Columbia Undergraduate Math Society

Spring 2018 << Summer 2019 Lectures >> Fall 2018

Wednesdays, 7:30pm; Room 507, Mathematics
Source: Computing the Continuous Discretely by Matthias Beck and Sinai Robins
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Date Speaker Title Abstract
12 June
Matthew Lerner-Brecher
Introduction
We will first introduce Summer UMS and then decide on the logistics for the ensuing weeks.
19 June
Matthew Lerner-Brecher
Generating Functions
and Ehrhart Polynomials
We will begin the talk by introducing an important tool in combinatorics known as the generating function and give some examples of how it can be used for problems such as coin counting and evaluating a formula for the fibonacci sequence. We will then introduce one of the most important tools that will guide us this summer: the Ehrhart Polynomial. For this lecture we will mostly focus on the example of the simplex, basic properties, and giving intuition for how the ehrhart polynomial relates the continuous and discrete. Lastly, we will finish the talk by introducing and proving Pick's theorem.
26 June
Koh Yamakawa
Ehrhart Theory and its Applications
We will explore the connection between generating functions and volumes of polygons again through the Ehrhart theorem. This theorem will allow us to explore another connection between the continuous and the discrete alongside understanding the basis of open problems in the field.
3 July
Ryan Abbott
Ehrhart-Macdonald Reciprocity
We will explore the topic of reciprocity in Ehrhart Theory, in particular proving Ehrhart-Macdonald reciprocity and providing applications to the theory of reflexive polytopes. Time permitting we will also give a brief introduction Finite Fourier Analysis.
10 July
Alex Gajewski
Finite Fourier Analysis
This week we’ll be exploring finite Fourier analysis, an analog of the continuous Fourier analysis you might already be familiar with, but for periodic functions defined on the integers. I’ll try to focus on examples and applications, rather than getting lost in detailed trigonometric proofs. We’ll start by studying representations of periodic functions in terms of finite Fourier series, and then move forward with a linear algebraic interpretation of these series, studying finite Fourier transforms and the finite Parseval identity. We’ll conclude with a discussion of finite convolutions, and time-permitting a brief introduction to Dedekind sums.
17 July
Destine Lee
The Enumeration of
Semimagic Squares
We will be tackling the problem of efficiently enumerating semimagic squares, square matrices whose entries are nonnegative integers and whose rows and column sum up to the same number. Along the way, we will be investigating properties of the Birkhoff-von Neumann polytope and use these properties to establish a connection between semimagic squares and the Ehrhart theory we have been exploring over the past few weeks. If time permits, we will close with a discussion of magic squares.
24 July
Matthew Lerner-Brecher
Face Numbers and More
The main goal of this talk will be to prove the Dehn-Somerville Relations, which are a nice set of equations in Ehrhart Theory with implications regarding the faces of a polytope. In whatever time is remaining, we will give a survey of topics in chapters 9,10, and 11 of Computing the Continuous Discretely.
31 July
 
 
 
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