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Simons Laufer Mathematical Sciences Institute (SLMath)

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

Simons Laufer Mathematical Sciences Institute (SLMath)

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

The Integers conferences are international conferences held for the purpose of bringing together mathematicians, students, and others interested in number theory and combinatorics.

In Honor of the 80th Birthdays of Melvyn Nathanson and Carl Pomerance

The series of Cumberland Conferences on Combinatorics, Graph Theory, and Computing began with the intention of meeting a regional need. The conferences bring together internationally known researchers, university and college professors from throughout the southeastern region, some industrial mathematicians and computer scientists, graduate students in these fields, and some undergraduate students. Particular efforts are made to include faculty and students from smaller institutions throughout the region. These conferences are typically held during the month of May, immediately after the end of the spring semester, and the location rotates among universities in the Cumberland region.

Simons Laufer Mathematical Sciences Institute (SLMath)

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

Simons Laufer Mathematical Sciences Institute (SLMath)

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

IPCO (Integer Programming and Combinatorial Optimization) is a conference that is sponsored by the Mathematical Programming Society. IPCO is a forum for researchers and practitioners working on various aspects of integer programming and combinatorial optimization. The aim is to present recent developments in theory, computation, and applications of integer programming and combinatorial optimization. The conference and summer school will take place at Johns Hopkins University in Baltimore, Maryland, USA.

Courses and Events for Math Students and Early Career Researchers,    Algorithms and Data Structures

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

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

Can machines prove theorems? Can they have mathematical ideas? On one hand, category theory offers a formalism for axiomatising ideas from machine learning. On the other hand, mathematicians are excited about the prospect of utilising machine learning techniques to spot new patterns in vast swathes of combinatorial data and hence formulate new conjectures. The purpose of this workshop is to bring together experts from across algebraic combinatorics, category theory, and machine learning in order to make headway on topics at the intersection of these fields.

American Institute of Mathematics, Pasadena, California (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to further developing the method of flag algebras and its applications. Flag algebras, developed by Razborov in 2007, allows one to solve problems in combinatorics via streamlined calculations that combine elements from computer engineering and optimization. It led to many recent breakthroughs on long-standing open problems of Erdős, Sós, Turán, Gromov and Zarankiewicz, to name a few. The technique is versatile and can be applied in other settings than graphs and hypergraphs including permutations, oriented graphs, point sets, embedded graphs, and phylogenetic trees.

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

This workshop encompasses three major aspects of computation within Representation Theory and Algebraic Combinatorics. One concerns the development of efficient algorithms to compute important quantities in order to understand and classify them better. Such problems include structure constants and representation theoretic multiplicities, mutation invariants in cluster algebras, computing dimensions of coinvariant rings, characters of finite-dimensional representations, coefficients of Kazhdan-Lusztig polynomials, web bases etc. This is closely related to understanding what optimality we could expect and in particular the computational complexity aspects of those problems. Their computational complexity class can also be used to understand the existence of combinatorial interpretations, in particular for major structure constants lacking positive formulas like Kronecker and plethysm coefficients. On the other hand, representation theory has seen important applications within computational complexity theory, in the context of Geometric Complexity Theory and Quantum Information Theory.

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

Webs are diagrammatic tools for representing complex calculations graphically. These diagrams first arose from the representation theory of classical groups, and they have since become important in disparate areas of mathematics. In representation theory, they encode morphisms of quantum groups. In topology, webs give rise to powerful link invariants. In algebra and geometry, Kuperberg's \(mathrm{sl}(3)\) web bases have important relationships with the theory of cluster algebras and affine buildings. In combinatorics, they explain certain dynamics on Young tableaux. Recent work by Gaetz--Pechenik--Pfannerer--Striker--Swanson introduced an \(\mathrm{sl}(4)\) web basis that has exploited and extended exciting connections between webs, plabic graphs, and crystals. There are further connections to total positivity, duality conjectures for cluster algebras and mirror symmetry.

Group Theory,    Algebra