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Simons Laufer Mathematical Sciences Institute (SLMath)

This school will introduce students to a range of powerful combinatorial tools used to understand algebraic objects ranging from the homogeneous coordinate ring of the Grassmannian to symmetric functions. The summer school will center around two main lecture series "Webs and Plabic Graphs" and "Vertex Models and Applications". While the exact applications differ, both courses will center on graphical models for algebraic problems closely related to Grassmannian and its generalizations. This school will be accessible to a wide range of students. Students will leave the school with a solid grasp of the combinatorics of webs, plabic graphs, and the six-vertex model, an understanding of their algebraic applications, and a taste of current research directions.

IAS/Park City Mathematics Institute

Extremal graph theory and Ramsey theory are two of the central branches of modern extremal combinatorics, which seeks to understand the size and structure of discrete objects under certain natural constraints. In this course we will explore these topics, seeing both some of the beautiful techniques developed to study such problems, as well as many innocent-looking problems that seem completely out of reach of the currently-known techniques. We will also see some of the many connections these questions have to other areas of mathematics, including geometry, number theory, probability, and theoretical computer science.

Research Theme: Extremal and Probabilistic Combinatorics

Simons Laufer Mathematical Sciences Institute (SLMath)

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 automatic differentiation, and apply these methods to establish rigorous results in otherwise intractable problems.

Automated Theorem Proving,    Applied Mathematics (in general)

Simons Laufer Mathematical Sciences Institute (SLMath)

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.

Courses and Events for Physics Students,    Mathematical Physics

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. These subgroups are central objects in the study of higher Teichmuller theory, convex projective geometry, and character varieties. In this summer school, there will be an emphasis on discussing some of the dynamical tools that have recently been successfully used to study Anosov subgroups. The required background in dynamics, hyperbolic geometry, and Lie theory will also be discussed. Aside from providing a stimulating academic environment for learning about Anosov subgroups, this summer school also aims to be a relaxed and friendly space for participants to interact with fellow aspiring mathematicians.

Geometry and Topology,    Group Theory

Organised in partnership with the Clay Mathematics Institute

Simons Laufer Mathematical Sciences Institute (SLMath)

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.

Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI (ICERM)

Linear Algebra over finite fields is a building block for several applications including data storage, error detection and correction, and public-key cryptography. These applications enable the security and possibility of our daily digital lives. This workshop aims to expose and engage junior researchers in the foundations and applications of linear algebra over finite fields.

The program will unite researchers from a number of areas: algebraic and complex geometry, arithmetic geometry, Hodge theory, and mathematical physics. It will bring theoretically and computationally oriented researchers together, expecting that computations will illuminate conjectures made by the theorists and that theory will enlarge the range of what can be computed. We intend to develop databases of certain types of K3 surfaces for the L-Functions and Modular Forms Database and promote the development of software for computations on K3 surfaces in Magma, SageMath, or other systems for public release.

Geometry and Topology,    Number Theory, Arithmetic