Mathematics

Aside from games of chance and a handful of textbook topics (e.g. opinion polls) there is little overlap between the content of an introductory course in mathematical probability and our everyday perception of chance. In this mostly non-mathematical talk I will give some illustrations of the broader scope of probability.

Why do your friends have more friends than you do, on average? How can we judge someone’s ability to assess probabilities of future geopolitical events, where the true probabilities are unknown? Were there unusually many candidates for the 2012 and 2016 Republican Presidential Nominations whose fortunes rose and fell? Why, in a long line at airport security, do you move forward a few paces and then wait half a minute before moving forward again? In what everyday contexts do ordinary people perceive uncertainty/unpredictability in terms of chance?

As the title indicates, this talk has two parts. In the first part I will describe the problem that homomorphic encryption is meant to solve, discuss some of the difficulties in creating such systems, and describe some of the (admittedly still slow) progress that has been made. In the second part I will explain what digital signatures are and why they are so important, followed by a description of a relatively new lattice-based digital signature scheme that is both quantum-resistant and impervious to the transcript attacks that be-deviled earlier digital signature schemes based on lattice problems.

The Langlands program predicts that for every n-dimensional Abelian variety over Q there is an automorphic representation of GSpin(2n+1) over Q whose L-function coincides with the L-function coming from the Galois representation on the Tate module of the Abelian variety. Recently, Gross has refined this prediction by identifying specific properties that one should find in a vector in the automorphic representation. In joint work with Lassina Dembele, we have found some examples of automorphic representations of GSpin(2n+1) over Q whose L-functions match those coming from certain n-dimensional Abelian varieties over Q, all built from certain Hilbert modular forms. We are in the process of checking if these examples contain vectors with the properties predicted by Gross. In this talk I will explain the lifting procedure we are using to manufacture GSpin automorphic representations and describe the examples we are focusing on as we hunt for the predicted vectors in the representation space.

Factor analysis is a core technique in applied statistics with implications for biology, education, finance, psychology and engineering. It represents a large matrix of data through a small number k of latent variables or factors. Despite more than 100 years of use, it remains challenging to choose k from the data. Ad hoc and subjective methods are popular, but subject to confirmation bias and they do not scale to automatic uses. There are many recent tools in random matrix theory (RMT) that apply to the factor analysis setting, so long as the noise has constant variance. Real data usually involves heteroscedasticity foiling those techniques. There are also tools in the econometrics literature, but those apply mostly to the strong factor setting unlike RMT which handles weaker factors. The best published method is parallel analysis, but that is only justified by simulations. We propose a bi-cross-validation approach holding out some rows and some columns of the data matrix, predicting the held out data via a factor analysis on the held in data. We also use simulations to justify the method, though our simulations are designed using recent findings from RMT. The new approach outperforms previous methods that we found, as measured by recovery of a true underlying factor matrix.

This is joint work with Jingshu Wang of Stanford University.

Biosketch: Art Owen is a professor of statistics at Stanford University. He is best known for developing empirical likelihood and randomized quasi-Monte Carlo. Empirical likelihood is an inferential method that uses a data driven likelihood without requiring the user to specify a parametric family of distributions. It yields very powerful tests and is used in econometrics. Randomized quasi-Monte Carlo sampling, is a quadrature method that can attain nearly O(n**-3) mean squared errors on smooth enough functions. It is useful in valuation of options and in computer graphics. His present research interests focus on large scale data matrices. Professor Owen's teaching is focused on doctoral applied courses including linear modeling, categorical data, and stochastic simulation (Monte Carlo).

Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$ a monic irreducible separable polynomial. Denote by $\theta\in K^{\mathrm{sep}}$ a root of $f$ and let $F=K(\theta)$ be the finite separable extension of $K$ generated by $\theta$. We consider $\mathcal{O}$ the integral closure of $A$ in $L$. For a given non-zero prime ideal $\mathfrak{p}$ of $A$ the Montes algorithm determines a parametrization (OM representation) for every prime ideal $\mathfrak{P}$ of $\mathcal{O}$ lying over $\mathfrak{p}$. For a field $k$ and $f\in k[t,x]$ this yields a new representation of places of the function field $F/k$ determined by $f$. In this talk we summarize some applications which improve the arithmetic in the divisor class group of $F$ using this new representation.