

*September 2*

**Title:** Random matrix theory for high-dimensional time series

Alexander Aue - UC Davis (Host: Didier, Gustavo)

**Abstract:**

This talk is concerned with extensions of the classical Marcenko–Pastur law to time series. Specifically, p-dimensional linear processes are considered which are built from innovation vectors with independent, identically distributed entries possessing zero mean, unit variance and finite fourth moments. Under suitable assumptions on the coefficient matrices of the linear process, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high-dimensional setting for which dimension p and sample size n diverge to infinity at the same rate, enabling the use of results from random matrix theory. The presented theory extends existing contributions available in the literature for the covariance case and is one of the first of its kind for the autocovariance case. Several applications are discussed to highlight the potential usefulness of the results. The talk is based on joint work with Haoyang Liu (New York Fed) and Debashis Paul (UC Davis).

**Location:** TBA

**Time:** TBA

*September 9*

**Title:** TBA

Gautam Iyer - Carnegie Mellon (Host: Glatt-Holtz)

**Abstract:**

TBA

**Location:** TBA

**Time:** TBA

*September 16*

**Title:** Our Place Among the Infinities

Bill Taber - JPL (Host: Glatt-Holtz)

**Abstract:**

We can see the planets and smaller bodies of the solar system with earth bound telescopes, but telescopes cannot answer the big questions. How do these bodies “work?” What is their chemistry, their dynamics, their evolution? Where is there water in the solar system? Is there now or has there ever been life anywhere but Earth. To answer the big questions, we cannot do so from the comfort of Earth; we have to go there. To go there requires machines that did not exist 100 years ago: rockets, ultra-stable oscillators, deep space communication antennae, computers, etc. But even more than these machines, it requires mathematics: mathematic to design trajectories from earth to distant bodies; mathematics to navigate the trajectories, mathematics to control the flight of spacecrafts, mathematics to communicate with spacecraft, and mathematics to arrive safely. This talk will sketch out in broad strokes the mathematics of deep space exploration and how it can help us to know our place among the infinities.

Bill Taber is group supervisor of the Mission Design and Navigation Software Group at NASA’s Jet Propulsion Laboratory in Pasadena, California where he has been since 1983. He holds the degrees of Masters in Business Administration from the Peter Drucker School of Management of the Claremont Graduate University at Claremont, California, a Ph. D. in Mathematics M.S. in Mathematic from the University of Illinois at Urbana-Champagne, Illinois, and a B.A. in Mathematics from Eastern Illinois University at Charleston, Illinois.

**Location:** TBA

**Time:** 3:30

*September 23*

**Title:** TBA

Vladimir Sverak - University of Minnesota (Host: Glatt-Holtz)

**Abstract:**

TBA

**Location:** TBA

**Time:** TBA

*September 30*

**Title:** The Statistical Finite Element Method

Mark Girolami - University of Cambridge (Host: Glatt-Holtz)

**Abstract:**

Science and engineering have benefited greatly from the ability of finite element methods (FEMs) to simulate nonlinear, time-dependent complex systems. The recent advent of extensive data collection from such complex systems now raises the question of how to systematically incorporate these data into finite element models, consistently updating the solution in the face of mathematical model misspecification with physical reality. This article describes general and widely applicable methodology for the coherent synthesis of data with FEM models, providing a data-driven probability measure that captures all sources of uncertainty in the pairing of FEM with measurements.

**Location:** TBA

**Time:** TBA

*October 14*

**Title:** Convergence results for Yule's "nonsense correlation" using stochastic analysis

Frederi Viens | Michigan State - Michigan State (Host: Glatt-Holtz)

**Abstract:**

We provide an analysis of the empirical correlation of two independent Gaussian processes in two cases: pure diffusion and mean-reverting diffusion. Included are an explicit formula for the variance in the former in discrete time, and some convergence theorems and numerical results in the long-time horizon and in-fill-asymptotics regimes.

This empirical correlation $\rho_n$, defined for two related series of data of length $n$ using the standard Pearson correlation statistic which is appropriate for i.i.d. data with two moments, is known as Yule's "nonsense correlation" in honor of the statistician G. Udny Yule. He described in 1926 the phenomenon by which random walks and other time series are not appropriate for use in this statistic to gauge independence of data series. He observed empirically that its distribution is not concentrated around 0 but diffuse over the entire interval (−1,1). This well-documented effect was roundly ignored by many scientists over the decades, up to the present day, even sparking recent controversies in important areas like climate-change attribution. Since the 1960s, probability theorists wanted to close any possible ambiguity about the issue by computing the variance of the continuous-time version $\rho$ of Yule's nonsense correlation, based on the paths of two independent Brownian motions. This problem eluded the best minds until it was finally closed by Philip Ernst and two co-authors 90 years after Yule's observation, in a paper published in 2017 in the Annals of Statistics.

The more practical question of what happens with $\rho_n$ in discrete time remained. We address it here by computing its moments in the case of Gaussian data, the second moment being explicit, and by estimating the speed of convergence of the second moment of $\rho - \rho_n$, which we find tends to zero at the rate $1/n^2$. The latter is an important result in practice since it could help justify using statistical properties of $\rho$ when devising tests for pairs of time series of moderate length. We also investigate what remains of the diffuse behavior of $\rho$ and $\rho_n$ in long-time asymptotics. The asymptotic self-similarity of pure random walks means that the distribution of $\rho$ is insensitive to the time scale in the pure-diffusion case, but this is far from being true in mean-reverting cases, as we show when the two processes are Ornstein-Uhlenbeck OU processes observed in discrete or continuous time. In that case, $\rho$ concentrates as time $T$ increases, and has Gaussian fluctuations: the asymptotic variance of $T^{1/2} \rho$ is the inverse of the rate of mean reversion, and we establish a Berry-Esseen-type result for the speed of this normal convergence in Kolmogorov distance, modulo a log correction. We prove that these results for the OU processes also hold for discete-observation case under sufficiently high-frequency assumptions.

In this presentation, we provide ideas of the tools used to prove these results, which come from two seemingly orthogonal directions: algebraically tractable trivariate moment-generating functions for the three components of $\rho$, leading to integro-differential representation formulas for the moments of $\rho$; and applications of the connection between Analysis on Wiener space and Stein's method to access the Kolmogorov distance between $\rho$ and a normal law. Time permitting, we will attempt to explain why these two apparently dissimilar mathematical methodologies are intimately connected because the three components of $\rho$ belong to the so-called second Wiener chaos, which has a remarkable Hilbert-space structure. We conjecture that the speed $1/n^2$ which we found for the convergence of the variance of $\rho$ in in-fill asymptotics only applies because of the random-walk structure (independence of increments), while for other types of time series, such as mean-reverting ones, the speed increases to $1/n$; this would be consistent with our Berry-Esseen result.

This work is partially supported by the US National Science Foundation award DMS-1811779, the Office of Naval Research award N00014-18-1-2192, and a US Fulbright Dissertation Scholarship. It is joint work with Soukaina Douissi (Cadi Ayyad University, Marrakech, Morocco), Philip Ernst and Dongzhou Huang (Rice University, Houston, TX, USA), and Khalifa es-Sebaiy (Kuwait University, Kuwait).

**Location:** TBA

**Time:** 3:30

*November 4*

**Title:** Stochastic analysis and geometric functional inequalities

Maria Gordina | UConn (Host: Glatt-Holtz)

**Abstract:**

We will survey different methods of proving functional inequalities for hypoelliptic diffusions and the corresponding heat kernels. Some of these methods rely on geometric methods such as curvature-dimension inequalities (due to Baudoin-Garofalo), and some are probabilistic such as couplings or Lyapunov function techniques. One of the recent applications is to ergodicity for Langevin dynamics. This is based on joint work with F. Baudoin, B. Driver, D. Herzog, T. Melcher, Ph. Mariano et al.

**Location:** TBA

**Time:** 3:30

*November 18*

**Title:** TBA

Frank Garvan | University of Florida (Host: Beckwith)

**Abstract:**

TBA

**Location:** TBA

**Time:** TBA

*December 2*

**Title:** Gravitational Collapse for Newtonian Stars

Juhi Jang | USC (Host: Glatt-Holtz)

**Abstract:**

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model to describe the dynamics of Newtonian stars is given by the gravitational Euler-Poisson system, which admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In particular, using numerics, the Euler-Poisson system in the super-critical regime has been widely used in astrophysics literature to describe the gravitational collapse, but its rigorous proof has been established only recently. The main challenge comes from the pressure, which acts against gravitational force. In this talk, I will discuss some recent progress on Newtonian dust-like collapse and self-similar collapse.

**Location:** TBA

**Time:** TBA

*December 9*

**Title:** Random matrix theory for high-dimensional time series

Alexander Aue | The Chair of the Statistics Department at UC Davis

**Abstract:**

This talk is concerned with extensions of the classical Marcenko–Pastur law to time series. Specifically, p-dimensional linear processes are considered which are built from innovation vectors with independent, identically distributed entries possessing zero mean, unit variance and finite fourth moments. Under suitable assumptions on the coefficient matrices of the linear process, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high-dimensional setting for which dimension p and sample size n diverge to infinity at the same rate, enabling the use of results from random matrix theory. The presented theory extends existing contributions available in the literature for the covariance case and is one of the first of its kind for the autocovariance case. Several applications are discussed to highlight the potential usefulness of the results. The talk is based on joint work with Haoyang Liu (New York Fed) and Debashis Paul (UC Davis).

**Location:** TBA

**Time:** TBA