Summer Graduate School

The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner's lemma and Brouwer's fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Shih-Lee, Komiya, Frick-Zerbib, and Soberon) and have been widely applied as well. In my lecture series we will prove several different KKM-type theorems and use them to solve a variety of problems in discrete geometry, combinatorics and game theory. We will also discuss open problems that may benefit from a similar topological approach. 

Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.

Various classes of problems from discrete and convex geometry, mathematical mechanics, algorithm complexity theory and approximation theory can be transformed into challenging, and often unsolved problems in equivariant topology. The focus of our lecture series is understanding where these topological problems come from and then demonstrating some of the diverse methods of algebraic topology which are critical to study and subsequently solve them.

The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner's lemma and Brouwer's fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Shih-Lee, Komiya, Frick-Zerbib, and Soberon) and have been widely applied as well. In my lecture series we will prove several different KKM-type theorems and use them to solve a variety of problems in discrete geometry, combinatorics and game theory. We will also discuss open problems that may benefit from a similar topological approach. 

Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.

Various classes of problems from discrete and convex geometry, mathematical mechanics, algorithm complexity theory and approximation theory can be transformed into challenging, and often unsolved problems in equivariant topology. The focus of our lecture series is understanding where these topological problems come from and then demonstrating some of the diverse methods of algebraic topology which are critical to study and subsequently solve them.

The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner's lemma and Brouwer's fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Shih-Lee, Komiya, Frick-Zerbib, and Soberon) and have been widely applied as well. In my lecture series we will prove several different KKM-type theorems and use them to solve a variety of problems in discrete geometry, combinatorics and game theory. We will also discuss open problems that may benefit from a similar topological approach. 

Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.

Various classes of problems from discrete and convex geometry, mathematical mechanics, algorithm complexity theory and approximation theory can be transformed into challenging, and often unsolved problems in equivariant topology. The focus of our lecture series is understanding where these topological problems come from and then demonstrating some of the diverse methods of algebraic topology which are critical to study and subsequently solve them.

Various classes of problems from discrete and convex geometry, mathematical mechanics, algorithm complexity theory and approximation theory can be transformed into challenging, and often unsolved problems in equivariant topology. The focus of our lecture series is understanding where these topological problems come from and then demonstrating some of the diverse methods of algebraic topology which are critical to study and subsequently solve them.

The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner's lemma and Brouwer's fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Shih-Lee, Komiya, Frick-Zerbib, and Soberon) and have been widely applied as well. In my lecture series we will prove several different KKM-type theorems and use them to solve a variety of problems in discrete geometry, combinatorics and game theory. We will also discuss open problems that may benefit from a similar topological approach. 

The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner's lemma and Brouwer's fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Shih-Lee, Komiya, Frick-Zerbib, and Soberon) and have been widely applied as well. In my lecture series we will prove several different KKM-type theorems and use them to solve a variety of problems in discrete geometry, combinatorics and game theory. We will also discuss open problems that may benefit from a similar topological approach. 

Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.