Jacopo

Long increasing subsequences and large homogenous subsets

The length of the longest increasing subsequence in a random permutation and the size of the largest homogeneous set (i.e. a clique or an independent set) in a random graph are two of the classical problems at the interface of combinatorics and probability theory, with connections to several other areas of mathematics. In this paper, …

[16] Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons (with William Da Silva and Ewain Gwynne). To appear in Advances in Mathematics.

The Brownian separable permutons are a one-parameter family – indexed by – of universal limits of random constrained permutations. We show that for each , there are explicit constants such that the length of the longest increasing subsequence in a random permutation of size sampled from the Brownian separable permuton is between and with probability …

Meanders and Meandric Systems

In 1912 Henri Poincaré asked the following simple question: “In how many different ways a simple loop in the plane, called a meander, can cross a line a specified number of times?” Despite many efforts, this question remains very open after more than a century. In this talk, I will present the conjectural scaling limit …

MATH 151 – Introduction to Probability | 2022-2023

This is the webpage for the course of Math 151 – Introduction to Probability. Lecture times: Tuesday and Thursday 9:00 AM – 10:20 AM (Room: 380-380X). Instructor: Jacopo Borga, jborga_at_stanford.edu. Office hours: Thursday 11:00-12:15 AM (Building 380, Room 382-Q2). Assistant: Zhihan Li, zhihanli@stanford.edu.  Office hours: Mondays and Fridays 11:30 AM-1:30 PM (Building 380, Room 381A). Description: This is a first …

MATH 159 – Discrete Probabilistic Methods | 2022-2023

This is the webpage for the course of Math 159 – Discrete Probabilistic Methods. Lecture times: Tuesday and Thursday, 1:30 PM – 2:50 PM (Mccullough Building (04-490), Room 122). Instructor: Jacopo Borga, jborga_at_stanford.edu. Office hours: Tuesday 11:00 AM-12:15 PM (Building 380, 382-Q2). Assistant: Pranav Nuti, pranavn_at_stanford.edu. Office hours: Monday 3:00 PM-5:00 PM and Thursday 11:00 AM-1:00 PM (Building 380, Room …

[14] On the geometry of uniform meandric systems (with Ewain Gwynne and Minjae Park). Communications in Mathematical Physics (2023), https://doi.org/10.1007/s00220-023-04846-y.

A meandric system of size is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on , one drawn above the real line and the other below the real line. A uniform random meandric system can be viewed as a random planar map decorated by a Hamiltonian path (corresponding to the real …

Permutations in Random Geometry

Random geometry and random permutations have been extremely active fields of research for several years. The former is characterized by the study of large planar maps and their continuum limits, i.e. the Brownian map, Liouville quantum gravity surfaces and Schramm–Loewner evolutions. The latter is characterized by the study of large uniform permutations and (more recently) …

MATH 106 – Functions of a Complex Variable | 2022-2023

This is the webpage for the course MATH 106 – Functions of a Complex Variable. IMPORTANT NEWS: / Lecture times: Tuesday+Thursday 9:00 AM – 10:20 AM (Room: 380-380X). Instructor: Jacopo Borga (jborga_at_stanford.edu) Office hours: Tuesday 11:00-12:00 AM (Room: 382-Q2). Assistant: Judson Otto Kuhrman (kuhrman_at_stanford.edu) Office hours: Monday Wednesday (Room: ). Description: Math 106 is an introductory course on complex analysis (focused on functions of …

MATH 136 – Stochastic Processes | 2022-2023

This is the webpage for the course MATH 136 – Stochastic Processes. IMPORTANT NEWS: / Lecture times: Tuesday+Thursday 1:30 PM – 2:50 PM (Room: 260-113). Instructor: Jacopo Borga (jborga_at_stanford.edu) Office hours: Thursday 11:00-12:00 AM (Room: 382-Q2). Assistant: Christian Serio (cdserio_at_stanford.edu) Office hours: Wednesday 1:00-3:00 PM (Room: 381-D). Assistant: Zhongren Chen (czrbear_at_stanford.edu) Office hours: Monday 4:30-6:30 PM (Room: TBA). Description: This course prepares students to a rigorous study of …

Permutons, meanders, and SLE-decorated Liouville quantum gravity

In 1912 Henri Poincaré asked the following question: “In how many different ways a simple loop in the plane, called meander, can cross a line a specified number of times?” Despite many efforts, this question remains open after more than a century. In this talk we construct and study the conjectural scaling limit of uniform …