Dr. M. Brad Henry
Assistant Professor -- Siena College -- Mathematics
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Talks
Undergraduates
"The Infinitude of the Primes: The beauty of many proofs," Siena College Math Colloquium, (2012).- Abstract
- Mathematicians derive immense pleasure from discovering elegant proofs of existing theorems. We are, after all, artists of reason. During this talk, we will investigate as many different proofs of the infinitude of primes as time allows. We will begin with Euclid's beautifully clever proof involving a few basic facts from arithmetic. From there we will venture into proofs involving counting arguments, calculus, infinite series, and group theory. Each proof is truly elegant and each gives us new insight into the complexity of the prime numbers.
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- There exist as many proofs of the Pythagorean Theorem as there do courses in the average math department. But how many of these proofs offer a unique perspective on the theorem and how many are repackaged forms of a single, fundamental idea? By creating and refining the web of connections between mathematical ideas, researchers are able to exploit the web to solve new problems. Today we go hunting for connections between two important ideas in the study of Legendrian knots.
A Legendrian knot in 3-dimensional Euclidean space is a smooth knot obeying a set of extra geometric conditions that limit our ability to stretch and move the knot. The extra conditions refine smooth knot theory so that, for example, there exist infinitely many Legendrian unknots that can not be deformed into each other without breaking our geometric conditions.
We will define two differential graded algebras central to the study of Legendrian knots. One is geometrically motivated by classical Morse Theory and the other by the more modern Floer Theory. On the surface, the foundations of these invariants appear very different. However, recent results suggest deep connections exist between these two algebras. This work is joint with Dan Rutherford.
- There exist as many proofs of the Pythagorean Theorem as there do courses in the average math department. But how many of these proofs offer a unique perspective on the theorem and how many are repackaged forms of a single, fundamental idea? By creating and refining the web of connections between mathematical ideas, researchers are able to exploit the web to solve new problems. Today we go hunting for connections between two important ideas in the study of Legendrian knots.
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What careers await me if I leave Siena with a math degree?
What have other students done with their math degrees?
What can I do during my time at Siena to put me in a good position to find a rewarding career?"
Who can I contact at Siena if I need advice?
This colloquium will help answer all these questions and hopefully shed some light on the ubiquity of mathematics in the working world. If you are interested or enrolled in a math degree program and curious of the career opportunities waiting beyond Siena, then come find out what's out there for you.
John O'Neill will speak on careers related to the actuarial sciences. Jim Matthews will focus on mathematics education. Nik Krylov and senior Lindsay Kulzer will talk about their experiences doing summer research at Siena. Jennifer Heptig will describe the services and programs provided by the Career Center to help with internships, graduate school, and job hunting. Jon Bannon will offer insight into math graduate school and the life of the research mathematician. Finally, Brad Henry will present a variety of online resources that offer a glimpse into the innumerable careers held by people with math degrees.
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What careers await me if I leave Siena with a math degree?
- Abstract
- For as long as humans have held secrets, some have sought to ensure their safety, while others have worked to discover their treasures. The yin and yang relationship between code makers (those who develop methods to secure our secrets) and code breakers (those who seek to undermine those methods) has been a constant catalyst for truly ingenious ideas. We will walk the path of history, revisiting the innovative code makers since the time of the pharaohs and their tenacious code breaking counter-parts. Along the way we will explore the Roman code makers that preserved the strength of Caesar's army, the English code breakers that helped bring down the Third Reich, and the Native American code makers who flummoxed the Japanese army. Modern computational power has given mathematics a prominent position in today's secrecy industry, but the goals are the same as they ever were: protect what's yours, get what's theirs.
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- In the 19th century, it was postulated that very tiny knotted circles formed fundamental building blocks within the structure of the universe. The discovery of the atom would eventually pull many scientists away from such knots, but mathematicians have persisted with their attempts to understand these beautiful and complex objects. In the last 25 years, a new type of knot has become popular. These are called Legendrian knots and they also originated as an attempt to understand our physical world.
The track of a roller coaster can be thought of as a knotted circle. Riding the coaster gives you a close-up view of the complexity of the knot. A Legendrian knot is a roller coaster where the cars may move in two additional directions. The cars may spin like the cups on the Tea Cup ride and twist around the track as they move forward. The resulting three motions (forward/backward, spinning, and twisting) create a ride to test even the strongest stomach. In this talk, we will explore knots and roller coasters and, with any luck, avoid giving ourselves vertigo.
- In the 19th century, it was postulated that very tiny knotted circles formed fundamental building blocks within the structure of the universe. The discovery of the atom would eventually pull many scientists away from such knots, but mathematicians have persisted with their attempts to understand these beautiful and complex objects. In the last 25 years, a new type of knot has become popular. These are called Legendrian knots and they also originated as an attempt to understand our physical world.
- Abstract
- Mathematicians derive immense pleasure from discovering elegant proofs of existing theorems. We are, after all, artists of reason. During this talk, we will investigate as many different proofs of the infinitude of primes as time allows. We will begin with Euclid's beautifully clever proof involving a few basic facts from arithmetic. From there we will venture into proofs involving calculus, infinite series, and topology. Each proof is truly elegant and each gives us new insight into the complexity of the prime numbers.
- Abstract
- Knot theory began as an attempt to understand the structure of the universe and dates back to the late 1800s. Though its origins are in the physical world, knot theory has been an active area of mathematical research for over a century. In the last 25 years, a new type of knot has become popular. These are called Legendrian knots and they also originated as an attempt to understand our physical world.
In this talk we will explore both regular knot theory and Legendrian knot theory. As we will see, both theories seem tantalizing easy and yet both have complexities that have baffled even the best mathematicians
- Knot theory began as an attempt to understand the structure of the universe and dates back to the late 1800s. Though its origins are in the physical world, knot theory has been an active area of mathematical research for over a century. In the last 25 years, a new type of knot has become popular. These are called Legendrian knots and they also originated as an attempt to understand our physical world.
- Abstract
- The concepts of "shape" and "dimension" have fascinated scientists, mathematicians, and daydreamers for millennia. In the past, seafarers looked at the ocean horizon and wondered what lay beyond. Today, stargazers dream of a spacecraft that will allow them to explore the depths of the universe. And all the while, mathematicians build sophisticated theories to try to explain our physical world without ever leaving their desks.
In this talk we will create and investigate shapes of dimension 1, 2, and 3 (and possibly 4!). We will explore Abbott's famous Flatland and consider the possible shape of our universe. Along the way we will come to understand the fundamental questions in the fields of Knot Theory and Low-dimensional Topology.
As a warm-up exercise to get your imagination going, consider the following:
Suppose you had a spaceship that could travel much faster than the speed of light and suppose you left Earth and flew towards a very distant star. Would you be surprised if you arrived at that star only to find that it was the Sun and you were back at Earth? What would this suggest about the shape of the universe?
- The concepts of "shape" and "dimension" have fascinated scientists, mathematicians, and daydreamers for millennia. In the past, seafarers looked at the ocean horizon and wondered what lay beyond. Today, stargazers dream of a spacecraft that will allow them to explore the depths of the universe. And all the while, mathematicians build sophisticated theories to try to explain our physical world without ever leaving their desks.
- Abstract
- Suppose the manager of a museum wants to make sure that at all times every point of the museum is watched by a guard. The guards are stationed at fixed posts, but they are able to turn around. How many guards are needed? We will see how a single clever idea can take this possibly difficult problem and make it easy. We will also give an introduction to "mathematical induction," which will help us turn the clever idea into an elegant proof.
- Abstract
- This presentation was the culmination of a semester of undergraduate research under the direction of Prof. Thomas Bengtson at Augustana College. The research was done as part of the Earl Beling Scholars program at Augustana.
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- The Illustrated Analyst was a project I completed while participating in the VIGRE REU in Geometric Visualization at the University of Illinois at Urbana/Champaign in the summer of 2002. The program was directed by Prof. George Francis and my mentor was Prof. Karen Shuman. The Illustrated Analyst is a piece of software that provides 3-dimensional, graphical information on a variety of functions, Fourier transforms and convolutions.
© 2011 Michael Brad Henry - mbhenry@siena.edu
Original design by Andreas Viklund