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Mathematics

A variational perspective on wrinkling patterns in thin elastic sheets

Speaker: 
Robert Kohn
Date: 
Fri, Apr 1, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium Series
Abstract: 
Thin sheets exhibit a daunting array of patterns. A key difficulty in their analysis is that while we have many examples, we have no classification of the possible "patterns." I have explored an alternative viewpoint in a series of recent projects with Peter Bella, Hoai-Minh Nguyen, and others. Our goal is to identify the *scaling law* of the minimum elastic energy (with respect to the sheet thickness, and the other parameters of the problem). Success requires proving upper bounds and lower bounds that scale the same way. The upper bounds are usually easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-independent. In many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. My talk will give an overview of this activity, and details of some examples.

Potential Infinity: A modal account

Speaker: 
Stewart Shapiro
Date: 
Thu, Mar 3, 2016
Location: 
PIMS, University of Calgary
Conference: 
Calgary Mathematics & Philosophy Lecture Series
Abstract: 
Beginning with Aristotle, almost every major philosopher and mathematician before the nineteenth century rejected the notion of the actual infinite. They all argued that the only sensible notion is that of potential infinity. The list includes some of the greatest mathematical minds ever. Due to Georg Cantor’s influence, the situation is almost the opposite nowadays (with some intuitionists as notable exceptions). The received view is that the notion of a merely potential infinity is dubious: it can only be understood if there is an actual infinity that underlies it. After a sketch of some of the history, Prof. Shapiro will analyze the notion of potential infinity, in modal terms, and assess its scientific merits. This leads to a number of more specific questions. Perhaps the most pressing of these is whether the conception of potential infinity can be explicated in a way that is both interesting and substantially different from the now-dominant conception of actual infinity. One might suspect that, when metaphors and loose talk give way to precise definitions, the apparent differences will evaporate. A number of differences still remain. Some of the most interesting and surprising differences concern consequences that one’s conception of infinity has for higher-order logic. Another important question concerns the relation between potential infinity and mathematical intuitionism. In fact, as will be shown, potential infinity is not inextricably tied to intuitionistic logic. There are interesting explications of potential infinity that underwrite classical logic, while still differing in important ways from actual infinity. However, on some more stringent explications, potential infinity does indeed lead to intuitionistic logic. (The lecture is based on joint work with Øystein Linnebo.)

Probability, Outside the Classroom

Speaker: 
David Aldous
Date: 
Fri, Mar 4, 2016
Location: 
PIMS, University of British Columbia
Conference: 
Hugh C. Morris Lecture
Abstract: 
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?

Coloring some perfect graphs

Speaker: 
Maria Chudnovsky
Date: 
Fri, Feb 26, 2016
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
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).

Homomorphic Encryption / Transcript-Secure Digital Signatures

Speaker: 
Joseph Silverman
Date: 
Thu, Mar 10, 2016
Location: 
Colorado State University
Conference: 
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.

Modular Jacobians and degenerate p-adic geometry

Speaker: 
Jan Vonk
Date: 
Thu, Mar 3, 2016
Location: 
PIMS, Simon Fraser University
Conference: 
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.

Lifts of Hilbert modular forms and application to modularity of Abelian varieties

Speaker: 
Clifton Cunningham
Date: 
Thu, Feb 18, 2016
Location: 
PIMS, University of Calgary
Conference: 
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.

The Distribution of J-invariants for CM Elliptic Curves defined over Zp

Speaker: 
Andrew Fiori
Date: 
Thu, Jan 28, 2016
Location: 
PIMS, University of Calgary
Conference: 
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.

Bi-cross-validation for factor analysis

Speaker: 
Art Owen
Date: 
Tue, Jan 19, 2016
Location: 
PIMS, University of British Columbia
Conference: 
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).

Abelian Varieties Multi-Site Seminar Series: Drew Sutherland

Speaker: 
Drew Sutherland
Date: 
Tue, Jan 12, 2016
Location: 
PIMS, University of Washington
Conference: 
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.
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