Programs & Events
Public Lecture: Inverse Problems and Harry Potter's Cloak
Apr 20, 2016
Dr. Uhlmann will begin his talk by describing several inverse problems that arise in several contexts. For example, human vision: from the measurements of scattered light that reaches our retinas, our brains construct a detailed three-dimensional map of the world around us. Solving inverse problems is in fact how we obtain a large part of our information about the world we live in.
In the second part of his talk, Dr. Uhlmann will discuss invisibility, addressing the question, "can we make objects invisible?" This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc., including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter stories. In the last 13 years or so there have been several scientific efforts to achieve invisibility. Dr. Uhlmann will describe a simple and powerful proposal, the so-called transformation optics and the progress that has been made in achieving invisibility.
Computation in Dynamics
Apr 4 - 8, 2016
Numerical computations have always played an important role in the development of the theory of Differential Equations and Dynamical Systems, more and more so as the availability and power of computers has increased dramatically over the last few decades. At the same time, the limitations of computer-assisted numerical calculations have also become increasingly apparent. Notwithstanding their enormous power, the intrinsic finite resolution of computers can lead to significant errors, especially as a result of a large number of calculations through which small errors can accumulate.
An important and growing approach to certain mathematical problems consists of developing rigorous numerical techniques in combination with more classical analytic methods in order to obtain rigorous qualitative and quantitative results. In some cases this leads to the proof of deep mathematical theorems and in other cases to quantitative, and thus more concrete and applicable, versions of abstract... (more)
Organizing Committee
- Denis Gaidashev
- Stefano Galatolo
- Stefano Luzzatto
- Warwick Tucker
- Michael Yampolsky
Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism
Mar 7 - 11, 2016
A surprising discovery of 20th century mathematics is that many deterministic systems exhibit random behavior. One early example of a chaotic system was Lorenz equation used by meteorologist Edward Lorenz as a simplified model of atmospheric convection. One of the most common mechanisms of stochasticity is the Smale horseshoe appearing near a homoclinic intersection.
The Lorentz attractor and Smale horseshoe are typical examples of fractal invariant sets for dynamical systems. Fractal objects are ubiquitous in dynamics, including invariant sets, invariant measures, invariant foliations et cetera.
Thermodynamical formalism is a powerful tool for studying dimensions of fractal objects. It originated in statistical mechanics, but currently it has applications to many areas of mathematics including spectral theory, hyperbolic geometry and probability theory.
The goal of this conference is to bring together experts studying fractal objects in dynamics in order to review recent progress... (more)
Organizing Committee
- Lorenzo Diaz
- Dmitry Dolgopyat
- Maarit Jarvenpaa
- Jörg Schmeling
- Masato Tsujii
- Amie Wilkinson
Public Lecture: The Beautiful Mathematically Ordered Colors of Birds
Feb 17, 2016
The blue and green colors we see in birds, and even some of the ultraviolet that we cannot see, are produced by the way in which light interacts with ordered structures in the tissues of the birds. This order in the structures can be measured using Fourier analysis, a powerful mathematical tool.
Like a prism that decomposes a beam of light into a rainbow of colors, Fourier analysis transforms the geometrical arrangements observed in electron microscope images of the tissues into a mathematical rainbow of basic components that quantify order. We will illustrate how Fourier analysis processes the images and helps decipher the colors of birds and other animals. We will use this application of Fourier analysis to present also some of its mathematical concepts and interest.
The talk will be accessible to all those who are curious about some of the physics behind the bright blue and green colors found in nature and how mathematics can be used to describe such coloration.
Ergodic, Algebraic and Combinatorial Methods in Dimension Theory
Feb 15 - 19, 2016
There are natural interactions between dimension theory, ergodic theory, additive combinatorics, metric number theory and analysis. Each of these fields provides different perspectives on, and complementary approaches to, the hierarchical structures which appear in fractal geometry.
The workshop will focus on recent advances at the interfaces of these fields, including:
- Classical fractals (self-similar and self-affine sets, random fractals)
- Dimension theory and additive combinatorics
- Diophantine approximation and equidistribution
- Schmidt games
- Rigidity phenomena
- Scenery flow methods
- Projection and slice theorems
Organizing Committee
- Michael Hochman
- Izabella Laba
- Pablo Shmerkin
- Barak Weiss
Dimension and Dynamics
Feb 1 - May 6, 2016
Since its introduction by Felix Hausdorff in 1919, the concept of the Hausdorff dimension of sets and measures has been a versatile and powerful tool in classical analysis, geometry and geometric measure theory, mathematical physics and their numerous applications. However, there has been a particularly important symbiosis between dynamical systems and dimension theory. This connection arises both from application of dimension theory to the classification and geometric analysis of dynamical systems (and their invariant sets and measures), and the fact that many classical objects of study in mathematics arise from (sometimes implicit) dynamical systems, which often play a role in the dimension theory of said objects.
Recently, there has been substantial progress on a number of central problems in dimension theory, and while many old problems remain, many new ones have also presented themselves. These include a deeper understanding of the relationship between dimension, entropy and... (more)
Organizing Committee
- Dmitry Dolgopyat
- Michael Hochman
- Izabella Laba
- Stefano Luzzatto
- Yakov Pesin
- Mark Pollicott
- Jörg Schmeling
- Boris Solomyak
- Warwick Tucker
Public Lecture: From Flapping Birds to Space Telescopes: The Mathematics of Origami
Nov 12, 2015
The last decade of this past century has been witness to a revolution in the development and application of mathematical techniques to origami, the centuries-old Japanese art of paper-folding. The techniques used in mathematical origami design range from the abstruse to the highly approachable.
In this talk, Robert Lang will describe how geometric concepts led to the solution of a broad class of origami folding problems - specifically, the problem of efficiently folding a shape with an arbitrary number and arrangement of flaps, and along the way, enabled origami designs of mind-blowing complexity and realism, some of which you'll see, too. As often happens in mathematics, theory originally developed for its own sake has led to some surprising practical applications.
The algorithms and theorems of origami design have shed light on long-standing mathematical questions and have solved practical engineering problems. Examples will be discussed of how origami has enabled safer airbags,... (more)
Computational Aspects of L-functions
Nov 9 - 13, 2015
This conference will revolve around several themes: the computational complexity of L-functions; statistical problems concerning L-functions, such as the distribution of their values, and zeros, moments of L-functions, statistics and size of ranks in families of elliptic curves; practical implementations of algorithms and their applications to testing various conjectures about L-functions; rigorous and certifiable computations of L-functions. One goal is to stimulate dialogue between theoreticians and computationally minded researchers regarding problems to which computation might provide insight or important confirmation of conjectures. In the other direction, we hope that discussions will lead to new ideas concerning algorithms for L-functions.
Organizing Committee
- Valentin Blomer
- John Conrey
- Chantal David
- David Farmer
- Michael Rubinstein
- Peter Sarnak
Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives
Oct 19 - 24, 2015
Only recently has it become feasible to do large scale verification of the predictions of the Langlands program in higher rank cases and to present the results in a way that is accessible widely to mathematicians. Moving from the understanding of Galois representations attached to elliptic curves to those attached to surfaces and other higher-dimensional varieties poses interesting problems in both arithmetic, algebra, geometry, and analysis.
In this workshop, we will consider computational and other explicit aspects of modular forms in higher rank. Topics covered will include: K3 surfaces and their connections to modular forms on orthogonal groups, algebraic modular forms associated to classical groups and their computation, and motives arising from general Calabi-Yau varieties accessible to explicit methods, including hypergeometric motives.
This workshop will bring together researchers who approach the Langlands program from a number of different perspectives, and our goal is to... (more)
Organizing Committee
- Fernando Rodriguez-Villegas
- Matthias Schuett
- Holly Swisher
- Yuri Tschinkel
- Bianca Viray
- John Voight
Public Lecture: Lotteries and Geometries
Oct 2, 2015
For seven years, a group of students from the Massachusetts Institute of Technology exploited a loophole in the Massachusetts State Lottery’s Cash WinFall to win game after game, eventually pocketing more than $3 million. I’ll explain how they did it, why they got away with it, and what it all has to do with the mathematics of finite geometries.
This lecture will take place on the Brown campus in MacMillan Hall (167 Thayer Street), Room 117/Starr Auditorium.
NOTE: This public lecture is co-sponsored by the Brown Mathematics Department and is affiliated with the Algebraic Geometry Northeastern Series (AGNES) fall 2015 conference being held at Brown University.
Modular Forms and Curves of Low Genus: Computational Aspects
Sep 28 - Oct 2, 2015
One of the crowning achievements of number theory in the 20th century is the construction of the modularity correspondence between elliptic curves with rational coefficients and modular forms of weight 2. The consequences of this result resound throughout number theory; for instance, it enables the resolution of certain problems of diophantine equations (e.g., Fermat's last theorem) as well as the systematic tabulation of elliptic curves, which in turn provides the basis for many new conjectures and results.
The aim of this workshop is to lay the groundwork for extending this correspondence to curves of small genus over number fields. The general framework for this correspondence is predicted by the Langlands program, but much remains to be made explicit. We will explore theoretical, algorithmic, computational, and experimental questions on both sides of the correspondence, with an eye towards tabulation of numerical data and formulation of precise conjectures.
Organizing Committee
- John Cremona
- Kiran Kedlaya
- Kristin Lauter
- Ralf Schmidt
- Joseph Silverman
Computational Aspects of the Langlands Program
Sep 9 - Dec 4, 2015
In the late 1960s, Robert Langlands discovered a unifying principle in number theory providing a vast generalization of class field theory to include nonabelian extensions of number fields. This principle gives rise to a web of conjectures called the Langlands program which continues to guide research in number theory to the present day. For example, an important first instance of the Langlands program is the modularity theorem for elliptic curves over the rational numbers, an essential ingredient in the proof of Fermat's last theorem.
Despite its many successes, the Langlands program remains vague in many of its predictions, due in part to an absence of data to guide a precise formulation away from a few special cases. In this thematic program, we will experiment with and articulate refined conjectures relating arithmetic-geometric objects to automorphic forms, improve the computational infrastructure underpinning the Langlands program, and assemble additional supporting... (more)
Organizing Committee
- Alina Bucur
- John Conrey
- David Farmer
- John Jones
- Kiran Kedlaya
- Michael Rubinstein
- Holly Swisher
- John Voight