SEG Seminar, special series: the Millennium Problems in mathematics
Compiled by the Clay Mathematics Institute at the turn of the millennium,
the list of 7 problems covers a broad swath of modern mathematics, posing
extremely difficult challenges that remain unsolved, with a single exception!
These problems are all arduous, and it is often even difficult just to understand how they are stated.
The goal of this series of talks is to familiarize the audience with the context of each problem and
to understand why they matter for the mathematical landscape of the 21st century.
Of course, we won't be able to avoid the technical aspects entirely, but the emphasis will also be placed on the historical side of these problems.
Spring 2022
Thursday, June 16, 2022 at 1:30 PM (B-1404)
The Hodge Conjecture
Clément Hyvrier (CIRGET and Collège de St-Laurent)
An encyclopedia whose name I will not mention describes Scotland in these terms: "hills, peat, few trees, skirts (kilts), an accent, very fine spirits (Scotch), tasty eggs to go with them (Scottish eggs), a culinary sense as brilliant as it is absurd (haggis and fried Mars bars, in whichever order you please), and 'million-dollar problems' (membership in Europe and the Hodge conjecture, among others)."
For the sake of concision, and for lack of time, we will focus on this conjecture, introduced in 1950, which states, in its modern form, that certain analytic classes (called Hodge classes) of a compact (complex) projective algebraic variety are algebraic over the rationals.
After clarifying the terms of this rather vague statement, while leaving out as many technical details as possible, we will explain its origin as well as some developments around this question, allowing us, I hope, to better grasp its significance.
Friday, June 3, 2022 at 1:30 PM (B-1506)
P vs. NP
Xavier Provençal (ÉTS)
Is it harder to find the solution to a problem than to convince others that the solution we found is correct? In some cases, our intuition leads us to believe it is. For example, finding a solution to a system of equations is usually harder than checking its validity.
Complexity theory is a branch of theoretical computer science concerned with this kind of question. It defines a hierarchy of problem classes based on the amount of work needed to solve them. Among others, it defines the classes P (problems for which a solution is "easy" to find) and NP (problems for which a solution is "easy" to verify). In 1971, Stephen Cook and Leonid Levin posed the question: is P = NP? In other words, is there a problem for which it is significantly easier to verify a solution than to find one?
In this talk, we will present an introduction to the theory of algorithmic complexity in order to formally define the classes P and NP. Next, we will present the theory of NP-completeness. This theory helps us understand the full scope of this question, as well as its implications for many concrete algorithmic problems.
Winter 2022
Friday, April 8, 2022 at 1:30 PM (B-1516)
The Poincaré Conjecture
Guillaume Roy-Fortin (ÉTS)
If you take an elastic band and place it on the surface of an apple, it is possible to stretch it and contract it down to a single point, simply by moving it gently, without ever letting it leave the surface of the apple. On the other hand, if you attached the same elastic band around a donut by threading it through the hole, it would be impossible to do the same thing. We say the apple is simply connected, while the surface of a donut is not. More than 100 years ago, Poincaré knew that this property characterized a two-dimensional sphere, and he wondered whether the same thing was true for a three-dimensional sphere. It would take more than 100 years before Perelman's spectacular papers provided an answer to the famous Poincaré conjecture.
In this talk, we will present the various mathematical objects needed to understand the statement of the conjecture, give a historical overview, and finally present some elements essential to Perelman's proof. We will try to keep the technical elements to a minimum!
Friday, April 1, 2022 at 1:30 PM (B-1516)
The Riemann Hypothesis
Michel Beaudin (ÉTS)
Giving a talk on the Riemann hypothesis is an interesting challenge for a teacher who is not a specialist on the subject.
But it is a wonderful opportunity to connect several events from history and to show that certain results obtained by Euler, Gauss, and several others paved the way for the paper written by Riemann in 1859.
In that paper, his famous hypothesis appears after he became interested, in turn, in the question of the number of primes less than a given quantity.
Indeed, Riemann extended to the complex plane the series defining the ζ function, which was already known to be linked to a product over primes.
This allowed him to make his famous hypothesis: the non-trivial zeros of the ζ function all have real part ½.
In other words, they would all lie on the critical line. We still do not know whether this hypothesis is true, but the 10 trillion non-trivial zeros of the ζ function found so far are all located on this critical line!
And if the Riemann hypothesis is true, it will improve our understanding of the distribution of prime numbers (and hundreds of other results will follow from it).
We will try to make the presentation interesting for everyone by defining the functions required and showing connections with some of our mathematics courses.
The technology available to us today should allow us to better illustrate our subject.
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