The Fold and Cut Problem
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Speaker: Christopher Perry 
Time: 5:30 PM on Wednesday, March 21st, 2018 
Location: Porter Hall 100 
Abstract: Have you ever cut out a chain of paper figures or hearts? Or perhaps, what shapes you could make with fewer cuts? The Fold and Cut Problem asks what shapes can be created by folding a piece of paper, and creating a single cut through it. This talk will include the discussion of several methods and approaches to solving this problem, related theorems in paper folding, and some activities surrounding folding and cutting paper. Please bring scissors if you have them! 
Where the Nonzero Things Are
Speaker: Jason Howell 
Time: 5:30 PM on Wednesday, March 7th, 2018 
Location: *** Wean Hall 7500 *** 
Abstract: The performance of direct solvers for large, sparse linear systems of equations depends on a symbolic factorization of the coefficient matrix, which employs graph algorithms and data structures to determine the amount of work and memory required to factor the matrix. The location of the nonzero entries can heavily influence the symbolic factorization, and the presence of a dense row in the matrix can lead to significant overestimates of the amount of memory and work required to complete the factorization. In this talk we describe how an unconfentional application of a null space method for saddle point problems can be utilized to eliminate small number of dense rows while preserving the overall sparsity of the matrix. This results in a prestructuring technique, i.e. a method that seeks to modify the nonzero structure of the matrix with the intent of realizing gains in direct solver performance. 
Talk Slides: Click here. 
Taming Combinatorial Chaos through Imparital Game
Speaker: Alp Muyesser 
Time: 5:30 PM on Wednesday, February 28th, 2018 
Location: *** Doherty Hall 1212 *** 
Abstract: Combinatorial chaos is everywhere: whether we are trying to bound Ramsey numbers, compute NPhard languages, or solve complex board games like Go, we have to deal with the fact that 2^x grows, really, really fast. But chaos can also arise in way simpler settings, say when we are merely picking up stones from a bunch of piles! We'll discuss how we came really close to a complete characterization of an embarrassingly simple Nimlike impartial game, yet how combinatorial chaos managed to elude us once again. This is a very active and accessible area of research. All you need to know is how induction and binary works! This is joint research with Calvin Beideman and Matt Bowen.

Modeling Cognition and Interdisciplinary Collaboration
Speaker: Dr. April Galyardt

Time: 5:30pm on Wednesday, February 21st, 2018 
Location: *** Doherty Hall 2315 *** 
Information:
Dr. Galyardt is a statistician and data scientist at Carnegie Mellon University's Software Engineering Institute. She received her PhD from Carnegie Mellon University's Statistics Department, but received her Bachelor's and Master's in Mathematics previously. In Fall 2017, she taught 36401 Modern Regression at CMU. She will speak on the modeling of cognition, and how interdisciplinary collaboration plays a role in learning and beyond.

Game Night <3
Activity: Games and heartshaped pizza!

Time: 5:30 PM, Wednesday, February 14, 2018

Location: Porter Hall 100

More Information: What an amazing coincidence! Valentine's Day has fallen on the day of a Math Club meeting! Come join most of exec for an exciting hour of games and thematically appropriate pizza  feel free to bring your own games as well.

Movie Night
Movie: N is a Number: A Portrait of Paul Erdos

Time: 5:30 PM, Wednesday, February 7, 2018

Location: Porter Hall 100

More Information: 2/7/18 is e Day (for the beloved mathematical constant, e=2.718...), and what better way to celebrate than with a movie about a certain mathematician whose name starts with the letter e  Erdos (sorry Euler)! This critically acclaimed movie is 57 minutes long and rated 7.5/10 on IMDb. There will also be plenty of Pittsburgh Popcorn to go around! Happy e Day!

Cycles
Speaker: Professor John Mackey

Time: 5:30pm on Wednesday, January 31, 2018 
Location: Porter Hall 100 
Abstract:
When n people play each other in a tournament (each person playing the other n1 people), we'd like to rank the participants after knowing the results. Cycles are an impediment to such a ranking.
If A beats B, B beats C, and C beats A, how should A, B, and C be ranked relative to one another?
I will explain just how many of these cycle impediments can exist in a tournament (for 3, 4, and 5 cycles), and then state some open problems related to this question. This is joint work with Natasha Komarov. 
Undergraduate Research Opportunities
Speaker: Professors Jason Howell and Irina Gheorghiciuc

Time: 5:30pm on Wednesday, November 29th, 2017 
Location: Porter Hall 100 
Slides : Here. 
Abstract:
During this meeting, Dr. Irina Gheorghiciuc and I will give a presentation about undergraduate research opportunities in mathematical sciences at CMU and beyond. We will describe what an undergraduate research project usually entails and how to search for opportunities. We will also discuss how to apply for summer programs and grants, highlighting important aspects of application materials. Some students who have participated in summer research in the past will share their experiences as well. 
Monstruous Functions
Speaker: Professor Ian Tice

Time: 5:30pm on Wednesday, October 25th, 2017 
Location: Doherty Hall A302 
Abstract:
Until the 19th century it was widely believed among
mathematicians that continuous functions of a real variable should in
fact be differentiable at "most" points. It thus came as quite a
surprise when Karl Weierstrass constructed a beast of a continuous
function that is differentiable nowhere. The purpose of this talk is to
construct such a monstrous function. 
Have Yourself a Merry Little Christmas (Theorem)
Speaker: Professor Kate Thompson, Visiting Shelly Professor

Time: 5:30pm on Wednesday, October 11th, 2017 
Location: Porter Hall 100 
Abstract:
One of the most classic results in number theory was originally stated by Fermat in a letter to Mersenne dated Christmas,1640. In "typical" Fermat fashion, the first proof didn't appear for over 100 years. Having said that, this result shows the breadth and beauty of number theory techniques. Time permitting, we will go through a (1) algebraic (2) geometric (3) analytic proof of this result. At the end, we will also discuss generalizations of this result, some proved as recently as 2013.

Talk Slides: Click here. 
Constant Functions and Sobolev Spaces
Speaker: Michael Spoerl, Senior Math Major

Time: 5:30pm on Wednesday, September 27th, 2017 
Location: Porter Hall 100 
Abstract:
In a 2002 paper, Brezis gave a (relatively) simple criterion for identifying constant functions. The proof, however, is not so simple  it requires a detour through Sobolev spaces. In this talk I will introduce the theory of Sobolev spaces and use the main theorem of Brezis' paper to prove his criterion. Only a conceptual understanding of calculus is needed.

Mathematical Contest in Modeling Interest Meeting
Time: 5:30pm on Wednesday, September 20th, 2017 
Location: Porter Hall 100 
Details:
The Mathematical Contest in Modeling (MCM) is a yearly contest for undergraduates interested in applied mathematics think of it like a cross between Putnam and a hackathon for math. A team has 96 hours to put together a solution to a problem in math modeling. Some examples of problems from the past:
 Estimate the global effects of a large asteroid impacting Antarctica
 Study the hunting strategies of velociraptor dinosaurs based on fossil data
 Develop a more efficient method of boarding passengers onto large commercial jets
If this sounds interesting to you, feel free to email Adrian Hagerty (Math Grad student, president of SIAM).

First General Body Meeting!
Time: 5:30pm on Wednesday, September 13th, 2017 
Location: Porter Hall 100 
Details:
Welcome to Math Club! We'll introduce ourselves, give a brief overview of the type of things we do, show a tentative schedule of our events this weekend, and advertise our first event: Going to a Pirates Game on Friday, September 22nd! If we have time, we'll wrap up by watching the short film Flatland . There will be Pizza!

Viscosity Solutions
Speaker: Giovanni Leoni, Professor of Mathematics, CMU

Time: 5:30pm on Wednesday, April 12th, 2017 
Location: Doherty Hall 2315 
Abstract:
In this talk I will try to motivate the notion of viscosity solutions for ordinary differential equations.

Star Numbers: from 17thcentury oranges to delivery robots and beyond
Speaker: Josh Laison, Associate Professor of Mathematics, Willamette University

Time: 5:30pm on Monday, Mar. 27th, 2017 
Location: Scaife Hall 125 
Abstract:
We discuss a new variation of a 400yearold problem from Johannes Kepler and Isaac Newton about how densely geometric shapes can be packed together. Along the way we'll encounter Amazon delivery robots, geometric networks, an iPhone game, computers that prove theorems, 24dimensional spheres, Tetris, cannonballs, and some exciting new theorems. Joint work with Andrew Bishop, Ben Gardiner, and David Livingston.

Counterintuitive Ideas in Analysis
Speakers: Joseph Zoller and Ani Sridhar, Juniors at CMU

Time: 5:30pm on Wednesday, Mar. 8th, 2017 
Location: Margaret Morrison 103 
Abstract: We will explore the basic notions in real analysis such as continuity, differentiability and integrability and completely annihilate any intuition you may have about them. We'll use and introduce basic tools from real analysis to construct strange functions. Memes will be provided.

Fermi Questions
Speaker: Liza Sulkin, Freshman Mathematics Major, CMU

Time: 5:30pm on Wednesday, Feb. 22nd, 2017 
Location: Doherty Hall 2315 
Abstract: Fermi Questions involve quickly determining the order of magnitude of absurd questions. We'll be answering several important questions such as:
 How many Cathedral of Learnings would it take to reach Philadelphia from Pittsburgh?
 How many wire cuts would it take to free the UC Stones?

Attenuated Tomography Old and New
Speaker: Nicholas Hoell, University of Toronto

Time: 4:30pm on Friday, Feb. 3rd, 2017 
Location: Wean 7500 
Abstract: We will explore the mathematics of the attenuated ray transform. The problem of recovering an unknown function based on measuring its weighted line integrals arose in medical imaging modalities and is connected to deep open questions in geometry. Some of the techniques used to address this problem involve ideas from harmonic analysis, representation theory, and gauge theory. No background knowledge of these fields or of inverse problems in medical imaging is assumed.


Some Applications of Differential Equations in the Social, Life, and Physical Sciences
Speaker: Jason Howell, College of Charleston 
Time: 4:30pm on Friday, Jan. 27th, 2017 
Location: Wean 7500 
Abstract: In this talk we will survey several different applications of differential equations in a variety of settings. Witha primary focus on the modeling process and the influence of problem parameters on solutions, we will discuss how basic mathematical models of population growth and interaction can be extended to a multitude of areas, including love affairs, the spread of communicable diseases, warfare and combat, marketing, the spread of ideas, and even a zombie apocalypse. I will also discuss some ongoing undergraduate research projects that focus on popularity dynamics and targeted advertising. If time permits, a brief discussion of some applications of partial differential equations will be discussed.

How to Multiply Big Numbers
Speaker: Boris Bukh, Associate Professor of Mathematics, CMU 
Time: 5:30pm on Wednesday, Nov. 16th, 2016 
Location: Gates 4307 
Abstract: All over the globe, young school children spend millions of hours multiplying numbers. I will demonstrate possible ways of multiplying numbers. While much improvement is possible, they offer no relief to the children. I will conclude the talk with cryptic remarks about college students multiplying matrices.

Triforce of Chaos Theory
Speaker: Zachary Singer, Junior Mathematics Major, CMU 
Time: 5:30pm on Wednesday, Oct. 19th, 2016 
Location: Wean 7500 
Abstract: What does a butterfly flapping its wings have to do with the Triforce? In this talk we introduce a few ideas of chaos theory and fractal geometry, as well as how different notions of dimension can lead to a set having nonintegral dimension. These ideas can be combined to help determine the behavior of how a biological virus spreads, which is one application I worked with over the summer.

Sequences, Series, and Subterfuge
Speaker: Misha Lavrov, Mathematics PhD Student, CMU 
Time: 5:30pm on Wednesday, Oct. 5th, 2016 
Location: Gates 4307 
Abstract: Click here for a pdf version. 
Talk Slides: Click for slides in a pdf version. 
Partial Differential Equations
Speaker: Sam Zbarsky, Senior Mathematics Major, CMU 
Time: 5:30pm on Wednesday, Sept. 21st, 2016 
Location: Gates 4307 
Abstract: This talk will be an introduction to two tools useful in studying partial differential equations: the Fourier transform and energy methods. I will introduce both of them, and show how they apply to several partial differential equations, including one I worked with over the summer.

Geometry, Combinatorics, Patterns and Fun
Speaker: Niraj Khare, Faculty Candidate for CMU Qatar 
Time: 5:00pm on Wednesday, Mar.23rd, 2016 
Location: DH 2315 
Abstract: Click here for a pdf version 
Talk Slides: slides in pdf version 
Trigonometry Revisited or: If Fourier Had a Quantum Computer
Speaker: Zijian Diao, Faculty Candidate for CMU Qatar 
Time: 5:00pm on Wednesday, March 2nd, 2016 
Location: DH 2315 
Abstract:
Is sine of 1 degree rational? At a glance this question seems light years away from quantum computing, a cuttingedge research area where computer science meets quantum mechanics. Surprisingly, this question matters not only in the mathematical world, but also in the quantum realm. Its answer and many other rudimentary facts of trigonometry have found their way into the study of various research problems in quantum computing. In this talk we will explore these intriguing connections through the interplay of quantum algorithms and topics from number theory and classical analysis. Along the way, we will solve a longstanding puzzle in quantum search and provide a quantum approach to the centuriesold Basel problem.

Thue Theorems and Avoidable Patterns
Speaker: Martin Juras, Faculty Candidate for CMU Qatar 
Time: 5:00pm on Wednesday, Feb.24th, 2016 
Location: WEH 8220 
Abstract: Click here for a pdf version 
Infinite Dimension in Linear Algebra: A Case for Functional Analysis
Speaker: Adrian Hagerty, PhD student in Math Dept at CMU 
Time: 5:30pm on Wednesday, Feb.17th, 2016 
Location: DH 2315 
Abstract: The study of Functional Analysis forms an important role in the modern fields of PDEs and Calculus of Variations. This involves taking the familiar notion of a vector space and bringing it to the realm of infinite dimensions. What new problems arise as we attempt to generalize our old tools to the infinite setting? 
What is Structure?
Speaker: Clive Newstead, PhD student in Math Dept at CMU 
Time: 5:30pm on Wednesday, Jan.27th, 2016 
Location: DH 2315 
Abstract:
The idea of equipping a set with a structure is ubiquitous in mathematics. For example, graphs are sets equipped with an edge relation, groups are sets equipped with an algebraic structure, topological spaces are sets equipped with a notion of openness, posets are sets equipped with a notion of order... the list goes on forever and ever and ever. This talk will introduce a candidate for defining 'structure' in the abstract. With the tools we develop, we can prove some pretty cool stuff in enumerative combinatorics. 
Probability and Intuition
Speaker: Peter Winkler, William Morrill Professor of Mathematics and Computer Science, Dartmouth 
Time: 5:30pm on Wednesday, Dec.2nd, 2015 
Location: DH 1112 
Abstract:
Supposedly our intuition about probabilityeven though humans invented the conceptis faulty. Lots of puzzles, naturally, are designed to lead us down the garden path to incorrect conclusions. But maybe our intuition is not as bad as we think. We'll take a fresh look at some paradoxes, old and new, to see whether we or our assumptions are at fault.

Poster: Poster

Fine Structure in Shapememory Alloys
Speaker: Winston Yin, senior at CMU, math and physics double major 
Time: 5:30pm on Wednesday, Nov.18th, 2015 
Location: PH 100 
Abstract:
The peculiar properties of a shapememory alloy are the result of the fine structure formed between its several crystalline forms. How do these fine structures arise from basic ideas about a crystal? In this talk, I will explain how infinitely fine structures are a natural consequence of integral minimization problems, as well as the project I worked on at Oxford this summer.

Variational Problems in Materials Science and Image Inpainting
Speaker: Irene Fonseca, Mellon College of Science University Professor of Mathematics,
Director of Center for Nonlinear Analysis

Time: 5:30pm on Wednesday, Nov.11th, 2015 
Location: PH 100 
Abstract: A plethora of physical and technological applications ranging from analyzing instabilities in materials science to image analysis in computer vision are subject to rigorous mathematical understanding through recently developed methods and articulation of ideas in the calculus of variations, geometric measure theory, and nonlinear partial differential equations. In this talk, these techniques will be used for recolorization of damaged images.

The Projective Heat Map
Speaker: Richard Schwartz, Chancellor`s Professor of Mathematics, Brown University

Time: 11:30 on Friday, Oct.16th, 2015 
Location: WEH 8200 
Abstract: Abstract and Slides

MeetandGreet with Paul Raff
Triples
Speaker: PoShen Loh, Math Professor at CMU, USA IMO Lead Coach

Time: 5:30 on Wednesday, Sept.30th, 2015 
Location: PH 100 
What's the longest sequence of triples (x1, y1, z1), (x2, y2, z2), ... that satisfies the following properties?
(1) Each number is an integer between 1 and N inclusive.
(2) For every j less than k, if we compare the triples (x_j, y_j, z_j) and (x_k, y_k, z_k), there are at least two coordinates in which the latter triple strictly exceeds the former triple.
It turns out that this simplesounding problem is equivalent to a question from Ramsey Theory, inspired by a question from kmajority tournaments, and related to deep question involving induced matchings and Szemeredi' s Regularity Lemma.
The talk will be a tour of Combinatorics which introduces many of these concepts along the way.
Sticky Particles
Speaker: Adam Williams, senior math major at
CMU 
Time: 5:30 on Wednesday, Apr 29th, 2015 
Location: Scaife 125 
Imagine you have a finite set of particles, moving along the x axis, with
fixed masses and initial velocities. When two particles collide, they
*stick* together to form a new particle, preserving the mass and momentum
of the original two. This is a simple setup, but seems potentially
difficult to work with mathematically. Adam shows you how you can use math
to study this system in a way that makes it much more understandable.
Unexpected distribution phenomenon resulting from Cantor
series expansions
Speaker: William Mance, PhD, CMU alumnus 
Time: 5:00 on Wednesday, Apr 22nd, 2015 
Location: Scaife 125 
We explore in depth the theoretic and statistical properties of
certain sets of numbers arising from their Cantor series expansions.
As a direct consequence of our main theorem we deduce numerous new
results as well as strengthen known ones. This work was coauthored
with Dylan Airey while he was still a high school student at the
Texas Academy of Mathematics and Science. Our main theorem
will touch on issues related to computability theory, ergodic theory,
fractal geometry, number theory, and probability theory.
Why the IRD cares about the Riemann Zeta Function and Number Theory
Speaker: Steven Miller, Professor at Williams
College 
Time: 5:30 on Wednesday, Mar 25th, 2015 
Location: Scaife 125 
Many systems exhibit a digit bias. For example, the first digit
base 10 of the Fibonacci numbers or of 2^n equals 1 about 30% of
the time; the IRS uses this digit bias to detect fraudulent corporate tax
returns.
This phenomenon, known as Benford s Law, was first noticed by
observing which pages of log tables were most worn from age – it is a good
thing there were no calculators 100 years ago! We will discuss the general
theory and application, talk about some fun examples (ranging from the 3x1
problem to the Riemann zeta function), and if time permits discuss
some joint results with my REU students.
The Birch and SwinnertonDyer conjecture
Speaker: Tomer Reiter, senior math major at CMU 
Time: 5:30 on Wednesday, Mar 4th, 2015 
Location: Scaife 125 
The Birch and SwinnertonDyer conjecture is one of the Millennium
Problems. The conjecture relates the group of rational points on an
elliptic curve to information associated with the equation for the
elliptic curve over finite fields.
In the talk, we will see some of setup for the conjecture, the
statement and some of the partial progress made. I will basically
assume David's talk last time as a prerequisite, but I will briefly
go over the facts about elliptic curves he mentioned that we will
need.
The Congruent Number Problem and Elliptic Curves
Speaker: David Mehrle, senior math major at CMU 
Time: 5:30 on Wednesday, Feb 18th, 2015 
Location: Scaife 125 
A positive integer N is called congruent if it is the area of a
right triangle whose sides have rational lengths. Determining
whether or not a given positive integer is congruent is a
millenniaold problem that is still open today. I will explore this
problem and several attempted solutions, including an interesting
modern approach using elliptic curves.
Missing the Point
Speaker: Misha Lavrov, PhD student at CMU 
Time: 5:30 on Wednesday, Jan 28th, 2015 
Location: Scaife 125 
If you are speaking with an alien from Mars who knows a lot of
math and has never heard the word Euclid, explaining what
geometry is can be frustrating. You will try very hard to come up
with foolproof ways to axiomatize geometry, and the alien will
persist in coming up with bizarre structures that look nothing
like what you imagined.
In 1899, Hilbert came up with a set of 16 alienproof axioms for
plane geometry. (That is not quite true. He had 17. But there’s
been some progress in the intervening century.) In this talk, I
will play the role of the alien in an effort to persuade you that
the task of Hilbert was not an easy one.
Special Guest Speaker
Speaker: Noam Elkies, Professor of Mathematics, Harvard University

Time: 5:30 to 6:30 on Wednesday, December 3rd

Location: WEH 7500

TBA
The fundamental theorem of algebra
Speaker: Giovanni Leoni, Professor of Mathematics 
Time: 5:30 to 6:30 on Wednesday, October 27th

Location: DH 1112

In this talk we will discuss the fundamental theorem of algebra and give an analytical proof.
You do not have to believe in transcendental numbers
Speaker: Boris Bukh, Assistant Professor of Mathematics 
Time: 5:30 to 6:30 on Wednesday, October 8th

Location: DH 1112

The transcendental numbers are the atoms of the number line  they are everywhere, but few have seen one. No longer you will have to believe in the existence of these halfmythical numbers! I will arm you with the knowledge that will enable you to write a number down, and prove to your cat that it is transcendental!
Pizza will be served.
When IMO meets Research
Speaker: Po Shen Loh, Assistant Professor of Mathematics 
Time: 5:30 to 6:30 on Wednesday, September 24th

Location: MM 103

This year, problem 6 on the International Math Olympiad turned out to be a
semiopen research problem. The originally proposed problem is solvable
using basic methods, but it is still open to determine the best possible
result. A set of lines in the plane is in general position if no two are
parallel and no three are concurrent. A set of lines in general position
cuts the plane into cells, some of which have finite area; we call these
finite cells. Prove that for all sufficiently large $n$, in any set of $n$
lines in general position it is possible to color at least $\sqrt{n}$ of
the lines blue in such a way that none of its finite cells has a completely
blue boundary. What's the best bound that you can get? We'll improve the
stated bound using techniques from probabilistic combinatorics and extremal
hypergraph theory.
Assessing Artifacts: Using Calculus of Variations to Segment Damaged Images
Speaker: Nick Takaki, Undergraduate 
Time: 5:30 to 6:30 on Wednesday, September 10th 
Location: DH 1112

A biphase approximation of an image is a twocolor piecewiseconstant approximation of the original image which maximizes regional fidelity while minimizing complicated boundaries. When it comes to segmenting images, human intuition is very strong. We often have strong gut feelings about which pixels belong in which regions, where object boundaries should be, and what should be discounted as noise or damage. Variational image segmentation seeks to formalize these "gut feelings" by creating a functional which quantifies the error of approximation, and then defining correct segmentations as local minima of this functional. In this talk, I will discuss the intuition behind variational image approaches, popular regionbased segmentation methods including the ChanVese model, and my group's summer research into a segmentation method which dynamically identifies artifacts and corrects shadows.
Chaos In Cellular Automata
Speaker: Carson Sestili, Undergraduate 
Time: 5:30 on Thursday, May 1, 2014 
Location: WEH 5415 
An elementary cellular automaton is a sort of computational model. States are sequences of bits. Transitions between states are specified by a simple, deterministic rule that updates each bit according to a small neighborhood around that bit in the previous state. Even though the model is simple to describe, many cellular automata have surprisingly beautiful and complicated behavior. In some cases their behavior is so unpredictable that it might be called "random," even though there is no randomness in the system. In this talk I will show why several automata exhibit chaotic behavior, for a popular definition of chaos.
The Estimathon
Speaker: Andy Niedermaier, Jane Street Capital

Time: 5:30pm on Wednesday, March 26, 2014

Location: DH 2210

They're called Fermi problems...
 how many traffic lights are there in Boston?
 how many social security numbers are primes?
 how many calories are in an average Cheesecake Factory?
 how many times will this activity get rescheduled due to bad weather?
Jane Street presents "The Estimathon." Attempt 13 estimation problems in 30 minutes, ranging from totally trivial to positively Putnamesque. Work in teams to come up with the best set of confidence intervals. Compete against your fellow students for fame* and fortune^!
(* fame is subjective)
(^ there will be prizes for the winners)
How Physics can help with the "Big Picture" in Math
Speaker: Ira Rothstein, Professor of Physics 
Time: 5:30 on Wednesday, February 19, 2014 
Location: DH 2315 
Why are differential equations so ubiquitous? It seems that all
physical systems are described by a set of differential equations.
Perhaps we should stop and ask why? In this talk I will explain
how a certain set of fundamental principle that apply to all physical
systems automatically lead to differential equations (DE's) as equations of
motion, as opposed to say integral equations. I will also discuss how
the symmetries of physical systems, and their corresponding DE's,
naturally lead to solutions via the use of group theory. Finally I will
discuss how the principle of observer independence leads to the
idea of differential geometry.