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The theory of dynamical systems, which in barest terms is the study of maps from a set to itself, permeates mathematics and science. It is used in diverse contexts, from proving the existence of solutions of equations to the modelling of complex natural phenomena. In more recent years dynamical methods have proven to be remarkably powerful in relation to problems in number theory, geometry and combinatorics which on the surface may seem to lack dynamical content. These various applications have ...
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Partition functions arise as fundamental quantities in physics, mathematics, statistics, computer science and many engineering disciplines. They are often notoriously hard to compute since they typically involve summations over an exponential number of terms. Different computational approaches have been developed in various mathematical disciplines to bound or estimate these important quantities. The aim of this semester is to bring together experts from optimization, theoretical computer science, information theory and probability with the goal of jointly investigating computational approaches to partition functions, ...
Descriptive set theory is classically defined as the study of definable (e.g. Borel, analytic) sets in Polish spaces -- separable, completely metrizable spaces. It has long thrived in its connections with classical areas of mathematics such as harmonic and functional analysis and logic (set theory and computability theory in particular). Over the course of the last 25 years, however, a vast number of new interactions have opened up the field to novel and exciting directions. The principal goal of the ...
The goal of this semester-long program is to gather the leading experts in the area of Euler systems and the Birch and Swinnerton-Dyer conjecture in order to initiate a more systematic study of Euler systems on higher rank reductive groups and their applications to generalizations of the BSD conjecture (Bloch–Kato–Beilinson conjectures). At the same time, we envision several introductory courses suitable for graduate students and post-doctoral assistants.
This program focuses on three important areas where stochastic dynamical models play a major role, namely, mathematical finance, econometrics, and actuarial sciences. Stochastic dynamical models are of major importance in mathematical finance where, for example, the quality of the non-arbitrage pricing and hedging instruments for derivative products that this theory produces depends strongly on the accuracy of the models that describe the dynamical behavior of the underlying assets. The interplay between finance and stochastic calculus is one of the most ...
Local representation theory, pioneered by Richard Brauer in the 1930s
had its first big successes in the classification of the finite simple
groups. Since then, important and deep connections to areas as varied as
topology, geometry, Lie theory and homological algebra have been
discovered and used. Very recent breakthrough results have now led to
the hope that some of the long standing and deep problems, some of which
have been open for over five decades, can finally be settled.
These recent
...
This semester-long program will be focused on the enumerative
invariants of moduli spaces of sheaves in low dimensions. The main
examples of the moduli spaces we will consider are the moduli spaces of
sheaves on surfaces and Calabi-Yau 3-folds, the moduli spaces of vector
and Higgs bundles on curves, and quiver varieties.
The program will
cover a wide range of aspects of this problem: enumerative geometry,
knot theory, singularity theory and arithmetic, aiming to facilitate
fruitful interactions between the various
...
Interplay between the probability theory and geometric group theory yielded many spectacular results enriching both areas. Asymptotic properties of various models coming from statistical physics and probabilities were connected to the geometry and combinatorics of the underlying discrete algebraic structure (for instance a group, a graph, a group action). The study started with simple random walks and their boundary theory and then extended to other models, such as percolation or the Ising model. Substantial progress was achieved on many different fronts, and several applications to group theoretical problems ...
The basic goal of this program is to advance research in geometric as well as stochastic methods in mechanics, control theory, and imaging, taking into account recent developments also in connected areas, such as infinite dimensional Lie groups and representation theory, quantum gauge field theory, infinite dimensional analysis. Applications ranging from mathematical physics to modeling in life sciences and engineering will be considered.
The issues of climate change and biodiversity loss are definitely the major concerns for 21st-century science. Ecology is one of the most involved sciences in this challenge. Undoubtedly, mathematics and computer simulations have played an important role in the development of theoretical ecology for almost one century. They are often designated as "modeling". But what are "modeling" and "theory" in ecology and what are the places occupied respectively by mathematics and computer science are still controversial issues.
The structure considered
...
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