Like many universities, the University of Central Florida (UCF) has had many guest speakers this semester. It is quite interesting to hear the thoughts of folks from other universities regarding mathematics and its role. Two of my favorite talks recently have been by Dr. John dePillis and Krishnaswami Alladi.

**Mathematical Conversation Starters: John dePillis – University of California, Riverside**

Dr. dePillis’ talk was aimed at both mathematicians and non-mathematicians. (Even some of my calculus students were in attendance and found it quite enjoyable.) He discussed many different topics including the scientific method, the birthday problem, Monty Hall problem, Sherlock Holmes, algebra of the mind, critical thinking, etc.

I think the talk’s message that most impacted my students is the algebra of the mind. Students many times think of algebra as something that occurs in mathematics and does not relate to how they think during their daily lives. dePillis used the distributive law to show this is not the case. He defined the following events:

- R: I will go to a restaurant
- S: I will order salmon
- T: I will order tuna

Then he said he will go to the restaurant and order salmon or tuna, explaining that it is the same as I will go to a restaurant and order salmon or I will go to a restaurant and order tuna. Everyone in the room could easily understand this thought process. Then he used mathematics and the distributive law. He gave Then he explained that the mathematical equation was representing the exact thought process you were just having. Unfortunately, oftentimes this lesson is lost on students.

I cannot wait to check out dePillis’ books *777 Mathematical Conversation Starters* and *Illustrated Special Relativity Through Its Paradoxes*.

**Paul Erdős – one of the most influential mathematicians of our times: Krishnaswami Alladi – University of Florida**

This talk was sponsored by the UCF Math Club and was given at a level understandable to undergraduate mathematics (or related fields) majors. Dr. Alladi gave some general information about Erdős who, in my opinion, is one of the most interesting mathematicians of all time.

Alladi told of a hyperbole that I had never heard in which Erdős was on a train and by the end of the train ride, he had jointly authored a paper with the railway collector. Although this story, as Alladi pointed out, is likely untrue, it serves to demonstrate Erdős’ willingness to collaborate with multitudes of people.

I particularly enjoyed listening to Alladi’s story of how he met Erdős. Alladi had done some work on a problem and he was not sure of its important or if it was already known. He sent his work to Erdős and quickly got a response asking if they could meet at a conference in India. unfortunately, Alladi was not able to attend, but his father was attending the conference and went in his place. Erdős enjoyed the work so much he replanned his next trip so that he could travel to where Alladi lived to speak with him in person. This began his collaboration with Erdős.

One of my favorite quotes from the talk came from Alladi himself,

If you are comfortable with the convergence and divergence of a sequence, you have developed into a more mature man.

Have you had any really interesting speakers come to your university? What did they speak about and why did you enjoy it?

]]>Dr. Arthur Benjamin is both a professor of mathematics and a magician. He has combined his two loves to create a dynamic presentation called “Mathemagics,” suitable for all audiences, where he demonstrates and explains his secrets for performing rapid mental calculations faster than a calculator. He has presented his high energy talk for thousands of groups throughout the world.

MAM has listed some challenges from Dr. Benjamin. Check them out at http://www.mathaware.org/mam/2014/calendar/mentalmath.html.

How would you stand in a mental math competition against Dr. Benjamin?

]]>We use the Internet for many things, from reading news articles, to keeping in touch with friends on social media, to shopping from the comfort of our own homes. Many of these tasks involve sending sensitive personal information (such as credit card numbers and our home address) to complete strangers. We would like to keep this information safe, making sure no malicious third party is able to intercept our messages. RSA is a cryptosystem which is known as one of the first practicable public-key cryptosystems and is widely used for secure data transmission. RSA has stood the test of nearly 40 years of attacks, making it the algorithm of choice for encrypting Internet credit-card transactions, securing e-mail, and authenticating phone calls.

One of the distinguishing techniques employed in public-key cryptography is the use of asymmetric keys. In this scheme, one key (the public key) is used to encrypt the message while a different key (the private key) is used to decrypt it. The keys are related mathematically, but the parameters are chosen so that calculating the private key from the public key is either impossible or prohibitively expensive.

At MIT in the fall of 1976, computer scientists Ronald Rivest and Adi Shamir, along with number theorist Leonard Adleman devised the public-key encryption code that bears their initials (RSA encryption), and has been in use ever since to secure electronic transactions. The RSA algorithm was the culmination of many months of work, which all started when Rivest read a paper by Diffie and Hellman, proposing that a good public-key encryption scheme would need to be based on what they called a “trap-door one-way function.” This function would be easy to compute, but hard to invert unless you knew the secret (the “trap door”). Rivest and Shamir would come up with numerous number theoretic schemes to fit this “trap door” idea, and then Adleman would try to poke holes in it (usually succeeding after a few minutes’ thought). However, one evening Rivest called Adleman with a new idea, which Adleman agreed was a good one because it seemed that only factoring would break the algorithm. Rivest wrote up a paper about the new algorithm and sent a copy to Adleman. The authors were listed in the standard alphabetic order: Adleman, Rivest, and Shamir. Adleman objected to this ordering, however, because he stated that he had not done enough work to be listed first. Adleman consented to being listed as an author only if his name was put last, reflecting what he considered his minimal contribution. Adleman said later, “I remember thinking that this is probably the least interesting paper I will ever write and no one will read it and it will appear in some obscure journal.” The paper in question was published in Communications of the ACM and so the RSA (instead of the ARS) algorithm was born.

The RSA algorithm involves three steps: key generation, encryption, and decryption. RSA involves two keys – a public key and a private key. As the names suggest, anyone can be given information about the public key, whereas the private key must be kept secret. Anyone can use the public key to encrypt a message, but only someone with knowledge of the private key can hope to decrypt the message in a reasonable amount of time. So how are these keys generated? The power and security of the RSA cryptosystem is based on the fact that the factoring problem is “hard.” That is, it is believed that the full decryption of an RSA ciphertext is infeasible because no efficient algorithm currently exists for factoring large numbers. The keys for the RSA algorithm are generated as follows:

1. Choose two distinct prime numbers and . In order for the system to be secure, the integers and should be chosen at random and should be of similar bit-length. To find large primes, the numbers can be chosen at random and, using one of several fast probabilistic methods, we can test their primality.

2. Compute . The product will be used as the modulus for both the public and private keys. Its length, usually expressed in bits, is the length of the key.

3. Compute , where is Euler’s totient function.

4. Choose an integer such that and gcd (that is, and are coprime). The number is released as the public key exponent.

5. Determine as mod . That is, is the multiplicative inverse of (modulo ). This is often computed using the extended Euclidean algorithm. The number is kept as the private key exponent.

The public key is formed by the pair , where is called the modulus and is called the public (or encryption) exponent. The private key is formed by the pair , where is called the private (or decryption) exponent. It is imperative that the decryption exponent is kept secret. In addition, the numbers , and must also be kept private because they can be used to calculate .

Once the keys are determined, secure messages can now be sent. Suppose Bob would like to send Alice a message. Alice transmits her public key to Bob, keeping her private key secret. In order to send Alice an encrypted message, Bob first has to turn the message into an integer , such that . This is done by using a previously agreed-upon reversible protocol known as a padding scheme. Bob then computes the ciphertext corresponding to his message. The ciphertext can be found by computing (mod ). Bob then transmits the encoded message to Alice. In order to recover the message, Alice uses her private key, computing (mod ). Given , she can recover the original message by reversing the padding scheme.

We will now look at a small (and insecure) example. Alice would like to create a public and private key to use for her secure internet transactions. In order to create these keys, she chooses and . Next, she computes as well as the totient . In order to find the public key exponent, Alice must choose a number which is also coprime to 11200. For this example, Alice will choose = 3533. Finally, Alice must compute (mod ) = 6597. Alice publishes the public key pair , while keeping and private. Now suppose that Bob would like to send the message to Alice. Bob would compute

(mod ) = 5761,

which he would then send to Alice. After receiving the ciphertext , Alice can decode the message using her private key

(mod ) = 9726.

The RSA algorithm has remained a secure scheme for sending encrypted messages for almost 40 years, earning Rivest, Shamir, and Adleman the Association for Computing Machinery’s 2002 Alan Turing Award, among one of the highest honors in computer science. Currently, the only way to completely break the RSA cryptosystem in use today (which is slightly more sophisticated than that described here) is to factor the modulus . With the ability to recover the prime factors and , an attacker can compute the secret exponent from the public key . Once they have the secret exponent, the attacker can decrypt any message sent using the public key. What keeps RSA safe from such an attack is the fact that no polynomial-time algorithm for factoring large integers on a classical computer has been found yet. However, it also has not been proven that no such algorithm exists. As of 2010, the largest known number factored by a general-purpose factoring algorithm was 768 bits long, using a state-of-the-art distributed implementation. RSA keys are typically 1024 to 2048 bits long, though some experts believe that 1024-bit keys could be broken in the near future. It is generally believed that 4096-bit keys are unlikely to be broken in the foreseeable future, meaning that RSA should remain secure as long as is chosen to be sufficiently large. It is currently recommended that be at least 2048 bits long.

In 1994, Peter Shor showed that a quantum computer could be used to factor a number in polynomial time, thus effectively breaking RSA. Stay tuned for my next article where we will look at Shor’s algorithm in depth!

Sources:

Robinson, Sara. “Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders.” SIAM News, Volume 36, Number 5, June 2003.

R. L. Rivest, A. Shamir, and L. Adleman. 1978. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21, 2 (February 1978), 120-126. DOI=10.1145/359340.359342 http://doi.acm.org/10.1145/359340.359342

]]>

Wolfram Mathworld says,

Analysis is the systematic study of real and complex-valued continuous functions. Important subfields of analysis include calculus, differential equations, and functional analysis. The term is generally reserved for advanced topics which are not encountered in an introductory calculus sequence, although many ideas from those courses, such as derivatives, integrals, and series are studied in more detail. Real analysis and complex analysis are two broad subdivisions of analysis which deal with real-values and complex-valued functions, respectively.

Typically (at least in my experience), in your first year graduate analysis course, you will cover topics like the Real and Complex number systems, basic topology, numerical sequences and series, continuity, differentiation, Riemann-Stieltjes integral, and sequences and series of functions (I took this from the table of contents of Rudin).

Evelyn Lamb just wrote an article *On Teaching Analysis. *Her post discusses Timothy Gowers’s blog during his time teaching analysis at The University of Cambridge. For learning analysis (and anything in grad school really), it is important to really understand the material completely. To do this, you may need to take a look at more sources. Gowers’s blog is a place to start. You may also want to take a look at the following resources:

- Blogs
- Vicki Neale’s blog (colleague of Gowers)
- Terence Tao’s blog (great resource for MANY topics)

- Course Notes
- Paul Seidel’s Course Materials (lecture summaries, practice exams, homeworks, etc.)
- Interactive Real Analysis

- Textbooks

If you have already taken analysis, what resources did you find helpful? If you are currently taking analysis, what resources do you wish you have that you cannot find? Happy studying!

]]>Have you ever wanted to screencast your work? This is especially helpful when trying to teach someone long distance. You want to write things and talk them through it, but if they are not right there with you, it can be tricky to clearly capture everything. I ran across this neat iPad app called Educreations. Their homepage says: **Teach** what you know. **Learn** what you don’t. Create and share amazing video lessons with your iPad or browser.

I haven’t played around with it much, but what I have seen looks amazing. You can see all of their math videos at http://www.educreations.com/browse/mathematics/. I am a TA for calculus 1 this semester, so I found the calculus videos quite interesting. It is helpful to see how other folks explain concepts.

Share your favorite discoveries on Educreation with us. Have you found other software that you like for creating screencasts?

]]>Despite the lingering snow, March is here and Pi Day is only a week away! I love Pi Day because it combines three of my favorite things: math, puns, and dessert. Baking a pie is always a fun (and tasty!) way to celebrate—if you’re looking for a recipe, Buzzfeed has some really creative options, collected specifically for the holiday. It’s not specifically pi-related, but I also really enjoyed Vi Hart’s video about mathematical baking. She and her friends use shortbread cookies to build everything from tessellations to Sierpiński tetrahedra.

Another fun thing to check out is Numberphile’s collection of pi visualizations. You probably know pi to at least five digits, or possibly more, but the video showcases a variety of diagrams that allow conceptualization of much longer digit strings. From color-coded dots to spirals to web-like rainbow graphs, these diagrams are both elegant and educational. They also allow for interesting visual comparisons between pi and random digit strings, or pi and other irrational numbers such as the constants e and φ.

If you have any favorite Pi Day activities, feel free to share them in the comments!

]]>In the same section, he explains why each step should be taken. To me, the first step seems crucial and the third very important and practical. For the second, I still suspect that all the relationships may not be clearly established, not because of laziness from the student but from a rather superficial acquaintance with the problem, which may change over time. In fact, maybe this is why Polya urges that at least the “main connection” be established.

In the next section, he reiterates the importance of understanding the problem. Another point he makes is that the problem needs to interest the student not just the teacher. I wonder what Polya would think of teachers who readily conclude that students are lazy or retarded (or both) if they fail to quickly provide a solution to problems that may be of no interest to them. A way he suggests such understanding can be checked is for the student to be able to answer to “What is the unknown? What are the data? What is the condition?.”

In the eighth section, Polya presents an example to illustrate his method; it is a problem that asks to find the diagonal of a parallelepiped, given the length, the width, and the height of the solid. An answer to “What is the unknown?” is the length of the diagonal; for “What are the data?”, an answer is the dimensions of the solid, and the condition is that the diagonal and the dimensions are for the same solid. Then, he claims the condition is sufficient to find the unknown, which is true, but I suspect an answer about the sufficiency of a condition of a problem may not be automatic for some problems. Furthermore, Polya mentions two interesting points in this section: one is that some knowledge is necessary to solve some problems, and I suppose that knowledge could be acquired by reading mathematics (or maybe hearing); the other is that a *dialogue* needs to be taken place between the teacher and the students.

He presents more examples in the subsequent sections, and I may discuss some of them in the next article.

]]>With the Oscar’s comes talk of fashion. One of the highlights of the Oscar’s is watching the celebrities on the red carpet to see who is wearing who. Perhaps it is that she is from Kentucky (my home state) or perhaps it is my obsession with red (maybe a little of both), but I would go with Jennifer Lawrence as best dressed. The Daily News says in a slideshow,

Do you hear that? It’s a collective intake of breath as Jennifer Lawrence reminds us once more why she’s fashion’s favorite darling. The “American Hustle” star pulled out all the stops in a red peplum Dior Couture gown worthy of a Hollywood princess.

But is fashion just about aesthetics or is there perhaps math behind it?

I have seen quite a bit of chatter on Twitter lately of the mathematics behind ties. When researching the mathematics behind fashion, I came across a book by Thomas Fink and Yong Mao entitled *The 85 Ways to Tie a Tie: The Science and Aesthetics of Tie Knots*. There are loads of resources out there, though. I was surprised to even find a school for fashion that offered courses relating mathematics and fashion. The Fashion Institute of Technology in New York offers classes like Geometry and the Art of Design that has the following description,

A contemporary primer of geometric topics that expand the concepts of shape and space, this course presents some of the established and emerging ways geometry can provide tools and insights for artists and designers. Included are a variety of visual phenomena such as fractals, knots, mazes, symmetry, and the golden ratio.

It seems that many designers are even aware of the relation between mathematics and fashion. Coco Chanel said

Fashion is architecture: it is a matter of proportion.

Along these same lines,

“All of the great fashion couturiers used the Divine Proportion, perhaps by instinct,” Antonio Gonzalez de Cosio, fashion editor and author, told USA TODAY College in an email. Take for example, “Balenciaga with his architectonical designs [or] Dior creating his new look with very small waist lines in proportion to the body.” Couture is about clothes with “harmony of form and of balance.”

See Caroline Slattery’s article, *How Fashion is Actually Just a Bunch of Math *for more on the golden ratio in relation to design (Slattery’s article also mentions the above quotes).

Below is a list of some more resources regarding fashion and mathematics. Have you thought about the mathematics behind fashion before?

- http://www.ams.org/news/ams-news-releases/thurston-miyake
- http://www.maa.org/news/math-news/applying-math-and-science-to-fashion
- http://www.wolfram.com/mathematica/customer-stories/japanese-fashion-designer-eri-matsui-puts-mathematica-on-the-tokyo-catwalks.html
- http://www.hindawi.com/journals/mpe/si/823857/
- http://www.dailymail.co.uk/home/you/article-1366562/Fashion-life-Trend-problem-Do-maths.html
- http://plus.maths.org/content/career-interview-fashion-designer