Mathematics

Crossing Numbers of Large Complete Graphs

Speaker: 
Noam Elkies
Date: 
Fri, Oct 2, 2020
Location: 
Zoom
PIMS, University of Calgary
Conference: 
The Unsolved Problems Conference: Celebrating the living legacy of the mathematics of Richard Guy
Abstract: 

TBA

Class: 

Aliquot sequences

Speaker: 
Carl Pomerance
Date: 
Fri, Oct 2, 2020
Location: 
Zoom
PIMS, University of Calgary
Conference: 
The Unsolved Problems Conference: Celebrating the living legacy of the mathematics of Richard Guy
Abstract: 

These are sequences formed by iterating the sum-of-proper-divisors function. For example: 12, 16, 15, 9, 4, 3, 1, 0. Of interest since Pythagoras, who remarked on the fixed point 6 (a perfect number) and the 2-cycle 220, 284 (an amicable pair), aliquot sequences were also one of Richard Guy's favorite subjects. The Catalan--Dickson conjecture asserts that every aliquot sequence is bounded (either terminates at zero or becomes periodic), while the Guy--Selfridge counter-conjecture asserts that many aliquot sequences diverge to infinity. It is interesting that Guy and Selfridge would make such a claim since no aliquot sequence is known to diverge, though the numerical evidence is certainly suggestive. The first case in doubt is the sequence beginning with 276. This talk will survey what's known about the problem and give evidence for and against the two countervailing views.

Class: 

The favorite elliptic curve of Richard

Speaker: 
Jaap Top
Date: 
Fri, Oct 2, 2020
Location: 
Zoom
PIMS, University of Calgary
Conference: 
The Unsolved Problems Conference: Celebrating the living legacy of the mathematics of Richard Guy
Abstract: 

Even in the title of one of his papers, Richard Guy called the elliptic curve with equation $y^2 = x^3 - 4x + 4$ his favorite. During the CNTA-XIV meeting in Calgary in 2016, I recalled some of his reasons for this (with Richard listening from the front row). The story as well as a few additional developments will also be the topic of the present lecture.

Class: 

Richard Guy and the Encyclopedia of Integer Sequences: A Fifty-Year Friendship

Speaker: 
Neil J. Sloane
Date: 
Fri, Oct 2, 2020
Location: 
Zoom
PIMS, University of Calgary
Conference: 
The Unsolved Problems Conference: Celebrating the living legacy of the mathematics of Richard Guy
Abstract: 

Richard Guy was a supporter of the database of integer sequences right from its beginning in the 1960s. This talk will be illustrated by sequences that he contributed, sequences he wrote about, and especially sequences with open problems that he would have liked but that I never got to tell him about.

Class: 

The Unity of Combinatorics: Connections and Wonders

Speaker: 
Bud Brown
Date: 
Fri, Oct 2, 2020
Location: 
Zoom
PIMS, University of Calgary
Conference: 
The Unsolved Problems Conference: Celebrating the living legacy of the mathematics of Richard Guy
Abstract: 

An account of how a great Guy and his Brown coauthor created a 300-page book entitled "The Unity of Combinatorics" out of a 30-page paper from 1995 of the same name. The latter was an outline of a proposed lecture series, whose purpose was to feature the many connections within the vast area of combinatorics, thereby dispelling the then prevalent notion that combinatorics is just a bag of tricks. In writing the book, we took this notion and ran with it --- and how!

I'll talk about a number of these connections and some topics that seem almost magical, including Beatty sequences, Conway worms, games played with turtles instead of coins, and a way of viewing the non-negative integers as a field. The book begins with a child playing with colored blocks on his living-room rug and ends with a description of the Miracle Octad Generator. Finally, I'll talk about working with this gentlemanly giant of the world of numbers and sequences and patterns and games.

Class: 

Unsolved Combinatorial Games Richard K. Guy liked and others he would have liked

Speaker: 
Richard Nowakowski
Date: 
Fri, Oct 2, 2020
Location: 
Zoom
PIMS, University of Calgary
Conference: 
The Unsolved Problems Conference: Celebrating the living legacy of the mathematics of Richard Guy
Abstract: 

Richard started the Unsolved Problems in Combinatorial Games Column. I'll consider some of his favourites, talk about some developments, and add a few reminiscences.

Class: 

A recurrence lemma and its applications and extensions

Speaker: 
Barak Weiss
Date: 
Mon, Oct 12, 2020
Location: 
Zoom
Conference: 
Online working seminar in Ergodic Theory
Abstract: 

I will discuss a lemma which is usually attributed to Atkinson, is very useful and apparently has been rediscovered multiple times. The lemma says that if (X, B, mu, T) is an ergodic system and f is in L^1(mu) with mean zero, then the Birkhoff sums of f do not diverge almost surely. Moreover the sums switch signs in a wide sense, infinitely many times. I will give the proof and discuss some applications of this result. Then I will discuss higher dimensional analogues, where the situation is significantly more complicated.

Class: 
Subject: 

Equidistribution of geodesic flow pushes via exponential mixing.

Speaker: 
Jon Chaika
Date: 
Mon, Oct 5, 2020
Location: 
Zoom
Conference: 
Online working seminar in Ergodic Theory
Abstract: 

Pick a point at random in a finite volume hyperbolic surface and simultaneously flow in all directions from it. For the typical starting point these expanding circles will equidistribute and this talk will present a (more general) argument of Margulis establishing this fact.

Class: 
Subject: 

Conformal field theories and quantum phase transitions: an entanglement perspective

Speaker: 
William Witczak-Krempa
Date: 
Wed, Sep 30, 2020
Location: 
Zoom
University of Sasktachewan
Conference: 
quanTA CRG Seminar
Abstract: 

Quantum phase transitions occur when a quantum system undergoes a sharp change in its ground state, e.g. between a ferro- and para-magnet. I will present a remarkable set of transitions, called quantum critical, that are described by conformal field theories (CFTs). I will focus on 2 and 3 spatial dimensions, where the conformal symmetry is powerful yet less constraining than in 1 dimension. We will probe these scale-invariant theories via the structure of their quantum entanglement. The methods will include large-N expansions, the AdS/CFT duality from string theory, and large-scale numerical simulations. Finally, we’ll see that certain quantum Hall states, which are topological in nature, possess very similar entanglement properties. This hints at broader principles that relate very different quantum states.

For other events in this series see the quanTA events website.

Class: 

Characterizing handle-ribbon knots

Speaker: 
Maggie Miller
Date: 
Sun, Sep 27, 2020
Location: 
Zoom
Conference: 
Cascade Toplogy Seminar
Abstract: 

Kauffman conjectured that a knot K is slice if and only if it bounds a genus-g Seifert surface containing a g-component slice link as a cut system. It’s very easy to show that a knot is ribbon if and only if it bounds a genus-g Seifert surface containing a g-component unlink as a cut system. Alex Zupan and I proved something in the middle of these statements: a knot is handle-ribbon (aka strongly homotopy-ribbon, aka something I will define in the talk) if and only if it bounds a genus-g Seifert surface containing a g-component R link L as a cut system—meaning that zero-surgery on L yields #_ g S^1 × S^2 . This gives a 3-dimensional definition of a 4-dimensional property. I’ll talk about these 3.5D knot properties and maybe how we use these techniques to extend a statement of Casson and Gordon. (The work in this talk is joint with Alexander Zupan from the University of Nebraska–Lincoln.)

Class: 

Pages