Abstract / résumé
Name / nom: Andrew Portolesi
School / école: Queen's University at Kingston
Length / durée:
Talk #1: 25 min
Talk #2: 50 min
Title / titre:
Talk #1: Algebraic Topology: Lost in Triangulation -- An Introduction into a Crazy Abstraction on Geometry
Talk #2: Dude, Where's My Public Key? An Exploration into the Workings of Modern-Day Cryptography
Abstract / résumé:
Algebraic Topology: Lost in Triangulation -- An Introduction into a Crazy Abstraction on Geometry
Topology is the study of shapes, surfaces, and spaces ("objects") where we ignore distance (that is, the space doesn't have a metric). Two objects are considered topologically equivalent if you can stretch, bend or twist one into the other without making any holes or filling up any (so a coffee mug with one handle is equivalent to a donut -- and you don't know which one to dunk into which). Unfortunately, trying to prove that one object "can be morphed into the other" is rather...well...hand-wavey.... That's where the algebraic part comes in. Algebraic topology uses (mostly) group theory to classify different objects to determine whether or not they are topologically equivalent. What we'll cover: a few topology definitions; fundamental groups; triangulation & homology groups (hopefully).
Dude, Where's My Public Key? An Exploration into the Workings of Modern-Day Cryptography
Cryptography is the science of encoding information so that others cannot decode it. Modern-day cryptography uses lots of number theory (modular arithmetic). What we'll look at is the mathematical mechanisms used in various cryptosystems (RSA, elliptic curves) -- how they work, why they work, and some attempts to crack them.... What we'll cover: a preamble on modular arithmetic; RSA cryptography; elliptic curve cryptography.
Prerequisites / choses nécessaires:
Talk #1: Group theory (just know what the different basic groups are: e, Z, etc.). Of help will be a bit of pre-thinking about abstract geometrical objects, such as a Möbius strip (a loop of paper that only has one twist in it) and n-spheres (a 1-sphere is a circle, a 2-sphere is a spherical shell, a 3-sphere is {(x,y,z,w) | x2 + y2 + z2 + w2 = 1}, ...).
Talk #2: Modular arithmetic: e.g. 7 + 5 is congruent to 3 (mod 9). Some knowledge of the properties of finite fields (like Fermat's Little Theorem) will help.
PDF format / format PDF: portolesi.pdf