# Mathematics

## Using physical metaphors to understand networks

Speaker:

Daniel A. Spielman
Date:

Mon, May 29, 2017
Location:

PIMS, University of British Columbia
Conference:

2017 Niven Lecture Abstract:

Networks describe how things are connected, and are ubiquitous in science and society. Networks can be very concrete, like road networks connecting cities or networks of wires connecting computers. They can represent more abstract connections such as friendship on Facebook. Networks are widely used to model connections between things that have no real connections. For example, Biologists try to understand how cells work by studying networks connecting proteins that interact with each other, and Economists try to understand markets by studying networks connecting institutions that trade with each other.
Questions we ask about a network include "which components of the network are the most important?", "how well do things like information, cars, or disease spread through the network?", and "does the network have a governing structure?".
Professor Spielman will explain how mathematicians address these questions by modeling networks as physical objects, imagining that the connections are springs, electrical resistors, or pipes that carry fluid, and analyzing the resulting systems.

## Using physical metaphors to understand networks

Speaker:

Daniel A. Spielman
Date:

Mon, May 29, 2017
Location:

PIMS, University of British Columbia
Conference:

2017 Niven Lecture Abstract:

## Multivariate (phi, Gamma)-modules

Speaker:

Kiran Kedlaya
Date:

Thu, May 18, 2017
Location:

PIMS, University of British Columbia
Conference:

Focus Period on Representations in Arithmetic Abstract:

The classical theory of (phi, Gamma)-modules relates continuous p-adic representations of the Galois group of a p-adic field with modules over a certain mildly noncommutative ring. That ring admits a description in terms of a group algebra over Z_p which is crucial for Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We describe a method for applying a key property of perfectoid spaces, the analytic analogue of Drinfeld's lemma, to the construction of "multivariate (phi, Gamma)-modules" corresponding to p-adic Galois representations in more exotic ways. Based on joint work with Annie Carter and Gergely Zabradi.

## Theory Reduction, Algebraic Number Theory, and the Complex Plane

Speaker:

Emily Grosholznt
Date:

Thu, Mar 16, 2017
Location:

PIMS, University of Calgary
Conference:

The Calgary Mathematics & Philosophy Lectures Abstract:

How does mathematical knowledge grow? According to an influential formulation due to philosopher Ernest Nagel, when a scientific theory "reduces" another, the reduced theory is deductively subsumed under the reducing theory: thus for example chemistry is deduced from quantum mechanics, and molecular biology from chemistry. Recent critics, using examples from science, argue that Nagel's criteria for theory reduction are both too strict, and too weak. Prof. Grosholz reviews Nagel's model and its difficulties, and argues that theory reduction faces similar problems in mathematics. Certain proofs of Fermat's conjectures about whole number solutions of quadratic and cubic polynomials, by means of the alliance of number theory with complex analysis, lead not deductively but abductively (adding content) to the study of algebraic number fields, and class field theory. This extension of number theory is at once too strong and too weak to look like Nagelian theory reduction, which is precisely why it turns out to be so fruitful.

## Counting Problems for Elliptic Curves over a Fixed Finite Field

Speaker:

Nathan Kaplan
Date:

Thu, Mar 30, 2017 - Sun, Apr 30, 2017
Location:

Colorado State University
Conference:

PIMS CRG in Explicit Methods for Abelian Varieties Abstract:

Let E be an elliptic curve defined over a finite field with q elements. Hasse’s theorem says that #E(F_q) = q + 1 - t_E where |t_E| is at most twice the square root of q. Deuring uses the theory of complex multiplication to express the number of isomorphism classes of curves with a fixed value of t_E in terms of sums of ideal class numbers of orders in quadratic imaginary fields. Birch shows that as q goes to infinity the normalized values of these point counts converge to the Sato-Tate distribution by applying the Selberg Trace Formula.
In this talk we discuss finer counting questions for elliptic curves over a fixed finite field. We study the distribution of rational point counts for elliptic curves containing a specified subgroup, giving exact formulas for moments in terms of traces of Hecke operators. This leads to formulas for the expected value of the exponent of the group of rational points of an elliptic curve over F_q and for the probability that this group is cyclic. This is joint with work Ian Petrow (ETH Zurich).
Please see the event webpage for more information.

## Managing Patients with Chronic Conditions

Speaker:

Mariel Lavieri
Date:

Thu, Feb 23, 2017
Location:

University of Calgary, Downtown Campus
Conference:

Lunchbox Lecture Series Abstract:

Chronic disease management often involves sequential decisions that have long-term implications. Those decisions are based on high dimensional information, which pose a problem for traditional modeling paradigms. In some key instances, the disease dynamics might not be known, but instead are learned as new information becomes available. As a first step, we will describe some of the ongoing research modeling medical decisions of patients with chronic conditions. Key to the models developed is the incorporation of the individual patient's disease dynamics into the parameterization of the models of the disease state evolution. Model conception and validation is described, as well as the role of multidisciplinary collaborations in ensuring practical impact of this work.

## Automorphism groups in algebraic geometry

Speaker:

Michel Brion
Date:

Fri, Mar 10, 2017
Location:

PIMS, University of British Columbia
Conference:

PIMS/UBC Distinguished Colloquium Abstract:

The talk will first present some classical results on the automorphisms of complex projective curves (or alternatively, of compact Riemann surfaces). We will then discuss the automorphism groups of projective algebraic varieties of higher dimensions; in particular, their "connected part" (which can be arbitrary) and their "discrete part" (of which little is known).

## Efficient Compression of SIDH Public Keys

Speaker:

David Jao
Date:

Thu, Mar 23, 2017
Location:

University of Waterloo
Conference:

PIMS CRG in Explicit Methods for Abelian Varieties Abstract:

Supersingular isogeny Diffie-Hellman (SIDH) is an attractive candidate for post-quantum key exchange, in large part due to its relatively small public key sizes. In this work we develop methods to reduce the size of public keys in isogeny-based cryptosystems by more than a factor of two, with performance cost comparable to that of a round of standalone SIDH key exchange, using a combination of techniques from the theory of elliptic curve descent, faster bilinear pairings, and windowed Pohlig-Hellman for discrete logarithms. Our results provide SIDH public keys of 330 bytes for the 128-bit quantum security level, far smaller than any other available alternative, and further strengthen the case for SIDH as a promising post-quantum primitive.
Joint work with Craig Costello, Patrick Longa, Micahel Naehrig, Joost
Renes, and David Urbanik

## Prison Guard’s Dilemma: Optimal Inmate Assignment by Multi-Objective MILO

Speaker:

Dr. Tamás Terlaky, Lehigh University
Date:

Tue, Jan 10, 2017
Location:

University of Calgary - Downtown Campus
Conference:

Lunchbox Lecture Series Abstract:

he Pennsylvania Department of Correction operates 29 correctional facilities (prisons) and about 50,000 prisoners (inmates) each year. The assignments of inmates to appropriate correctional facilities is a complex task. Well over 80 rules need to be considered. Many of them, such as the security of prison units, yield hard constraints; while others, such assigning the inmates to prisons close to their home, are arranged in a preference hierarchy because it is impossible to satisfy all for all inmates.
We are giving an overview of the complexity of the problem; discuss the data/rule collection phase of the project by using decision trees; discuss a how the MILO model is developed by using a weighted penalty objective function. Finally we discuss further extensions of the model, including waiting lists for mental/educational/job training programs, and transfers between facilities.
Based on joint work with L. Plebani, M. Shahabsafa, G. Wilson and K. Bucklen.

## Distinguished models of intermediate Jacobians

Speaker:

Jeff Achter
Date:

Thu, Feb 9, 2017
Location:

Colorado State University
Conference:

PIMS CRG in Explicit Methods for Abelian Varieties Abstract:

Consider a smooth projective variety over a number field. The image of the associated (complex) Abel-Jacobi map inside the (transcendental) intermediate Jacobian is a complex abelian variety. We show that this abelian variety admits a distinguished model over the original number field, and use it to address a problem of Mazur on modeling the cohomology of an arbitrary smooth projective variety by that of an abelian variety.
(This is joint work with Sebastian Casalaina-Martin and Charles Vial.)
See the event webpage for more details.