Billingsley Lectures on Probability

We are pleased to have Prof. Nina Holden, Courant Institute of Mathematical Sciences, New York University, as our honored speaker.

Thursday, May 8, 2025, at 3:30 PM, Jones 303, 5747 South Ellis Avenue.                                                              Reception will immediately follow the lecture at 5:20 pm, in the Stevanovich Center for Financial Mathematics, 5727 South University Avenue.

Title: Scaling limits of random planar maps

Abstract: Planar maps are graphs embedded in the sphere such that no two edges cross, where we view two planar maps as equivalent if we can get one from the other via a continuous deformation of the sphere. In this talk we will present scaling limit results (i.e., convergence results) for random planar maps and we will focus in particular on a notion of convergence known as convergence under conformal embedding.

We are pleased to have Prof. Sourav Chatterjee, Department of Statistics, Stanford University, as our honored speaker.

• Thursday, May 2, 2024, at 4:30 PM, Math/Stat 112, 5727 South University Avenue
  Reception immediately following the lecture at 5:30 pm, in the Reading Room, Math/Stat 101, 5727 South University Avenue
  "A definition of spectral gap for nonreversible Markov chains"
  
  Abstract: 
  While the notion of spectral gap is a fundamental and very useful feature of reversible Markov chains, there is no standard analogue of this notion for nonreversible chains. In this talk I will present a simple proposal for spectral gap of nonreversible chains, and show that it shares all of the nice properties of the reversible spectral gap. The most important property of this spectral gap is that its reciprocal gives an exact characterization, with upper and lower bounds, of the time required for convergence of empirical averages. This works even if there is no contraction, such as in dynamical systems.
• Poster
   
• Prof. Chatterjee will also give a talk at the following (please note room change):
  Probability and Statistical Physics Seminar
  Friday, May 3, 2024, at 2:30 PM, in Ryerson 276, 1100 E. 58th Street
  "Chaos in lattice spin glasses"
  
  Abstract: 
  In spite of tremendous progress in the mean-field theory of spin glasses in the last forty years, culminating in Giorgio Parisi’s Nobel Prize in 2021, the more “realistic” short-range spin glass models have remained almost completely intractable. In this talk, I will show that the ground states of short-range spin glasses are chaotic with respect to small perturbations of the disorder, settling a conjecture made by Daniel Fisher and David Huse in 1986.

We are pleased to have Prof. Russell Lyons, Department of Mathematics, Indiana University, as our honored speaker.

• Thursday, May 18, 2023, at 4:30 PM, Math/Stat 112, 5727 South University Avenue
  Reception following the lecture at 5:30 pm, in the Reading Room, Math/Stat 101, 5727 South University Avenue
  "Voronoi Tessellations without Nuclei"
  
  Abstract:  Given a discrete set of points in a metric space, called nuclei, one associates to each such nucleus its Voronoi cell, which consists of all points closer to it than to other nuclei. This construction is widely used in mathematics, science, and engineering; it is even used in baking. In Euclidean space, one commonly uses a homogeneous Poisson point process to assign the locations of the nuclei. As the intensity of the point process tends to 0, the nuclei spread out and disappear in the limit, with each pair of points eventually belonging to the same cell. Surprisingly, this does not happen in other settings such as hyperbolic space; instead, one obtains a Voronoi tessellation without nuclei! We describe properties of such a limiting tessellation, as well as analogous behavior on Cayley graphs of finitely generated groups. We will illustrate results with many pictures and several animations. The talk is based on work of Sandeep Bhupatiraju and joint work in progress with Matteo d'Achille, Nicolas Curien, Nathanael Enriquez, and Meltem Unel. We will not assume knowledge of Poisson point processes or of hyperbolic space.

• Poster

• Prof. Lyons will also give a talk at the following:
  Probability and Statistical Physics Seminar
  Friday, May 19, 2023, 2:30 PM, in Eckhart 202, 5734 S. University Avenue
  "Monotonicity for Continuous-Time Random Walks"
  
  Abstract:  Variable-speed, continuous-time random walk on a graph is given by an assignment of nonnegative rates to its edges. There are independent Poisson processes associated to the edges with the given rates. When a walker is at a vertex, it jumps to a neighbor at the time of the next event that occurs for the corresponding incident edges. In the case of a Cayley graph of a finitely generated group, we are particularly interested in the setting where the edge rates depend only on the corresponding generators. Our lecture is concerned with monotonicity in the rates for various fundamental properties of random walks. We will survey results, counterexamples, and open questions. We will give general ideas of proofs, but avoid technicalities. Most of the talk will be devoted to two questions on Cayley graphs: On infinite graphs, we ask about the limiting linear rate of escape, i.e., the limit of the distance divided by the time. Does this increase when the rates are increased? On finite graphs, we ask about the convergence to the stationary (uniform) distribution. Does this happen faster when the rates are increased? It turns out that both questions have surprising answers. This is joint work with Graham White. 

We are pleased to have Professor Fredrik Viklund, Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden, as our honored speaker.

• Thursday, May 5, 2022, at 4:30 PM, in Kent 120, 1024 E. 58th Street
  Reception following the lecture at 5:30 PM, in Eckhart 209, 5734 S. University Avenue
  "Interface dynamics and conformal maps"
  
  ABSTRACT:  A range of phenomena in nature can be modelled as a monotone evolution of an interface in the plane, with the dynamics being in some way related to harmonic measure. Examples include (deterministic) Hele-Shaw flow and (random) aggregation models such as DLA. While such models are often very hard to analyze, some progress has been achieved for simplified models using conformal maps and the Loewner differential equation. I will survey some of the models, tools, and results along with open questions. Based in part on joint works with Amanda Turner (Lancaster) and Alan Sola (SU) and with Yilin Wang (MIT).
• Poster
• Directions for Kent
• Directions for Jones

Alexei Borodin, Department of Mathematics, Massachusetts Institute of Technology

We are pleased to have Professor Allan Sly, Department of Mathematics, Princeton University as our honored speaker.

• Thursday, November 8, 2018, at 4:00 PM, in Kent 120, 1024 E. 58th Street
  Reception following the lecture in Jones 111, at 5:00 PM
  "Phase Transitions of Random Constraint Satisfaction Problems"
  
  ABSTRACT:  Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo.
• Poster
• Directions for Kent
• Directions for Jones

We are pleased to have Professor Yuval Peres, Theory Group, Microsoft Research, Redmond, as our honored speaker.

• Thursday, October 5, 2017, at 4:30 PM, in Kent 120, 1028 E. 58th Street
  Reception following the lecture in Jones 111, at 5:30 PM
  "Gravitational Allocation to Uniform Points on the Sphere"
  
  ABSTRACT:  Given uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984).
  Joint work with Nina Holden and Alex Zhai.
• Poster
• Directions for Kent
• Directions for Jones

We are pleased to have Professor Jean-Francois Le Gall, Universite Paris-Sud Orsay, as our honored speaker.

• Thursday, October 6, at 4:30 PM, in Eckhart 133, 5734 South University Ave.
  Reception following the lecture in Eckhart 209
  "Random Planar Geometry"
  
  ABSTRACT:  Much recent work has been devoted to the definition and study of a canonical random geometry in the plane. The basic idea is to study metric properties of large random graphs drawn in the plane or on the sphere, which are also called random planar maps. Starting from a triangulation of the sphere with a given number of faces (triangles), and chosen uniformly at random, one considers the metric space consisting of the vertex set of the triangulation equipped with the graph distance. When the size of the triangulation tends to infinity, this suitably rescaled random metric space converges in distribution, in the Gromov-Hausdorff sense, to a random compact metric space called the Brownian map. We discuss various properties of the Brownian map, and survey recent results showing that this random metric space is indeed a universal model of random geometry in two dimensions.
• Poster

We are pleased to have Professor Wendelin Werner, Department of Mathematics, Eidgenössische Technische Hochschule Zürich, as our honored speaker.

• Thursday, April 16, 2015, at 4:30 PM, in Eckhart 133, 5734 S. University Avenue
  Reception following the lecture in Eckhart 209
  "Random Phenomena and Conformal Invariance Within Fractal Gaskets"
  
  ABSTRACT:  We will survey recent and ongoing development dealing with the existence of continuous random models within fractal carpets (of the Sierpinski-carpet type), the (mostly conjectural) role played by conformal invariance within these fractals, and the relation to the Schramm-Loewner evolution (SLE).
• Poster

We are pleased to have Professor Scott Sheffield, Department of Mathematics, Massachusetts Institute of Technology, as our honored speaker.

• Thursday, May 29, 2014, at 4:30 PM, in Eckhart 133, 5734 S. University Avenue
  Reception following the lecture in Eckhart 209
  "Snowflakes, Slot Machines, Chinese Dragons, and QLE"
  
  ABSTRACT:  What is the right way to think of a "random surface" or a "random planar graph"? How can one explain the dendritic patterns that appear in snowflakes, coral reefs, lightning bolts, and other physical systems, as well in as toy mathematical models inspired by these systems? How are these questions related to random walks and random fractal curves (in particular the famous SLE curves)?
  To begin to address these questions, I will introduce and explain the "quantum Loewner evolution," which is a family of growth processes closely related to SLE. I will explain through pictures and animations and some discrete arguments how QLE is defined and what role it might play in addressing the questions raised above. This is a collaboration with Jason Miller.
• Poster

We are pleased to have Professor Persi Diaconis, Mary V. Sunseri Professor of Statistics and Mathematics at Stanford University, as our honored speaker.

• Thursday, May 2, 2013, at 4:00 PM, in Breasted Hall, at the Oriental Institute, 1155 E. 58th Street
  Reception following the lecture in Eckhart 209, 5734 S. University Avenue
  "On Coincidences"
  
  ABSTRACT:  Coincidences amaze us. They can affect where we live and with whom. I will review early work of Jung, Freud, and Kammerer, and then introduce some skeptical tools. Sometimes, a little bit of quantitative thinking shows that things are not so surprising after all.
• Poster
• Scenes from the lecture

We are pleased to have Professor S. R. S. Varadhan, of the Courant Institute of Mathematical Sciences at New York University, as our honored speaker.

• Thursday, May 31, 2012, at 4:00 PM
  Eckhart 133, 5734 South University Avenue
  Reception following the lecture in Eckhart 209.
  "Large Deviations with Applications to Random Matrices and Random Graphs"
• Poster
• Scenes from the lecture

Introduction to the First Billingsley Lecture on Probability Theory

Welcome to the first Billingsley Lecture on Probability Theory. This is the first in what will be an annual series of special lectures to commemorate the contributions of Patrick Billingsley to the field of probability theory and to the University of Chicago, where Pat spent nearly all of his academic career, from 1958 until his retirement in 1994.

Pat Billingsley was a man of diverse talents: he excelled not only as a scholar and a teacher but also as a writer, an actor, and even a black belt in judo. Pat is remembered for his research in weak convergence, in ergodic theory and its connections with information theory, in probabilistic number theory, and as the inventor of the "Billingsley dimension" of a measure. He advised a number of Ph.D. students during his years on the UC faculty, several of whom went on to distinguished careers in mathematics, including Rabi Bhattacharya and Richard Gundy. He played leading roles in more than 20 productions of the Court Theater, and also appeared in a number of movies and television shows. Pat once remarked to me that he suspected he was one of the few people alive whose Erdös number and his Kevin Bacon number summed to less than ten. (He did not know of Wendelin Werner another distinguished probabilist who like Pat has an Erdös-Bacon number of 6. In fact, Pat shares 12th place in this category, along with Werner, Richard Feynman, John Nash, and Tom Lehrer.)

But what we in the fields of probability and statistics mainly remember Pat for are his superb monographs and textbooks on weak convergence, ergodic theory and information, inference for Markov chains, and the measure-theoretic foundations of probability. These books are all models of mathematical exposition, and "Probability and Measure" remains a widely used and frequently cited graduate textbook. A generation of probabilists and statisticians in the United States learned about the measure-theoretic underpinnings of probability from this book.

Pat would have been pleased to be honored by a lecture series such as this, and especially proud that the first lecture would be delivered by Raghu Varadhan.