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?
Perfect graphs are a class of graphs that behave particularly well with respect to coloring. In the 1960's Claude Berge made two conjectures about this class of graphs, that motivated a great deal of research, and by now they have both been solved.
The following remained open however: design a combinatorial algorithm that produces an optimal coloring of a perfect graph. Recently, we were able to make progress on this question, and we will discuss it in this talk. Last year, in joint work with Lo, Maffray, Trotignon and Vuskovic we were able to construct such an algorithm under the additional assumption that the input graph is square-free (contains no induced four-cycle). More recently, together with Lagoutte, Seymour and Spirkl, we solved another case of the problem, when the clique number of the input graph is fixed (and not part of the input).
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
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.
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
p-Adic analogues of triangulations of Riemann surfaces give us a very concrete way of understanding degenerate parts of modular Jacobians. In this talk, I will discuss how this yields a flexible way to understand the action of Hecke operators on modular curves, and functoriality of canonical integral "hidden" structures on de Rham cohomology. Finally, I will discuss progress on a strategy for defining p-adic L-functions of special modular forms via such degenerate techniques, proposed by Schneider.
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
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.
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Let K be a quadratic imaginary field, and p be a prime which is inert in K. It is known that the mod p reductions of the j-invariants of elliptic curves defined over the algebraic closure of Qp which admit CM by an order of K are equidistributed among the supersingular values in F{p2}. By contrast, if we replace this algebraically closed field by Qp, the j-invariants for many natural families of orders do not share this same distribution and are simply not uniformly distributed among all the supersingular values in Fp.
In this talk I will explain why this occurs, and some of the computations which led me to consider this question.
Constance van Eeden Invited Speaker, UBC Statistics Department
Abstract:
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).
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
Let A be an abelian variety of dimension g over a number field K. The Sato-Tate group ST(A) is a compact subgroup of the unitary symplectic group USp(2g) that can be defined in terms of the l-adic Galois representation associated to A. Under the generalized Sato-Tate conjecture, the Haar measure of ST(A) governs the distribution of various arithmetic statistics associated to A, including the distribution of normalized Frobenius traces at primes of good reduction. The Sato-Tate groups that can and do arise for g=1 and g=2 have been completely determined, but the case g=3 remains open. I will give a brief overview of the classification for g=2 and then discuss the current state of progress for g=3.
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract:
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.
The presentation will take us along the road to the ozone standard for the United States, announced in Mar 2008 by the US Environmental Protection Agency, and then the new proposal in 2014. That agency is responsible for monitoring that nation’s air quality standards under the Clean Air Act of 1970. I will describe how I, a Canadian statistician, came to serve on the US Clean Air Scientific Advisory Committee (CASAC) for Ozone that recommended the standard and my perspectives on the process of developing it. I will introduce the rich cast of players involved including the Committee, the EPA staff, “blackhats,” “whitehats,” “gunslingers,” politicians and an unrevealed character waiting in the wings who appeared onstage only as the 2008 standards had been formulated. And we will encounter a couple of tricky statistical problems that arose along with approaches, developed by the speaker and his coresearchers, which could be used to address them. The first was about how a computational model based on things like meteorology could be combined with statistical models to infer a certain unmeasurable but hugely important ozone level, the “policy related background level” generated by things like lightning, below which the ozone standard could not go. The second was about estimating the actual human exposure to ozone that may differ considerably from measurements taken at fixed site monitoring locations. Above all, the talk will be a narrative about the interaction between science and public policy - in an environment that harbors a lot of stakeholders with varying but legitimate perspectives, a lot of uncertainty in spite of the great body of knowledge about ozone and above all, a lot of potential risk to human health and welfare.