Monday, July 13th, 2026
Workshop Mini Course: Rick Kenyon
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Dimers and webs Part 1
Speaker: Rick Kenyon
Abstract: We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges
Title: Dimers and webs Part 1
Speaker: Rick Kenyon
Abstract: We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges
Workshop Mini Course: Persi Diaconis
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Markov Chains for Enumeration under Symmetry Lecture 1
Speaker: Persi Diaconis
Abstract: Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution. Problems abound: How do you actually carry out the two steps in problems of interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open. Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group). Lecture 3: Careful proof in a simple case: the binary Burnside process, from Tchebychev polynomials to Schur-Weyl duality. I expect the audience to know something about Markov chains (at the level of Levin- Peres' book) and something about groups (a good undergraduate course should suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.
Title: Markov Chains for Enumeration under Symmetry Lecture 1
Speaker: Persi Diaconis
Abstract: Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution. Problems abound: How do you actually carry out the two steps in problems of interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open. Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group). Lecture 3: Careful proof in a simple case: the binary Burnside process, from Tchebychev polynomials to Schur-Weyl duality. I expect the audience to know something about Markov chains (at the level of Levin- Peres' book) and something about groups (a good undergraduate course should suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.
Tuesday, July 14th, 2026
Workshop Mini Course: Rick Kenyon
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Dimers and webs Part 2
Speaker: Rick Kenyon
Abstract: We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges
Title: Dimers and webs Part 2
Speaker: Rick Kenyon
Abstract: We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges
Workshop Mini Course: Persi Diaconis
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Markov Chains for Enumeration under Symmetry Lecture 2
Speaker: Persi Diaconis
Abstract: Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution. Problems abound: How do you actually carry out the two steps in problems of interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open. Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group). Lecture 3: Careful proof in a simple case: the binary Burnside process, from Tchebychev polynomials to Schur-Weyl duality. I expect the audience to know something about Markov chains (at the level of Levin- Peres' book) and something about groups (a good undergraduate course should suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.
Title: Markov Chains for Enumeration under Symmetry Lecture 2
Speaker: Persi Diaconis
Abstract: Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution. Problems abound: How do you actually carry out the two steps in problems of interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open. Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group). Lecture 3: Careful proof in a simple case: the binary Burnside process, from Tchebychev polynomials to Schur-Weyl duality. I expect the audience to know something about Markov chains (at the level of Levin- Peres' book) and something about groups (a good undergraduate course should suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.
Workshop Mini Course: Amol Aggarwal
Time: 2:00 PM - 3:30 PM
Location: SCGP 102
Title: Asymptotics for the Toda Lattice Part 1
Speaker: Amol Aggarwal
Abstract: The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Title: Asymptotics for the Toda Lattice Part 1
Speaker: Amol Aggarwal
Abstract: The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Wednesday, July 15th, 2026
Workshop Mini Course: Amol Aggarwal
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Asymptotics for the Toda Lattice Part 2
Speaker: Amol Aggarwal
Abstract: The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Title: Asymptotics for the Toda Lattice Part 2
Speaker: Amol Aggarwal
Abstract: The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Workshop Mini Course: Vadim Gorin
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Random lozenge tilings via Nekrasov equations, Part 1
Speaker: Vadim Gorin
Abstract: This course studies the macroscopic behavior and fluctuations of uniformly random lozenge tilings of polygonal domains. We identify tilings with their height functions and focus on central limit theorem-type results for these functions. Our main results establish Gaussian Free Field fluctuations for the associated two-dimensional fields, as well as discrete Gaussian asymptotics for height differences between boundary components of the tiled domain. The class of domains we consider consists of gluings of elementary building blocks, called trapezoids, along a single vertical line. Such gluings may have complicated topology and, in some cases, may even be non-orientable. Exact enumeration results and algebraic properties of lozenge tilings of trapezoids provide a rich set of tools for the analysis. A key ingredient is the identification of the model with a discrete log-gas, which allows for the use of Nekrasov equations (also known as discrete loop or discrete Dyson–Schwinger equations). The course is based on a research monograph written jointly with G. Borot and A. Guionnet.
Title: Random lozenge tilings via Nekrasov equations, Part 1
Speaker: Vadim Gorin
Abstract: This course studies the macroscopic behavior and fluctuations of uniformly random lozenge tilings of polygonal domains. We identify tilings with their height functions and focus on central limit theorem-type results for these functions. Our main results establish Gaussian Free Field fluctuations for the associated two-dimensional fields, as well as discrete Gaussian asymptotics for height differences between boundary components of the tiled domain. The class of domains we consider consists of gluings of elementary building blocks, called trapezoids, along a single vertical line. Such gluings may have complicated topology and, in some cases, may even be non-orientable. Exact enumeration results and algebraic properties of lozenge tilings of trapezoids provide a rich set of tools for the analysis. A key ingredient is the identification of the model with a discrete log-gas, which allows for the use of Nekrasov equations (also known as discrete loop or discrete Dyson–Schwinger equations). The course is based on a research monograph written jointly with G. Borot and A. Guionnet.
Thursday, July 16th, 2026
Workshop Mini Course: Rick Kenyon
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Dimers and webs Part 3
Speaker: Rick Kenyon
Abstract: We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges
Title: Dimers and webs Part 3
Speaker: Rick Kenyon
Abstract: We'll discuss connections between the dimer model and representation theory of SL_nand other Lie groups. The basic result is a generalization of Kasteleyn's theorem (which counts dimer covers of planar graphs using the Pfaffian of an adjacency-type matrix), to the setting of a graph with a matrix-valued connection on edges
Workshop Mini Course: Amol Aggarwal
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Asymptotics for the Toda Lattice Part 3
Speaker: Amol Aggarwal
Abstract: The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Title: Asymptotics for the Toda Lattice Part 3
Speaker: Amol Aggarwal
Abstract: The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this course we describe several results explaining such asymptotics under certain invariant initial data. The proofs proceed by finding a way to interpret the Toda lattice as a dense collection of "quasi-particles" that behave similarly to solitons, and providing a framework to study how these quasi-particles asymptotically evolve in time. In this analysis, arguments from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices, play an important role.
Workshop Mini Course: Vadim Gorin
Time: 2:00 PM - 3:30 PM
Location: SCGP 102
Title: Random lozenge tilings via Nekrasov equations, Part 2
Speaker: Vadim Gorin
Abstract: This course studies the macroscopic behavior and fluctuations of uniformly random lozenge tilings of polygonal domains. We identify tilings with their height functions and focus on central limit theorem-type results for these functions. Our main results establish Gaussian Free Field fluctuations for the associated two-dimensional fields, as well as discrete Gaussian asymptotics for height differences between boundary components of the tiled domain. The class of domains we consider consists of gluings of elementary building blocks, called trapezoids, along a single vertical line. Such gluings may have complicated topology and, in some cases, may even be non-orientable. Exact enumeration results and algebraic properties of lozenge tilings of trapezoids provide a rich set of tools for the analysis. A key ingredient is the identification of the model with a discrete log-gas, which allows for the use of Nekrasov equations (also known as discrete loop or discrete Dyson–Schwinger equations). The course is based on a research monograph written jointly with G. Borot and A. Guionnet.
Title: Random lozenge tilings via Nekrasov equations, Part 2
Speaker: Vadim Gorin
Abstract: This course studies the macroscopic behavior and fluctuations of uniformly random lozenge tilings of polygonal domains. We identify tilings with their height functions and focus on central limit theorem-type results for these functions. Our main results establish Gaussian Free Field fluctuations for the associated two-dimensional fields, as well as discrete Gaussian asymptotics for height differences between boundary components of the tiled domain. The class of domains we consider consists of gluings of elementary building blocks, called trapezoids, along a single vertical line. Such gluings may have complicated topology and, in some cases, may even be non-orientable. Exact enumeration results and algebraic properties of lozenge tilings of trapezoids provide a rich set of tools for the analysis. A key ingredient is the identification of the model with a discrete log-gas, which allows for the use of Nekrasov equations (also known as discrete loop or discrete Dyson–Schwinger equations). The course is based on a research monograph written jointly with G. Borot and A. Guionnet.
Friday, July 17th, 2026
Workshop Mini Course: Vadim Gorin
Time: 9:15 AM - 10:45 AM
Location: SCGP 102
Title: Random lozenge tilings via Nekrasov equations, Part 3
Speaker: Vadim Gorin
Abstract: This course studies the macroscopic behavior and fluctuations of uniformly random lozenge tilings of polygonal domains. We identify tilings with their height functions and focus on central limit theorem-type results for these functions. Our main results establish Gaussian Free Field fluctuations for the associated two-dimensional fields, as well as discrete Gaussian asymptotics for height differences between boundary components of the tiled domain. The class of domains we consider consists of gluings of elementary building blocks, called trapezoids, along a single vertical line. Such gluings may have complicated topology and, in some cases, may even be non-orientable. Exact enumeration results and algebraic properties of lozenge tilings of trapezoids provide a rich set of tools for the analysis. A key ingredient is the identification of the model with a discrete log-gas, which allows for the use of Nekrasov equations (also known as discrete loop or discrete Dyson–Schwinger equations). The course is based on a research monograph written jointly with G. Borot and A. Guionnet.
Title: Random lozenge tilings via Nekrasov equations, Part 3
Speaker: Vadim Gorin
Abstract: This course studies the macroscopic behavior and fluctuations of uniformly random lozenge tilings of polygonal domains. We identify tilings with their height functions and focus on central limit theorem-type results for these functions. Our main results establish Gaussian Free Field fluctuations for the associated two-dimensional fields, as well as discrete Gaussian asymptotics for height differences between boundary components of the tiled domain. The class of domains we consider consists of gluings of elementary building blocks, called trapezoids, along a single vertical line. Such gluings may have complicated topology and, in some cases, may even be non-orientable. Exact enumeration results and algebraic properties of lozenge tilings of trapezoids provide a rich set of tools for the analysis. A key ingredient is the identification of the model with a discrete log-gas, which allows for the use of Nekrasov equations (also known as discrete loop or discrete Dyson–Schwinger equations). The course is based on a research monograph written jointly with G. Borot and A. Guionnet.
Workshop Mini Course: Persi Diaconis
Time: 11:15 AM - 12:45 PM
Location: SCGP 102
Title: Markov Chains for Enumeration under Symmetry Lecture 3
Speaker: Persi Diaconis
Abstract: Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution. Problems abound: How do you actually carry out the two steps in problems of interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open. Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group). Lecture 3: Careful proof in a simple case: the binary Burnside process, from Tchebychev polynomials to Schur-Weyl duality. I expect the audience to know something about Markov chains (at the level of Levin- Peres' book) and something about groups (a good undergraduate course should suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.
Title: Markov Chains for Enumeration under Symmetry Lecture 3
Speaker: Persi Diaconis
Abstract: Let G be a finite group acting on finite set X. This divides X into orbits and we ask ?how many orbits are there?, ?What is the typical size of an orbit?, ?do the orbits have 'nice names'?, ?do they fit together into some kind of 'moduli space'?. The Burnside process addresses the first two topics. It allows uniform sampling of a random orbit. It runs the following Markov chain on X: From x, choose g fixing x (uniformly) and then y fixed by this g (uniformly). The chain moves from x to y. This chain has stationary distribution pi(x) proportional to the size of the orbit containing x. Thus simply reporting the size of the current orbit gives a Markov chain with a uniform stationary distribution. Problems abound: How do you actually carry out the two steps in problems of interest? How can you convert the output into a useful estimate of the number and size of orbits? What is the running time and typical behaviour of this Markov chain? There is progress in special cases, but in general, all problems are open. Lecture 1: An introduction to Polya Theory, the Burnside process and auxiliary variables algorithms Lecture 2: Three real examples: Polya trees, contingency tables with fixed row and column sums, The commuting graph process for generating random partitions of n (and counting the number of conjugacy classes in a finite group). Lecture 3: Careful proof in a simple case: the binary Burnside process, from Tchebychev polynomials to Schur-Weyl duality. I expect the audience to know something about Markov chains (at the level of Levin- Peres' book) and something about groups (a good undergraduate course should suffice). These topics are based on my current research and you can look at papers on my homepage for the past year or two.