Distinguished Lecture Series : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Jan. 15, 2013
Curtis T. McMullen :
4 p.m. in Student Center East Tower 302
Jan. 16, 2013
Curtis T. McMullen :
4 p.m. in Lecture Center D2
Jan. 17, 2013
Curtis T. McMullen :
4 p.m. in Lecture Center D2
March 5, 2014
Michael Hopkins :
4 p.m. in BSB 250
Abstract
In mathematics and in science an "invariant" of a system is
a quantity, like the total energy, that does not change as the system
evolves. The discovery and understanding of invariants is a
significant part of what drives the development of mathematics. In
this talk I will describe some simple mathematical invariants and the
deep mathematics that has evolved from trying to understand them.
March 6, 2014
Michael Hopkins :
4 p.m. in LCD 005
Abstract
The Kervaire invariant is subtle and important invariant
lying at the interface of algebraic and differential topology. I
will describe the history of this invariant, the problem it raised,
and its eventual solution.
March 7, 2014
Michael Hopkins :
3 p.m. in LCD 005
Abstract
Nature does not come to us with a coordinate system. In
writing down the equations describing the evolution of physical
systems, it is therefore important that the mathematical entities that
arise do not depend on a choice of coordinates. The Chern-Weil
invariants are important examples of such entities. In this talk I
will explain the Chern-Weil invariants and how one is led by by
thinking carefully about them to modern day abstract homotopy theory.
Oct. 15, 2014
Avi Wigderson :
4 p.m. in 2LCC C6
Abstract
The "P vs. NP" problem, formulated by computer theorists in the 1970s, quickly became a central outstanding problem
of science and mathematics. In this talk I will attempt to describe its mathematical, scientific and philosophical
content. I will discuss its status, and the implications of its resolution on science and technology (making clear
that the \$1M prize on solving it pales in comparison with these implications).
No special background will be assumed.
Oct. 16, 2014
Avi Wigderson :
4 p.m. in 2LCC C4
Abstract
Is the universe inherently deterministic or probabilistic? Perhaps more importantly - can we tell the difference
between the two?
Humanity has pondered the meaning and utility of randomness for millennia.
There is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage:
in statistics, cryptography, game theory, algorithms, gambling... Indeed, randomness seems indispensable!
Which of these applications survive if the universe had no randomness in it at all? Which of them survive
if only poor quality randomness is available, e.g. that arises from "unpredictable" phenomena like the
weather or the stock market?
A computational theory of randomness, developed in the past three decades, reveals (perhaps counter-intuitively)
that very little is lost in such deterministic or weakly random worlds. In the talk I'll explain the main ideas
and results of this theory.
The talk is aimed at a general scientific audience.
Oct. 17, 2014
Avi Wigderson :
3 p.m. in 2LCC C3
Abstract
The determinant is undoubtedly the most important polynomial function in mathematics.
Its lesser known sibling, the permanent, plays very important roles in
enumerative combinatorics, statistical and quantum physics, and the
theory of computation. In this lecture I plan to survey some of the
remarkable properties of the permanent, its applications and impact on
fundamental computational problems, its similarities to and apparent
differences from the determinant, and how these relate to the P vs. NP
prolem.
This lecture is intended to a general Math & CS audience.
March 14, 2016
Robert Lazarsfeld :
3 p.m. in Lecture Center D4
Abstract
See http://kftucker.people.uic.edu/DLS2016/ for more information.
March 16, 2016
Robert Lazarsfeld :
4 p.m. in Lecture Center D4
Abstract
See http://kftucker.people.uic.edu/DLS2016/ for more information.
March 18, 2016
Robert Lazarsfeld :
3 p.m. in Lecture Center D4
Abstract
See http://kftucker.people.uic.edu/DLS2016/ for more information.
April 13, 2022
W. Hugh Woodin :
4 p.m. in LC D5
Abstract
The modern mathematical story of infinity began in the period 1879-84 with a series of papers by Cantor that defined the fundamental framework of the subject. Within 40 years the key ZFC axioms for Set Theory were in place and the stage was set for the detailed development of transfinite mathematics, or so it seemed. However, in a completely unexpected development, Cohen showed in 1963 that even the most basic problem of Set Theory, that of Cantor's Continuum Hypothesis, was not solvable on the basis of the ZFC axioms. The now nearly 60 years since Cohen's work has seen a vast development of Cohen's method and the realization that the occurrence of unsolvable problems is ubiquitous in Set Theory. This arguably challenges the very conception of Cantor on which Set Theory is based. Thus a fundamental dilemma has emerged. On the one hand, the discovery, also over the last 60 years, of a rich hierarchy axioms of infinity seems to argue that Cantor's conception is fundamentally sound. But on the other hand, the developments of Cohen's method over this same period seem to strongly suggest there can be no preferred extension of the ZFC axioms to a system of axioms that can escape the ramifications of Cohen's method. But this dilemma was itself based on a misconception and recent discoveries now strongly suggest there is a resolution.
April 14, 2022
W. Hugh Woodin :
4 p.m. in LC D1
Abstract
The determinacy axiom, AD, was introduced by Mycielski and Steinhaus over 60 years ago as an alternative to the Axiom of Choice for the study of arbitrary sets of real numbers. The modern view is that determinacy axioms concern generalizations of the borel sets, and deep connections with large cardinal axioms have emerged. Further a specific technical refinement of AD, this is the axiom AD+, has also been isolated. The further connections with large axioms have implicitly led to a duality program, which is the AD+ Duality Program. The central open problems here are intertwined with those of the Inner Model Program, and this is distilled into a specific conjecture, the Ultimate-L Conjecture. This conjecture, and its related problems, are now arguably the key problems in both AD+-theory and the Inner Model Program.
April 15, 2022
W. Hugh Woodin :
3 p.m. in LC D5
Abstract
There are many open problems in the model theory of well founded models of ZFC. Cohen's method of forcing, including the generalization to class forcing, is the main technique for building models of ZFC. But consider the following problem. Suppose ZFC + ϕ has a unique well founded model. Must that model satisfy V = L? No counterexample can be a nontrivial class forcing extension of any other model, and moreover that model of ZFC + ϕ must be countable and belong to L (by Shoenfield absoluteness). This suggests that any such model must satisfy V = L. Rephrased, if ZFC + ϕ is both β-consistent and β-categorical then the natural conjecture is that
ZFC + ϕ⊢β V=L. However relativising to a real and assuming large cardinals, there are counterexamples, and in the strongest possible sense. This is joint work with Peter Koellner.