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Scientific

Extrema of 2D Discrete Gaussian Free Field - Lecture 2

Speaker: 
Marek Biskup
Date: 
Tue, Jun 6, 2017
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-CRM Summer School in Probability 2017
Abstract: 
The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

Graphical approach to lattice spin models - Lecture 2

Speaker: 
Hugo Duminil-Copin
Date: 
Tue, Jun 6, 2017
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-CRM Summer School in Probability 2017
Abstract: 
Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by √ 2 + √ 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

Graphical approach to lattice spin models - Lecture 1

Speaker: 
Hugo Duminil-Copin
Date: 
Mon, Jun 5, 2017
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-CRM Summer School in Probability 2017
Abstract: 
Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proof that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by √ 2 + √ 2 , and that the magnetisation of the three-dimensional Ising model vanishes at the critical point.

Extrema of 2D Discrete Gaussian Free Field - Lecture 1

Speaker: 
Marek Biskup
Date: 
Mon, Jun 5, 2017
Location: 
PIMS, University of British Columbia
Conference: 
PIMS-CRM Summer School in Probability 2017
Abstract: 
The Gaussian free field (GFF) is a fundamental model for random fluctuations of a surface. The GFF is closely related to local times of random walks via relations that originated in the study of spin systems. The continuous GFF appears as the limit law of height functions of dimer covers, uniform spanning trees and other models without apparent Gaussian correlation structure. The GFF is also a simple example of a quantum field theory. Intriguing connections to SLE, the Brownian map and other recently studied problems exist. The GFF has recently become subject of focused interest by probabilists. Through Kahane's theory of multiplicative chaos, the GFF naturally enters into models of Liouville quantum gravity. Multiplicative chaos is also central to the description of level sets where the GFF takes values proportional to its maximum, or values order-unity away from the absolute maximum. Random walks in random environments given as exponentials of the GFF show intriguing subdiffusive behavior. Universality of these conclusions for other models such as gradient systems and/or local times of random walks are within reach.

Abelian varieties with good reduction everywhere

Speaker: 
Rene Schoof
Date: 
Thu, May 25, 2017
Location: 
PIMS, University of Calgary
Conference: 
PIMS CRG in Explicit Methods for Abelian Varieties
Abstract: 

Over TeX Embedding failed! there do not exist any non-zero abelian varieties with good reduction everywhere. However, over all but six real quadratic fields they actually do exist. In this talk we determine, for certain small real quadratic fields TeX Embedding failed!, the set of abelian varieties over TeX Embedding failed! with good reduction everywhere.

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: 
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.

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.
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