| High-dimensional multivariate data is commonly encountered nowadays in a variety of disciplines, including genomics, finance and economics, information technology systems, and biomedical engineering. Understanding the structure of and uncovering relationships among variables measured by these data will have crucial impacts in the corresponding scientific areas.
Though some heuristic algorithms and intuitive methods have been designed for and widely applied in both industrial and scientific applications, as of now, our understandings of them are still limited. The advances of random matrix theory provide a tool set for researchers to study behaviors of many practical algorithms. For example, establishing estimation rates of the algorithms of interest is very helpful in understanding when they should be used in practice. |
Programs
During the 2017-2018 academic year, the CMSA will be hosting the following ongoing programs. These programs will have various workshops and conferences associated with them. Visit the program pages for more information.
Spacetime and Quantum Mechanics, Total Positivity and Motives |
| Recent developments have poised this area to make serious advances in 2019, and we feel that bringing together many of the relevant experts for an intensive semester of discussions and collaboration will trigger some great things to happen. To this end, the organizers will host a small workshop during fall 2019, with between 20-30 participants. They will also invite 10-20 longer-term visitors throughout the semester.
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Noncommutative Real Algebraic Geometry and Analysis |
| There has been great progress over a decade and a half in understanding equations and inequalities for functions having matrix variables. Three experts in these areas– Bill Helton, Igor Klep, and Jurij Volcic–will be at CMSA for the Fall 2019. They will extend basic theory and seek ideas in related areas of science and mathematics from the rich Cambridge environment and see how the last decade’s worth of new developments in free analysis might apply to them. In addition to regular seminars and meetings with scholars in residence, the program will also feature a workshop on the subject of noncommutative convexity and applications, organized by Boaz Barak (Harvard SEAS), William Helton (UCSD), and Pablo Parrilo (MIT)..
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| In Fall 2018, CMSA will focus on a program that aims at recent mathematical advances in describing shape using geometry and statistics in a biological context, while also considering a range of physical theories that can predict biological shape at scales ranging from macromolecular assemblies to whole organ systems
In the Fall of 2018, the CMSA will be hosting two workshops as part of this program. The Workshop on Morphometrics Morphogenesis and Mathematics will take place on October 22-26. The workshop on Morphogenesis: Geometry and Physics will take place on December 3-6, 2018. |
| During Academic year 2018-19, the CMSA will be hosting a Program on Topological Aspects of Condensed Matter.
New ideas rooted in topology have recently had a big impact on condensed matter physics, and have highlighted new connections with high energy physics, mathematics and quantum information theory. Additionally, these ideas have found applications in the design of photonic systems and of materials with novel mechanical properties. The aim of this program will be to deepen these connections by foster discussion and seeding new collaborations within and across disciplines. |
| The Simons Collaboration on Homological Mirror Symmetry brings together a group of leading mathematicians working towards the goal of proving Homological Mirror Symmetry (HMS) in full generality, and fully exploring its applications.
Mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of progress in mathematics. At the 1994 ICM, Kontsevich reinterpreted mirror symmetry as a deep categorical duality: the HMS conjecture states that the derived category of coherent sheaves of a smooth projective variety is equivalent to the Fukaya category of a mirror symplectic manifold (or Landau-Ginzburg model). We envision that our goal of proving HMS in full generality can be accomplished by combining three main viewpoints:
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Previous Programs:
| During Academic year 2017-18, the CMSA will be hosting a Program on Combinatorics and Complexity. This year will be organized by Noga Alon, Boaz Barak, Jacob Fox, Madhu Sudan, Salil Vadhan, and Leslie Valiant.
Combinatorics and Computational Complexity have enjoyed a rich history of interaction leading to many significant developments in the two fields, such as the theories of NP-completeness, expander graphs, pseudorandomness, and property testing. Lately these fields have seen many new points of intersection such as in the development of the polynomial method (used, for example, in recent advances on the cap-set problem as well as in development of optimal list-decodable codes), the method of interlacing families of polynomials (yielding Ramanujan graphs and the resolution of the Kadison-Singer problem), and the theory of randomness extractors (yielding explicit constructions of Ramsey graphs). This special program will bring together experts in the fields to collaborate, to learn about the latest advances in the area, and to forge new connections. |
| Large random matrices provide some of the simplest models for large, strongly correlated quantum systems. The statistics of the energy levels of ensembles of such systems are expected to exhibit universality, in the sense that they depend only on the symmetry class of the system. Recent advances have enabled a rigorous understanding of universality in the case of orthogonal, Hermitian, or symplectic matrices with independent entries, resolving a conjecture of Wigner-Dyson-Mehta dating back 50 years. These new developments have exploited techniques from a wide range of mathematical areas in addition to probability, including combinatorics, partial differential equations, and hydrodynamic limits. It is hoped that these new techniques will be useful in the analysis of universal behaviour in matrix ensembles with more complicated structure such as random regular graph models, or 2D matrix ensembles, as well as more physically relevant systems such as band matrices and random Schroedinger-type Hamiltonians. For some of these models, results in the direction of universality have already been obtained. |