Periodic Reporting for period 2 - CARPE (Combinatorial Applications of Random Processes and Expansion)
Berichtszeitraum: 2023-01-01 bis 2024-06-30
The concepts of Randomness and Expansion are pervasive throughout Mathematics and its applica- tions to many areas of Science and Engineering. The mathematical study of Expansion can be traced back to the ancient Greeks and of Probability to the analysis (e.g. by Fermat and Pascal) of games of chance. In the modern era, both concepts are influential in many areas of Mathematics (this proposal will emphasise Combinatorics and Probability, and also touch on Analysis, Geometry, Topology, Number Theory and Theoretical Computer Science). Within Science and Engineering, topics related to the mathematical problems covered in this proposal include Approximation Algorithms (Counting and Sampling), Statistical Physics (Magnetism, Lattice Gases, Polymer Models), Mathematical Biology (Epidemiology), Control Theory and Fluid Flow.
My recent and ongoing research has generated several exciting new ideas and methods. The most recent of these, the Cluster Expansion Method (work with Jenssen), is a far-reaching program to apply a classical tool from Statistical Physics to my objective of developing methods for describing the typical structure of models such as random homomorphisms from a discrete torus. Another exciting recent technique, Global Hypercontractivity (work with Lifshitz, Long and Minzer), is a structural refinement of the classical hypercontractivity theorem; I will generalise many of its applications to Mathematics and Computer Science and give several new applications, e.g. in Extremal Combinatorics (via the Junta Method). I will also develop new Absorption tech- niques to answer constructive mathematical questions that seem beyond the reach of Randomised Algebraic Construction (a method I developed to solve Steiner’s 1852 question on the Existence of Designs) such as the existence of Steiner Triple Systems of high girth or bounded degree high-dimensional expanders.
My recent and ongoing research has generated several exciting new ideas and methods. The most recent of these, the Cluster Expansion Method (work with Jenssen), is a far-reaching program to apply a classical tool from Statistical Physics to my objective of developing methods for describing the typical structure of models such as random homomorphisms from a discrete torus. Another exciting recent technique, Global Hypercontractivity (work with Lifshitz, Long and Minzer), is a structural refinement of the classical hypercontractivity theorem; I will generalise many of its applications to Mathematics and Computer Science and give several new applications, e.g. in Extremal Combinatorics (via the Junta Method). I will also develop new Absorption tech- niques to answer constructive mathematical questions that seem beyond the reach of Randomised Algebraic Construction (a method I developed to solve Steiner’s 1852 question on the Existence of Designs) such as the existence of Steiner Triple Systems of high girth or bounded degree high-dimensional expanders.
Major progress has been achieved across all 3 objectives of the project, as follows.
Objective 1: New methods for analysing random processes and absorption have led to solutions of several problems raised in the proposal, as follows. Problem 1.6 (Polya’s toroidal n-queens problem) was solved by Bowtell and the PI. Conjecture 1.7 (Erdos’ problem on high girth Steiner Triple Systems) was solved by Kwan, Sah, Sawhney and Simkin. Conjecture 1.8 (towards Ryser’s Conjecture) was solved by the PI with Pokrovskiy, Sudakov and Yepremyan. The PI with Sah and Sawhney solved the existence problem for subspace designs.
Objective 2: The first paper of the PI with Jenssen solved Problem 2.1 and has many further applications that set the stage for the cluster expansion and container approaches in this objective. Work in progress makes progress on Problem 2.3.
Objective 3: A series of 3 papers with Lifshitz, Long and Minzer has greatly advanced the theory of global hypercontractivity and its applications. We have solved problems 3.6 3.7 and 3.8. Recently we extended our scope to extremal problems on groups and solved a conjecture of Babai and Sos from 1985 on the largest product-free subsets of alternating groups. The theory of thresholds has been revolutionised by a series of papers culminating in the solution of the Kahn-Kalai Conjecture by Park and Pham. In particular this solved Problem 3.5. The PI recently found an unexpected connection to Objective 1 (constructing spread measures via constrained random processes) and thus resolved Johannson’s question on thresholds for latin squares (a variant of Problem 1.13).
Objective 1: New methods for analysing random processes and absorption have led to solutions of several problems raised in the proposal, as follows. Problem 1.6 (Polya’s toroidal n-queens problem) was solved by Bowtell and the PI. Conjecture 1.7 (Erdos’ problem on high girth Steiner Triple Systems) was solved by Kwan, Sah, Sawhney and Simkin. Conjecture 1.8 (towards Ryser’s Conjecture) was solved by the PI with Pokrovskiy, Sudakov and Yepremyan. The PI with Sah and Sawhney solved the existence problem for subspace designs.
Objective 2: The first paper of the PI with Jenssen solved Problem 2.1 and has many further applications that set the stage for the cluster expansion and container approaches in this objective. Work in progress makes progress on Problem 2.3.
Objective 3: A series of 3 papers with Lifshitz, Long and Minzer has greatly advanced the theory of global hypercontractivity and its applications. We have solved problems 3.6 3.7 and 3.8. Recently we extended our scope to extremal problems on groups and solved a conjecture of Babai and Sos from 1985 on the largest product-free subsets of alternating groups. The theory of thresholds has been revolutionised by a series of papers culminating in the solution of the Kahn-Kalai Conjecture by Park and Pham. In particular this solved Problem 3.5. The PI recently found an unexpected connection to Objective 1 (constructing spread measures via constrained random processes) and thus resolved Johannson’s question on thresholds for latin squares (a variant of Problem 1.13).
We have developed several new methods in the solutions of the problems mentioned above, particularly for absorption within Objective 1, analysing cluster expansions within Objective 2, and for increasing the applicability of the junta method within Objective 3. We anticipate that further developments of these methods will resolve several other problems targeted by the project.