GEMS December 7th Session

This GEMS session will be facilitated by Professor Lenny Fukshansky from Claremont McKenna College.

Title:  From Knapsacks and Changing Coins to Geometry

Abstract:  Suppose you have a bag that can hold a fixed amount of weight, and you are trying to fill it with several types of objects of different weights and prices. The goal is to maximize the value of your bag. How do you do it? This is a notoriously difficult optimization problem, which often arises in resource allocation with financial constraints. Another famous optimization problem asks what amount of change can you give with coins of prescribed denominations? Surprisingly, not only are the two closely related to each other, they both can be restated in the geometric language of polygons, points with integer coordinates, and their higher-dimensional generalizations! We will discuss these important problems and their beautiful connection to a classical problem in geometry: how can we count the number of integer points in a fixed polygon? The answer is given by the celebrated 19th century theorem of Georg Alexander Pick, who proved a remarkable formula for this number in terms of the area and perimeter of the polygon. We will talk about Pick’s theorem and perform a hands-on exploration of this fascinating area of geometry.