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Computer Assisted Proofs in Applied Mathematics (SLMath)
Organizers: LEAD Jonathan Jaquette (New Jersey Institute of Technology), Evelyn Sander (George Mason University)One of the core elements of applied mathematics is mathematical modeling consisting of nonlinear equations such as ODEs, and PDEs. A fundamental difficulty which arises is that most nonlinear models cannot be solved in closed form. Computer assisted proofs are at the forefront of modern mathematics and have led to many important recent mathematical advances. They provide a way of melding analytical techniques with numerical methods, in order to provide rigorous statements for mathematical models that could not be treated by either method alone. In this summer school, students will review standard computational and analytical techniques, learn to combine these techniques with more specialized methods of interval arithmetic, and apply these methods to establish rigorous results in otherwise intractable problems
Updated on May 21, 2025 10:28 AM PDT -
Principled Scientific Discovery with Formal Methods (IBM, Yorktown)
Organizers: Kenneth Clarkson (IBM Research Division), Cristina Cornelio (Samsung AI), Claudia D Ambrosio (Centre National de la Recherche Scientifique (CNRS); École Polytechnique), Sanjeeb Dash (IBM Thomas J. Watson Research Center), Lior Horesh (IBM Thomas J. Watson Research Center)<p>The traditional scientific method cycle, with Francis Bacon and Rene Descartes, its concievers in the center, alongside formal and statistical AI machinary, as a propsective evolution of the method. <br /> </p>The summer school aims to expose participants to formal methods that can facilitate principled scientific discovery. The school will cover some of the basic automated statistical inference (in the form of machine learning techniques) and reasoning methods that are commonly used in scientific discovery, as well as novel techniques developed to tackle open questions and issues. This summer school will address novel computational methods for scientific discovery and focus on fusing axiomatic knowledge and experimental data to enable principled derivations of models of natural phenomena along with certificates of the consistency of these models with background knowledge specified as axioms.
Updated on May 23, 2025 08:55 AM PDT -
Geometry and Dynamics in Higher Rank Lie Groups (UC Berkeley)
Organizers: Richard Canary (University of Michigan), Sara Maloni (University of Virginia), Wenyu Pan (University of Toronto; University of Toronto), Cagri Sert (University of Warwick), LEAD Tengren Zhang (National University of Singapore)<p>Flats and hyperbolic planes in a higher rank symmetric space</p> Drawn by Steve Trettel.Lie groups are central objects in modern mathematics; they arise as the automorphism groups of many homogeneous spaces, such as flag manifolds and Riemannian symmetric spaces. Often, one can construct manifolds locally modelled on these homogeneous spaces by taking quotients of their subsets by discrete subgroups of their automorphism groups. Studying such discrete subgroups of Lie groups is an active and growing area of mathematical research. The objective of this summer school is to introduce young researchers to a class of discrete subgroups of Lie groups, called Anosov subgroups.
Updated on May 22, 2025 01:11 PM PDT -
Topological and Geometric Structures in Low Dimensions (SLMath)
Organizers: LEAD Kenneth Bromberg (University of Utah), Kathryn Mann (Cornell University)<p>Laminations arise naturally in hyperbolic geometry and (pseudo-) Anosov flows [Image by Jeffrey Brock]</p>This school will serve as an introduction to the SLMath semester “Topological and Geometric Structures in Low-Dimensions”. The school consists of two mini-courses: one on Teichmüller Theory and Hyperbolic 3-Manifolds and the other on Anosov Flows on Geometric 3-Manifolds. Both topics lie at the interface of low-dimensional geometric topology (specifically, surfaces, foliations, and 3-manifolds) and low-dimensional dynamics. The first course will be targeted towards students who have completed the standard first year graduate courses in geometry, topology, and analysis while the second course will geared towards more advanced students who are closer to beginning research. However, we expect that all students will benefit from both courses.
Updated on Apr 21, 2025 03:17 PM PDT
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