William Cosgrove Creative Portfolio

  • Virtual Reality Block/Puzzle Game using Optitrack camera rig – Created at USC’s World Building Media Lab:
    • Merges haptic feedback with virtual reality
      • Uses Oculus Rift. I attached Optitrack sensors to the Rift as well as to a set of wooden blocks. I then used Unity to program the blocks to map to virtual  puzzle pieces that the player works with.  It also features Alex McDowell’s “Chair in VR”, which the play can sit in.
    • Using same set of blocks for multiple games
      • I created multiple game types that could be played with the same set of blocks. One type was a standard puzzle game where you arrange the blocks as you would a puzzle, and it took place in various historical settings (Ancient Greece, Egypt, etc.). Another game was a color matching variant where you needed to observe your surroundings to decipher the right arrangement of the blocks. Lastly, one was based off of the World Building Institute’s renowned project The Leviathan, in which Gerty, a delicate floating jellyfish-like creature, is trapped and you must connect the circuits to free her.ext (1).jpgext.jpg

 

  • Conway’s Game of Life in 3D – Made for UCSB’s Allosphere (http://www.allosphere.ucsb.edu)
    • The Allosphere is a three story sphere with a metal bridge going through the middle of it. It has 26 projectors that project onto the interior of the sphere. People on the bridge where 3D glasses and get 360 degree immersion. It is meant to be a catalyst for collaboration among scholars.
      • I sat in on a graduate level course taught by the UCSB grad, Karl Yerkes, who was largely responsible for creating the “Allosystem”  coding language (primarily based off C++).  This language is exclusively used for programming for the Allosphere.
      • I created a 3 dimensional version of Conway’s Game of Life, which is the famous example of cellular automation created by mathematician John Horton Conway, the idea of which originated from John von Neumann in the 1940s. It is originally done in two dimensions. It was coded using Allosystem. It works by starting off with a randomized number of cells being “alive”. Then, based on the set of rules, which may be tweaked by the user, the cell colony develops on its own until it peters out, grows out of control, or finds a nice balanced state.                                                            Allosphere0018.jpgallosphere.jpg

Last Daily Writing

It feels great to have completed the mock grant proposal. I believe that it was a very valuable experience and will help me in the near future since I plan to apply for graduate school. I feel confident about the tone used throughout the paper. We did not stick to the most rigorous and formal language, but we were technical enough to establish our credibility. This was similar to our examples. There was a hint of personality to each of our example proposals that made them enjoyable to read.

I also feel confident in the content that is included in out grant proposal. We did our research to make sure our equations/facts were correct and up to date. The research process was actually very interesting. Had I not been assigned this project, I would almost surely not have started researching the Riemann Hypothesis. There were many aspects to it that related to my current math class in complex analysis. This assignment was very cool for that reason. While it teaches us to write, it also drives us to learn more about our disciplines.

If I were to revise the assignment again, I would include less generalities like, “This problem is the most infamous problem in the world.” I believe most mathematicians would find a statement like that unnecessary. All in all, I would try to make it more concise and make sure my information is relevant. I personally love learning about math history, and not for the purpose of boosting my ethos, I just enjoy learning about it. I am not sure, however, how commonplace such knowledge is to astute mathematicians. I chose to include multiple historical facts because I like them, but in case I ever write an actual grant proposal, I would first make sure this would not annoy my reader.

This entire process has relaxed my attitude on academic/professional writing. Before this class, academic articles were intimidating. They were these complicated bodies of words that I did not think I could understand. Now, I feel much more at ease about reading them as well as writing them. What helped a lot was finding sources relevant to our topic. I came across many confusing journal articles, most of what was inside I would not understand, but as I read on I found that there was a good amount I actually could understand. This was an exciting realization, that this level of research is right around the corner for me.

Daily Writing #11

Abstract

We are setting out to prove the infamously elusive Riemann Hypothesis. It has been about 150 years since Bernhard Riemann, a professor at University of Gottingen in Germany, proposed the hypothesis. Being the busy man he was, he left it aside for other mathematicians to prove as he continued on with other work. Many have since tried, and while nobody has proved it, there is now a wealth of knowledge on the topic. The central function, Riemann zeta function, is understood much more thoroughly than it once was, especially with the increase in visualization technology. We believe that a proof for this hypothesis is right on the horizon. A proof of this problem could also have very important implications. It would give us an equation to approximate the distribution of prime numbers, a central building block of mathematics. Albert Einstein built his famous theory of general relativity upon some of Riemann’s other work called “Riemann Geometry”. We can never know what will stem from new mathematical discoveries, but history has shown us that, often times, extraordinary scientific discoveries soon emerge. Using graphical representation, multi-dimensional modeling, and number analytics, we believe that we can construct a valid proof of the Riemann Hypothesis.

Daily Writing #10

The math grant proposals we found do not have an explicit methods section. My example has a section titled “Proposed Research” which is similar, in a way. The proposed research section is sort of like a rough sketch of a traditional scientific methods section mixed with a background section. The author describes areas of research that he will be undertaking (just like a background) and then discusses generally what tools he will be using to conduct the research (rough method description. This section is much less rigorous than it would be for a traditional science proposal due to the fact that, in math, it is hard to know exactly what method of research will yield the desired results. For this reason, our methods section will be somewhat vague. Here is an example, “I propose to compute hundreds (or thousands) of such discriminants, and then look for such certificates for their positivity.” (Sottile p.9) Here, the author explicitly states what he will be doing. However, he can’t say exactly what methods he will employ to “look for such certificates for their positivity” or if he will “compute discriminants” in different ways. In math, there are often numerous ways to go about analysis, and it would be a draining task to attempt to describe them all. By stating his plan the way he does, Sottile describes his methods as specifically as is practical. This is what we will attempt to do in our proposal. It is still important to lay out your plans for research because the grant organization needs to see some evidence that you will actually be working. Therefore, we will attempt to be specific in describing our proposed methods, but also include a sensible degree of generality.

The authors do, however, develop credibility with reassuring statements and rhetorical tones. One author mentions how the majority of the grant would go towards students, visitors, and their travel. This establishes a trustworthy quality for him, and shows that he has planned ahead in regards to his spending. In this same paragraph, he emphasizes the importance of training students so that they will become competent and skilled researchers in the future. He also mentions the importance of interdisciplinary work that will arise out of working with other professors and visitors, saying that there are many tools mathematicians use which could greatly benefit scientists. All of these reasons are aimed at reassuring the NSF that their money will be going towards important work, and that the author is responsible enough to follow through with his promises.

Intro/Background Draft

William Cosgrove

Sam Nasreldine

WRIT109ST

Dr. Fancher

Introduction and Background Draft

 

Proposal: Funding for research to prove the Riemann Hypothesis

Introduction

    The Riemann Hypothesis is one of the most famous mathematical problems that is yet to be solved. It is one of the “Millennium Problems”, which are a set of unsolved problems that were chosen by the Clay Mathematics Institute of Cambridge as the most difficult problems facing mathematicians at the turn of the millennium [4]. It has been over 150 years since the hypothesis has been proposed and yet nobody has proved it. This being said, there are countless published articles which conduct mathematical research on the assumption that the Riemann Hypothesis is true. These authors must, of course, state that they are researching under this assumption. Mathematical community members generally believe that the hypothesis is true, but if by some chance it were not, then all of these published works would be invalid, and so on to the “child” published works that stemmed from those ones. It would be a bit of a disaster. Luckily, this is probably not the case. Regardless, it is important that a valid proof is discovered for the Riemann Hypothesis for a couple of reasons. First of all, it is essential that science and math are grounded in proven facts. A proof of the Riemann Hypothesis would ground all the afore mentioned articles in fact. This is the most important reason for proving it. Two other and more general reason are that proving it could result in the discovery of a new method of analysis, which could in turn produce more mathematical breakthroughs, and the research which we will undergo will aid in educating university students to better prepare them for their future research.

    Our research team will use various methods of analysis to construct a proof of the Riemann Hypothesis. We are currently and will continue to undertake extensive research on methods that have been used by other mathematicians, which have shown to have some degree of success. We will be researching past attempts to prove the hypothesis like the Bloch-Kato Conjecture, various manipulations to the Riemann zeta function (which is the main piece of the hypothesis), Leonhard Euler’s related functions, and more. These examples of past work are within fields ranging from number theory, graph theory, geometry, fractals strings, to possibly more. Innovation will also be key, so a main goal of ours will be to pioneer our own analytic methods. Novel techniques in mathematics are often adopted by other mathematicians in hopes that these will assist in their research. Just as we are using equations formulated by Leonhard Euler in the 1700’s, it is possible mathematicians will someday use methods that we invent in their work. This process is integral to the advancement of mathematical knowledge, which is our number one concern.

    Graduate students will assist in research. We will also allow undergraduate students to volunteer if they desire. Training a competent workforce for the future is important across all disciplines. We all want our scientists to be able to excel and we want them to have the world’s best interests at heart. A study conducted by C. Madan and B. Teitge at the University of Alberta concluded that undergraduate students who were part of research teams had a clearer understanding of the publication process in their field and gained a passion for research. Including students in our mathematics research process will ensure that our field continues to advance for generations to come.

 

Background

    Since the Riemann Hypothesis has not been proven, no one knows which techniques will lead to its solution. For this reason, it is important that we examine many different areas of research. The following list is not an exhaustive list of all the fields which relate to the Riemann Hypothesis; they are just a few central ones. Sections “Analytic Number Theory” and “Fractal Geometry” contain only material that is relevant to our research.

Early Work (Pre 1950) and the Riemann Hypothesis Defined:

    Defining the Riemann Hypothesis is not a simple task. First we begin with the equation it is built upon. This equation is called the Basel Problem and was solved about 100 years earlier by Leonhard Euler. The equation is as follows:

1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + … + 1/n^2   as n increases to infinity.

It was proved by Euler that this series converges to 2/6. There were other problems similar to the Basel Problem where instead of squaring the denominator it was taken to other powers (3, 4, …), so for some number x, these equations would look like:

  1.       f(x) = 1/1^x + 1/2^x + 1/3^x + 1/4^x + … + 1/n^x   as n increases to infinity.

    Bernhard Riemann wanted see what happens when you change the domain of this function to include complex numbers. Complex numbers have a real component and imaginary component. The imaginary component has a coefficient i = -1. It does not make sense when working with real numbers to take the square root of a negative number, so, long ago, mathematicians imagined what would happen if you could. Imaginary numbers are of the form iy, where y is a real number. Complex numbers are of the form z = x + iy, so they have a real part and imaginary part (x and iy respectively). Out of this idea arose the complex plane:

plane.gif

Complex numbers are similar to ordered pairs in the commonly used xy-plane since z can also be written as (x,y).

    Now let’s look at Riemann’s famous function which extends the domain of (1) to include complex numbers. It looks exactly the same as (1) but s is a complex number. The function is famously known as the Riemann zeta function:

r(s) = 1/1s + 1/2s + 1/3s + 1/4s + … + 1/ns as n goes to infinity.

This function only has solutions when the real part of the complex number s is greater than 1 (Re(s) > 1). However, Riemann wanted to analyze this function for all complex numbers, so he analytically continued this function so that it converged for all complex numbers s 1(any s that is not equal to one, in complex form: 1+0i). This continuation is as follows, known as the Riemann functional equation:

r(s) = 22s-1sin(s/2)(1-s)(1-s)

Notice that (1-s) is part of the right side of the equation. This is what makes it a functional equation (points in the equation are defined in relation to other points).

    It turns out that this equation always equals zero when s is a negative even real number, and that is simply because sin(2n/2) = sin(n) = 0. These are the “trivial” zeros and are not the concern of the Riemann Hypothesis. When Riemann was working with his function, he proved that the non-trivial zeroes can only be obtained with s where 0 < Re(s) < 1. The hypothesis, which it is our goal to prove, takes this a step further and states, “All non-trivial zeroes are obtained by complex numbers s with Re(s) = 1/2.” The following shows the zeroes of the zeta function plotted on the complex plane (the dots to the left of the imaginary axis are the trivial zeroes):

Analytic Number Theory:

    The Riemann Hypothesis is considered to be one of the most difficult problems in number theory. Riemann specialized in complex analysis, so his hypothesis ended up making a fascinating bridge between the two fields. Moreover, it has implications for prime numbers and how often they occur, a phenomenon which has puzzled mathematicians for thousands of years.

    Researching how the hypothesis and zeta function are related to analytic number theory will help us in our development of a proof. Analytic number theory is a field in which various methods of analysis are employed to discover properties of the whole numbers. Multiplicative number theorists are interested in the Riemann zeta function because of its implications regarding the frequency of prime numbers.

    Let start by describing the Prime Number Theorem. It offers an equation which approximates the number of primes which occur before a given integer N. Here is that equation:

 

r(N) = N/log(N)

    where (N) is the actual number of primes less than the integer N and log is the natural log.

The Theorem claims that:

lim (as N goes to infinity) r(N) / (N/log(N)) = 1

This implies that their approximation equation gets closer and closer to the actual value as you increase the value of N.

    This theorem does not appear to be related to the Riemann zeta function. Actually, this theorem was not proven until after Riemann’s time (there were other approximations, but they were not as accurate). It was Riemann’s Hypothesis and its surprising connections to the distribution of prime numbers which encouraged mathematicians to use complex analysis in their research on the (N) function. It turns out that if it is true that the locations of the non-trivial zeroes in the Riemann zeta function do indeed lie on the line Re(s) = 1/2, then we would know much more precisely how primes numbers are distributed. There are other approximations which, like the above equation, depend on the truth of the Riemann Hypothesis in order to be accurate. Therefore, it is essential that we obtain a solid proof to solidify our knowledge of prime number distribution.

Graph Theory:

    Examining how the Riemann zeta function alters lines in the complex plane provides interesting visual results. Unlike functions in 2, complex functions are not lines. Instead, they alter the entire complex plane. To visualize this, we must add lines to the complex plane, then apply the function to see how the lines are altered after the function is applied. Here is an example of how the Riemann zeta function, before it is analytically continued, alters the complex plane:

Screen Shot 2017-02-17 at 4.32.24 PM.png          Screen Shot 2017-02-17 at 4.32.52 PM.png

We see an interesting pattern emerging. Now here is the analytic continuation:

Screen Shot 2017-02-17 at 4.39.27 PM.png           Screen Shot 2017-02-17 at 4.39.48 PM.png

Clearly there are benefits to analyzing the zeta function using graphs. It allows us to see patterns that would never be apparent by simply manipulating the function. One important thing to note about analytic continuations is that they are unique, meaning that if a function can be analytically continued, then it only has one analytic continuation. So the above graph represents the only analytic continuation of the zeta function that exists. However, it is possible to transform any line in the complex plane with a function. So the number of lines that can be altered with the zeta function are literally infinite. This will be an important method of research which we will combine with our number analysis.

3 Questions:

  • Is introduction understandable?
  • Does it fulfill the three rules adequately?
  • Is the background too long-winded? (it is less so than our examples)

 

References

  1. E. Bombieri, Problems of the millennium: The Riemann Hypothesis, CLAY, 2000.
  2. P. Borwein, S. Choi, B. Rooney, and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, Springer, New York, 2008.
  3. L. A. Bunimovich and C. P. Dettmann, Open circular billiards and the Riemann Hypothesis, Physical review letters 94.10, 2005.
  4. Clay Mathematics Institute, The Millennium Prize Problems, Oxford, United Kingdom, 2017. http://www.claymath.org/millennium-problems/millennium-prize-problems
  5. J. Coates, A. Raghuram, A. Saikia, and R. Sujatha, The Bloch-Kato conjecture for the Riemann Zeta Function, Cambridge University Press, Cambridge, 2015.
  6. J. Conrey, The Riemann Hypothesis, Notices of the AMS Volume 50, number 3, 2003.
  7. H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society: Colloquium Publications volume 53, Providence, RI, 2004.
  8. M. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, Springer, New York, 2013.
  9. C. Madan and B. Teitge, The Benefits of Undergraduate Research: The Student’s Perspective, Alberta, Canada, 2013.
  10. B. Mazur and W. Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, New York, 2016.
  11. B. Riemann, On the Number of Prime Numbers less than a Given Quantity, University of Dublin, Ireland, Translated 1998 (original 1859).
  12. Sanderson, Grant, Visualizing the Riemann zeta function and analytic continuation, Stanford, California 2016. https://www.youtube.com/watch?v=sD0NjbwqlYw
  13. A. Terras, Zeta Functions of Graphs: A Stroll Through the Garden, Cambridge University Press, United Kingdom, 2011.
  14. B. Wachsmuth, Interactive Complex Analysis, Seton Hall University, South Orange, New Jersey, 2007.

 

Screen Shot 2017-02-17 at 4.39.48 PM.png

Daily Writing #9

Our (Sam and I) plan is to put most of the emphasis on the importance of proving the Riemann Hypothesis. We will start by stating that proving the hypothesis is our sole research goal. It is easy to see that there is a gap in this area of research, hence the unproven hypothesis. We will then go on to describe the important impact this would have on the math community, by pointing out how many other journal articles in mathematics depend on assuming that the hypothesis is true. A proof would mean that all of this other articles are officially valid (currently they are assumed to be valid since most people believe the hypothesis to be true).

Secondly, we will research various organizations that would be willing to fund such research, and word our introduction to express how our research is inline with the organization’s values. This was a tactic used in both of our model grant proposals.

Another tactic that was used in my grant proposal fits in with emphasizing the importance of the research. The author included the fact that he would be working with graduate students and educating them on how to conduct research properly would follow naturally. The author claimed that this was extremely important since these graduate students will likely be the researchers, professors, and doctors of the future. We may include reasoning like this in our proposal (although we will not actually be employing graduate students). This can’t really be used as a central reason, however, because most grant proposals, regardless of the discipline, can make this claim.

Daily Writing #8

There are numerous proofs in mathematics that begin with “Suppose the Riemann Hypothesis is true…”. This is acceptable since most people already assume that the Riemann Hypothesis is true. All of this is well and good, but what if, by some minuscule chance, someone were to prove that the Riemann Hypothesis were false. Then, all of those proofs would be invalid. It would be a minor disaster.

This is why it is important that somebody proves this hypothesis. While it is probably the case that it is true, a valid proof of it would solidify the base from which so many other proofs have been constructed. Proving the Riemann Hypothesis will not only make it concrete knowledge, but also solidify all those proofs that start out by having the reader suppose the hypothesis is true.

There are many methods to analyzing prime numbers and the Riemann Hypothesis. Using binary numbers, matrix construction, etc. Our team will employ many methods to diversify our research and work towards a valid proof.

Daily Writing #7

  1. “What is Riemann’s Hypothesis?”

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7787E510886722FFBEB9EB0113C4B011/9781316182277c13_p40-41_CBO.pdf/what-is-riemann-s-hypothesis.pdf

This is a chapter from the book Prime Numbers and the Riemann Hypothesis by Barry Mazur (Harvard University, Massachusetts) William Stein, (University of Washington). It summarizes the Riemann hypothesis to put it into more understandable terms.

 

  1. http://site.ebrary.com/lib/ucsb/detail.action?docID=11036284

This source is a link to a book titled London Mathematical Society Lecture Note Series : The Bloch–Kato Conjecture for the Riemann Zeta Function written by John Coates, A. Raghuram, and Anupam Saikia. It goes into thorough detail on a conjecture (a statement which appears to be true but is not proven) regarding the Reimann Hypothesis.

 

  1. http://link.springer.com/book/10.1007%2F978-1-4614-2176-4

This is a link to a book titled Fractal Geometry, Complex Dimensions and Zeta Functions written by Michel L. Lapidus and Machiel van Frankenhuijsen. I will mainly focus on chapter 9 of this book which focuses on how the Riemann Hypothesis relates to complex dimensions and fractal sprays. Seeing how the hypothesis related to other areas will aid in my understanding.

 

  1. https://www.amazon.com/Zeta-Functions-Graphs-Cambridge-Mathematics/dp/0521113679

The book is titled Zeta functions of graphs : a stroll through the garden by Audrey Terras. It is not available online. It discusses how the Riemann Hypothesis relates to graph theory.

 

  1. http://link.springer.com/book/10.1007%2F978-0-387-72126-2

Titled The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, by Peter Borwein, Stephen Choi, Brendan Rooney, and Andrea Weirathmueller, this book will be a great resource as its sole topic is the Riemann Hypothesis. It provides an introduction to the subject, original papers written on it, and relevant theories and applications.