In algebra the concepts of a finitely generated and finitely presentable object work well: an algebra A in a variety has finitely many generators iff it is a finitely generated object (the hom-functor of A preserves directed unions). And A is presented by finitely many generators and...
Adding numbers and shuffling cards
Persi Diaconis (Stanford Univ., USA)
When numbers are added in the usual way, 'carries' accrue along the way. How do the carries go? if there was just a carry, is it more or less likely to have a carry in the next place? It turns out that carries form a Markov chain with an 'Amazing' transition matrix (really? are any...
Recurrent Neural Networks and Ordinary Differential Equations
Deep learning has become a prominent method for many applications, for instance computer vision or neural language processing. Mathematical understanding of these methods is yet incomplete. A recent approach has been to view a neural network as a discretized version of an ordinary...
Permutahedra and associahedra are classical structures at the interface between combinatorics, discrete geometry, algebra, and algorithms. I will first survey some of their properties and then present the generalization of quotientopes, a family of polytopes realizing any lattice quotient of...
Individual variation in susceptibility or exposure to SARS-CoV-2 lowers the herd immunity threshold
Gabriela Gomes (University of Strathclyde, Glasgow, UK)
Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) emerged in China in late 2019 and spread worldwide causing the ongoing pandemic of coronavirus disease (COVID-19). Scientists throughout the world have engaged with governments, health agencies, and with each other, to address this...
From differential equations to deep learning for image analysis
Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation....
In this talk I will present a survey of arithmetic Ramsey theory. This fascinating subject, which focuses on patterns which arise in arbitrary finite colourings of the natural numbers, combines ideas and tools from diverse areas of mathematics, such as graph theory, Fourier analysis,...
Computation, Statistics, and Optimization of random functions
When faced with a data analysis, learning, or statistical inference problem, the amount and quality of data available fundamentally determines whether such tasks can be performed with certain levels of accuracy. Indeed, many theoretical disciplines study limits of such tasks by investigating...
The search for simple modules
Stephen Donkin (The University of York, UK)
We trace some lines of the search for simple modules (equivalently irreducible representations) in the Lie type setting of highest weight theory. The story starts in the 1890s and 1900s with the work of Frobenius and Schur on symmetric groups and general linear groups. It continues with the...
The history of continued fractions and Padé approximants
Claude Brezinski (Univ. of Sciences and Technologies of Lille, France)
Continued fractions have an history as long as the history of mathematics. I will begin by several short histories showing where continued fractions appear. Then, I review some mathematical problems involving them such as the golden ratio, the GCD, the square root, and Diophantine equations....
In the years 1928 to 1937 a small group of scientists and engineers from different countries of Western Europe moved to the Soviet Union. They were hired to help in tasks anticipated by the early Five-Year Plans. That is, helping to achieve the rapid industrialization objectives promoted by...
S-structures on Lie algebras have been defined recently by Vinberg, as an extension of gradings by abelian groups. In this talk it will be shown how the exceptional simple Lie algebras of types E7 and E8are endowed with some SL2n-structures, and are thus described very nicely in terms of the...
Representing posets, groups, monoids and categories by other structures is associated with Prague school of categories. In this lecture we survey recent development which puts this classical topic in a yet another (combinatorial and model theoretic) context. There will be coffee and...
Kepler's problem (around 1706) concerns the motion of a planet around an immovable planet in the presence of the gravitational force. Based on empirical evidence, Kepler stated that the moon moves on an elliptical path around the earth (known as Kepler's first law, and secondly he stated...
From oscillations to integrable systems and Symplectic Geometry
Oscillations can be observed in many natural physical and biological phenomena. The pendulum can be used as an introduction to several mathematical concepts. Actually Jacobi created the theory of elliptic functions to solve the motion of the pendulum. His work can be revisited using...
Minimal surfaces are ubiquitous in geometry and applied science but their existence theory is rather mysterious. For instance, Yau in 1982 conjectured that any 3-manifold admits infinitely many closed minimal surfaces but the best one knows is the existence of at least three.After a brief...
The talk will be on statistical modelling for problems involving functional data. Functional data are elements of infinite dimensional spaces, and their statistical analysis has been popularized in the last twenty years (mainly with Ramsay-Silverman's books, ), leading to a new field of...
The understanding of liquid crystal behaviour has traditionally been an area with a fertile interaction between science and mathematics. The lecture will describe different theories of liquid crystals and some recent results concerning their predictions and the connection between them....
I shall consider a rhombus tiling model (equivalently, a dimer model on a hexagonal graph) with a free boundary. The correlation of a small triangular hole in this model will be determined. As I shall explain, this kind of problem features phenomena which are parallel to phenomena in...
The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic property of this group being left-orderable is related to two other aspects of 3-dimensional topology, one geometric and the other essentially analytic....
Computational modelling of problems of solid deformation, with applications of goal oriented techniques to thermoforming processes
The reliability of computational models of physical processes has received much attention and involves issues such as validation of the mathematical models being used, the error in any data that the models need and the accuracy of the numerical schemes being used, verification. These issues...
Leibniz is still alive - and so are his mathematical ideas in the third millennium
Marcel Erné (Leibniz Universität Hannover, Germany)
In the current year 2016, we are celebrating the third centennial of Leibniz's death. This is a good occasion to present some highlights in the immense mathematical work of the great universal genius under a modern point of view, and to add some surprising interpretations that might have...
In this talk I want to present my point of view on what (higher) representation theory is and why it is useful. I would like to show, using a baby example, how classical representation theory transforms into higher representation theory, in particular, explaining what the latter one is. At...
O vídeo da conferência pode ser visto aqui. Resumo: "Ninguém pode ser Matemático se não tiver a alma de um poeta", dizia Sonia Kovalevskaya. No entanto, a lógica matemática dificilmente parece deixar lugar à...
Normal forms in differential geometry and metrics on Lie groupoids
In this lecture I will recall some classical and not so classical normal form theorems in differential geometry and I will explain a new, far reaching, metric approach to these results using the language of Lie groupoids....
The sensitivity of interest rate options to monetary policy decisions: a regime-shift pricing approach
René Ferland (Université du Québec à Montréal, Canada)
We look at whether monetary decisions constitute a significant macro-finance risk for interest rate options and related implied volatilities. We devise an option-pricing model based on the dynamics of the Federal Reserve’s target rate via a regime-shift approach modeled as discrete...
In 1963, Erdos and Rényi proved the paradoxical result that there is a graph R with the property that, if a countable graph is chosen by selecting edges independently with probability 1/2, the resulting graph is almost surely isomorphic to R. Their proof was a beautiful example of a...
Introduced by R. Schwartz more than 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely...
I will outline the main ideas of the new approach to foundations of practical mathematics which we call univalent foundations. Mathematical objects and their equivalences form sets, groupoids or higher groupoids. According to Grothendieck's idea higher groupoids are the same as homotopy...
Robust optimization: the need, the challenge, the success
Aharon Ben-Tal (Technion-Israel Institute of Technology, Israel)
We list and illustrate by examples various sources of uncertainty associated with optimization problems. We then explain the difficulties arising when solving such uncertainty affected problems due to lack of full information on the nature of the uncertainty on one hand, and the likelihood...
Minimizing greenhouse emissions in vehicle routing
The Vehicle Routing Problem holds a central place in distribution management. In the classical version of the problem the aim is to distribute goods to a set of geographically dispersed customers under a cost minimization objective and various side constraints. In recent years a number of...
Variations on a conjecture of Nagata
Ciro Ciliberto (Univ. degli Studi di Roma Tor Vergata, Italy)
The classical multivariate Hermite interpolation problem asks for the determination of the dimension of the vector space of polynomials of a given degree d in a given number r of variables having n assigned, sufficiently general, zeroes with given multiplicities. This is trivial for n=1, but...
From Numerics to Computational Science and Engineering
Rolf Jeltsch (ETH Zurich, Switzerland)
The birth of the discipline numerics is due to the invention of modern computers in the forties of the last century. The main task was to learn to live with the deficiencies of computers, which are the limited memory space, that only a finite number of arithmetical operations can be carried...
For a long time there have been two kinds of mathematical computation: symbolic and numerical. Symbolic computing manipulates algebraic expressions exactly, but it is unworkable for many applications since the space and time requirements tend to grow combinatorially. Numerical computing...
Kazhdan-Lusztig polynomials in algebra, geometry, and combinatorics
We consider the inverse problem of discovering the location of point sources from very sparse point measurements in a bounded domain that contains impenetrable obstacles. The sources spread according to a class of linear equations, including the Laplace, heat, and Helmholtz equations, and...
In this talk I present the history and proof of the four-colour theorem: Can every map be coloured with just four colours so that neighbouring countries are coloured differently? The proof took 124 years to find, and used 1200 hours of computer time. But what did it involve, and is it really...
A fundamental difficulty in understanding and predicting large-scale fluid movements in porous media is that these movements depend upon phenomena occuring on small scales in space and/or time. The differences in scale can be staggering. Aquifers and reservoirs extend for thousands of...
Compared to the case of functions of one variable, or to the case of symmetric functions, there are not many algebraic tools to manipulate polynomials in several variables. The symmetric group can greatly help in that matter. Newton found how to transform a discrete set of data into an...
In multiscale processes different phenomena interact on different scales in time and space. Computer simulations of such processes are challenging since the smallest scales should be accurately represented over domains that cover the largest scales. This results in a very ...
Évariste Galois was born 200 years ago and died aged 20, shot in a mysterious early-morning duel in 1832. He left contributions to the theory of equations that changed the direction of mathematics and led directly to what is now broadly described as 'modern' or 'abstract' algebra. In...
Calculus of operator functions
Rajendra Bhatia (Indian Statistical Institute, New Delhi, India)
The basic problem of calculus is how much does a change in the variable x effects the function f(x). The question when the variable x is a matrix is of interest in several contexts. Noncommutativity of matrices leads to special problems and techniques have been invented to...
Exploring the Fourth Dimension: "Flatland: the Movie" and the geometric art of Salvador Dali and Ramon Lull
In the new movie version of "Flatland", the central geometric figure in that film is a cube that rotates about its center in three-space producing a number of central slices in the plane including a square, a rectangle, a rhombus, and a hexagon. What are the analogous central...
Alternating sign matrices have been introduced by combinatorists in the 80's as an extension of the notion of permutation matrices. A simple formula for the number of such matrices of size n has been conjectured. The resolution of this conjecture has been a very active subject in the 80's...
One of the central tenets of signal processing is the Shannon/Nyquist sampling theory: the number of samples needed to
reconstruct a signal without error is dictated by its bandwidth-the length of the shortest interval which contains the support of the spectrum of the signal under study. ...
On the geometry of interest rate models
Tomas Björk (Stockholm School of Economics, Sweden)
The purpose of this talk is to give an overview of some recent work concerning the structural and geometric properties of the
evolution of the forward rate curve in an arbitrage free bond market. The main problems to be discussed are as follows.
1. When is a given forward rate model...
The uses of mathematical images in the seventeenth century, from Galileo to Newton
From long ago, historians have shown the importance of images (emblems books, jesuit propaganda, etc.) during the seventeenth century. Historians of science also emphasized the role played by Optics and its representations. But there has been quite an absence about the role played by images...
Flow control in the presence of shocks
Enrique Zuazua (IMDEA-Matemáticas & Universidad Autónoma de Madrid, Spain)
Flow control is one of the most challenging and relevant topics connecting the theory of
Partial Differential Equations (PDE) and Control Theory. On one hand the number of
possible applications is huge including optimal shape design in aeronautics. On the other
hand, from a purely...
What's an infinite dimensional manifold and how can it be useful in anatomy?
David Mumford (Division of Applied Mathematics, Brown University, USA)
There has been a huge explosion in medical
imaging, in spatial and temporal resolution, in imaging many processes as well as organs.
New mathematical tools have entered the game, which are not so elementary. It's important to have some basic understanding of them, to know how what to...
Singular perturbation systems related to segregation
(University of Texas at Austin, USA)
I will discuss a singular perturbation system related to segregation and the regularity properties of the limiting solution and its free boundaries.
What geometry in the early history of architecture?
It is commonly said that Euclidean geometry played a large part in the development of architecture, but is this really true? An examination of the two earliest and best known treatises of architecture, On architecture by Vitruvius and the Sketchbooks of Villard de Honnecourt reveal no trace...
Variational Analysis and Generalized Differentiation: New Trends and Developments
(Dep. Mathematics, Wayne State Univ.)
Nonsmooth functions, sets with nonsmooth boundaries, and set-valued mappings naturally and frequently appear in various aspects of analysis. Constrained optimization, calculus of variations and its modern form of optimal control, stochastic and statistical problems, mathematical economics,...
Homomorphisms of the alternating group A_5 into semisimple groups
George Lusztig (Dept. Math., MIT)
Usually in representation theory one studies homomorphisms of a complicated group G into a simpler group: the general linear group of a vector space. But it is also interesting to replace the general linear group by other algebraic groups such as
symplectic, orthogonal or even exceptional...
Several transport-diffusion systems arise as simple models in chemotaxis (motion of bacterias or amebia interacting through a chemical signal) and in angiogenesis (development of capillary blood vessels
from an exhogeneous chemoattractive signal by solid tumors). These systems describe the...
Neuronal Calcium Signaling
Steve Cox (Computational and Applied Math., Rice University)
Calcium, the most important of the second messengers, regulates the synaptic plasticity that is underlies our ability to learn. Calcium enters cells through single-protein channels in the cells' outer membrane. We exploit the ability to dynamically monitor
cytosolic calcium, throughout...
Lorenz Strange Atractors
Marcelo Viana (IMPA, Brasil)
The Lorenz strange attractor, introduced in the sixties in the context of thermal convection and weather prediction, became a
paradigm chaotic behavior, and a crucial model to try and describe this type of dynamics.
In the nineties there were two fundamental developments:
On the one...
The Poincare Conjecture and Algorithmic Problems in Algebra
Alexey Sossinsky (Indep. Univ. Moscow)
It has been announced that the Poincare
Conjecture (claiming that any simply connected closed compact three-dimensional manifold is homeomorphic to the sphere), one of the most famous problems in mathematics,
has been solved by Grisha Perelman of St.Petersburg.
Although no complete...
Color printers, mailboxes, fish, and Homer Simpson - or - Centroidal Voronoi tessellations: algorithms and applications
M. D. Gunzburger (Florida State Univ.)
One of the beauties of mathematics is that it can uncover connections between seemingly disparate applications. One of the most fertile grounds for unearthing connections is computational algorithms where one often discovers that an algorithm developed for one application is equally useful...