This biennial lecture series honors the memory of Arthur B. Coble (1878-1966), Professor of Mathematics at the University of Illinois from 1918 to 1947. The late Professor Coble's family established a fund to endow a series of public lectures on mathematics to be delivered by outstanding mathematicians. The Coble Memorial Lecture series is offered in alternate years with the Tondeur Lecture series.

**February 24-26, 2020: **Frank Calegari, University of Chicago

## Monday, February 24, 2020

**4:00 p.m., 314 Altgeld Hall
Title: Digits**

**Abstract:**We discuss some results concerning the decimal expansion of 1/p for primes p, some due to Gauss, and some from the present day. This talk will be accessible to undergraduates.

## Tuesday, February 25, 2020

**4:00 p.m., 245 Altgeld Hall**

**Title: Counting**

**Abstract**: What can one say about a system of polynomial equations with integer coefficients simply by counting the number of solutions to these equations modulo primes? We begin with the case of polynomials in one variable and relate this to how the polynomial factors and to Galois theory. We then discuss what happens in higher dimensions, and are led to a conjectural notion of the "Galois group" of an algebraic variety. This will be a colloquium style talk and will be independent of the first talk.

## Wednesday, February 26, 2020

**4:00 p.m., 245 Altgeld Hall**

**Title: Coble**

**Abstract**: Coble is known (in part) for his work on invariant theory and the geometry of certain of exceptional moduli spaces in low dimension. We discuss the quest to find explicit equations for one particular family of moduli spaces. An important role is played by a number of exceptional geometrical coincidences and also the theory of complex reflection groups. This will be a colloquium style talk and will be independent of the first two talks.

## Previous Lectures

Year | Lecturer | Lecture Title |

1970-71 | Irving Kaplansky | The Hilbert Problems |

1971-72 | Samuel Eilenberg | Theory of automata Topology and automata Categories and automata |

1972-73 | Derrick Lehmer | Unusual Mathematical uses of computers Some interesting infinite series Permutation structure and partitions |

1973-74 | Lipman Bers | Some aspects of uniformization |

1974-75 | Richard Duffin | Optimization problems of mathematical science |

1975-76 | Alonzo Church | Set theory with a universal set |

1976-77 | Andrew M. Gleason | Coding theory and combinatorial accidents |

1977-78 | George W. Mackey | Noncommutative harmonic analysis and its applications |

1978-79 | Saunders MacLane | The development of abstract mathematics Abstract algebra and mathematics today Possibilities for the future |

1979-80 | Paul Erdös | Recent results and open problems in number theory Recent results and open problems in combinatorics Recent results and open problems in set theory |

1980-81 | Shiing-Shen Chern | Exterior differential systems |

1981-82 | Ron L. Graham | Bin packing: A paradigm in the analysis of algorithms Combinatorial number theory Euclidean Ramsey theory |

1982-83 | Richard V. Kadison | Non-commutative analysis |

1983-84 | Jean-Louis Verdier | Solitons, symmetries and geometry |

1984-85 | H. Jerome Keisler | Logical areas in probability theory |

1985-86 | J. William Helton | Operator theory, complex variables and optimization problems in electrical engineering |

1986-87 | Chas. T. Fefferman | Mathematical problems of quantum mechanics |

1987-88 | Wolfgang Schmidt | The number of solutions of diophantine equations A p-adic method in diophantine equations Quadratic forms and the geometry of numbers |

1989-90 | Donald A. Martin | A chronological survey of Determinacy |

1990-91 | Burgess J. Davis | Lifetimes of conditioned Brownian Motion, and the shapes of the heat kernels |

1991-92 | Israel Gelfand | What is a discriminant and resultant of algebraic equations A new proof of the oscillation theorem for Sturm-Lionville equations Cohomology of Poisson algebras |

1992-93 | Robert Steinberg | Central extensions of finite groups, Lie groups, and others |

1993-94 | Béla Bollobás | Projections of bodies and hereditary properties Oriented percolation in two dimensions Discrete isoperimetric inequalities |

1994-95 | Nigel Hitchin | Monopoles revised Symmetric Monopoles The geometry and topology of monopole moduli spaces |

1995-96 | Kenneth Ribet | Fermat's Last Theorem Perfect Powers in arithmetic progression Component groups and degrees of modular parametrizations |

1996-97 | Imre Bárány | Affine Perimeter and limit shape On the convex hull of the integer points in a ball A colourful version of Carathéodory's theorem with applications |

1997-98 | Freeman Dyson | Life in the Universe I. The Paradox of Order and Disorder Life in the Universe II. The Search for Life Does the Electron Charge Stay Constant as the Universe Evolves? |

1998-99 | Yuri Manin University of Moscow |
Quantum Computing Quantum Cohomology and Mirror Pattern I Quantum Cohomology and Mirror Pattern II |

1999-00 | George E. Andrews Pennsylvania State University |
Ramanujan's Lost Notebook The Rogers-Ramanujan Identities: Some History and Recent Developments Partitions and Positivity |

2000-01 | Michael Artin MIT |
Noncommutative Algebra in Dimension 2 Noncommutative Projective Surfaces Orders over Surfaces |

2002 | Noga Alon Tel Aviv University and Institute for Advanced Study |
Methods in Modern Combinatorics: Dimension, Distances, and Information Theory Polynomials, Addition and Coloring The Probabilistic Method |

2003 | Walter Neumann Columbia University |
The Volume of Three-Manifolds |

2004 | David Eisenbud Director, Mathematical Sciences Research Institute and Professor, University of California, Berkeley |
Aspects of Elimination Theory: I. Resultants, from 1693 to now II. Exterior algebra methods in algebraic geometry III. Instant elimination, and torsion in symmetric algebras |

2005 | Peter Sarnak Princeton University |
Zeta functions and spectral theory |

2006 | Efim Zelmanov University of California, San Diego |
Infinite dimensional Lie (super)algebras On Golod-Shafarevich groups Some Ring - Theoretic problems inspired by Combinatorial Group Theory |

2007 | I.M. Singer MIT |
Index Theory: Past, Present, and Future |

2008 | Ib Madsen University of Aarhus, Denmark |
Moduli Spaces and Topology |

2008 | Peter Ozsváth Columbia University |
Heegaard Floer homology and knots I and II Bordered Heegaard Floer homology |

2011 | Ngo Bao Chau University of Chicago |
Geometry of the Hitchin fibration |

2014 | Ian Agol University of California, Berkeley |
The virtual Haken conjecture |

2016 | Sorin Popa UCLA |
I. Classification of II_1 factors arising from free groups acting on spaces II&III. Approximating freeness under constraints |

2018 | Andrea Bertozzi UCLA |
cancelled |

2020 | Frank Calegari University of Chicago |
"Digits" / "Counting" / "Coble" |