This is our former Doxie Otto and me
This is my love Dana and our Brittany Cody
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This is our former Doxie Otto and me | This is my love Dana and our Brittany Cody |
As a graduate student at the University of Arizona, I've been an instructor of college courses with full teaching duties (creating curriculum, lecturing, grading, office hours), a mentor to newer graduate students (running problems sessions, organizing qualifying exam reviews, advising research projects), and a G-TEAMS fellow working in a high school classroom (developing creative projects, co-teaching lessons, after-school tutoring). Please see below (or in my CV) for a complete list of courses taught. These experiences have contributed in my personal growth and development of a number of effective techniques like (1) example-driven/project-based lesson plans, (2) cooperative learning, (3) student discovery, and (4) goal oriented curriculum creation.
In general, I've built up an arsenal of ideas to properly motivate and implement lesson plans in a way that is interesting, effective, and interactive. I love to learn and share what I've learned in a creative and personable fashion. This passion gives me a high level of enthusiasm while my experience ensures that my energy is well spent.
I really enjoy using and creating visual demonstrations of mathematical or scientific concepts. Animations or interactive worksheets are especially successful at not only relaying information, but also getting students focused and excited. Learning in a classroom environment can be fun and insightful when the instructor lets the pretty pictures speak.
The Formation of a Black Hole |
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If a massive star burns out but cannot support its matter against the force of gravity, a black hole forms. This means that the star collapses to a single point called the singularity. The singularity is a place of infinite gravitational forces and is surrounded by a boundary called the event horizon, inside of which light cannot escape. The animation shows an embedded spacetime diagram of a spherical star shrinking in radius. If the star keeps the same mass, then the geometry outside the star remains the same; what we are seeing is more of the curvature being revealed. In other words, the mass (not the radius) of the star determines the how spacetime is warped around it. We get a black hole when the star is so dense (think small and heavy) that the star can "fit" through its own ripple in the fabric of the universe. |
Click on the pic to see animation |
The Pentagonal Numbers |
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Consider the sequence: P1 = 1, P2 = 5, P3 = 12, P4 = 22, ..., and, in general, Pn = Pn - 1 + 3n - 2 for n ≥ 2. These are called pentagonal numbers because they count points in nested arrangements of pentagons as seen in the picture to the right. Likewise, there are triangular numbers T1 = 1, T2 = 3, T3 = 6, ..., and Tn = Tn - 1 + n, and square numbers S1 = 1, S2 = 4, S3 = 9, ..., and Sn = Sn - 1 + 2n - 1. These sequences count points in nested arrangmenets of triangles and squares. It is easy to notice (and prove algebraically) that a pentagonal number is, in fact, the sum of a triangular number and a square number: 5 = 1 + 4, 12 = 3 + 9, 22 = 6 + 16, ..., and Pn = Tn - 1 + Sn for n ≥ 2. The animation shows a geometric verification of this fact via rays from the corner vertex and stretching angles to deform a pentagon into a triangle and a square. Pentagonal numbers are related to the partition function p(n): if we extend to nonpositive indices by P0 = 0, P-1 = 2, P-2 = 7, P-3 = 15, ..., and then define f(x) = ··· + xP-2 - xP-1 + xP0 - xP1 + xP2 - ···, we have 1/f(x) = 1 + p(1)x1 + p(2)x2 + p(3)x3 + ··· . |
Click on the pic to see animation |
The Resolution of a Cusp |
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The image to the right is the graph of the curve y3 = x2 in the real plane. This curve has a "cusp" at the origin, and, in particular, does not have a well-defined tangent line there since the partial derivatives both vanish. One way to resolve this singularity is to view the curve as the shadow of a smooth curve in a higher dimensional space. The animation shows a rotating view of the twisted cubic. Here is another nice visualization and description of the resolution of a cusp via blowup from Donu Arapura at Purdue. |
Click on the pic to see animation |
Heron Triangles and Elliptic Curves |
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As explained below in my research section, triangles with some given fixed area and perimeter can be parameterized by an elliptic curve. In particular, since elliptic curves are equipped with a group law, we can add known triangles together to generate new ones. Moreover, rational points on the curve correspond to Heron triangles (i.e., triangles with rational side lengthes), so we can add Heron triangles together to get Heron triangles. The animation shows the family of triangles with area 6 and perimeter 12 and the associated points on the corrseponding elliptic curve; this family includes, for example, the (3, 4, 5) right triangle (which is roughly where the animation begins and ends). It was created from a Geogebra file provided to my advisor Bill McCallum by Tomas Reccio. |
Click on the pic to see animation |
In my first two semesters of graduate school at the University of Arizona (FALL 2006 and SPRING 2007), I taught Math 110 (College Algebra). In my third semester (FALL 2007), I taught Math 120 (Precalculus). In my fourth semester (SPRING 2008), I taught Math 113 (Elements of Calculus). Next (SUMMER 2008), I ran the 523A-B qual prep sessions. In my third year (FALL 2008 - SPRING 2009), I had a VIGRE fellowship which allowed me to finish all non-thesis requirements, during which time I was the super TA for the core graduate algebra sequence (MATH 511A-B). Next (SUMMER 2009), I taught Math 215 (Linear Algebra) and ran the 511A-B qual prep sessions. Here is my 215 course page. At the beginning of my fourth year (FALL 2009), I taught Math 105 (Math in Modern Society) and served again as the super TA for Math 511A. Here is my 105 course page. After that (SPRING 2010), I had a research assistantship, but I continued to be the super TA for Math 511B.
In the academic year (FALL 2010 - SPRING 2011), I was a G-TEAMS Fellow working in Rincon High School (co-teaching Stats, Calc, PreCalc) to bring creative and socially/environmentally responsible teaching tools into the classroom; the program as a whole was an effort to marry the passion of mathematicians with need for improving math education in the US on the K-12 levels. I kept a journal of classroom activities (including both successes and failures) via The Rincon Blog; I spent a lot of time and energy putting The Rincon Blog together, so please take a look and enjoy. It contains projects, links, media, and many interesting facts. Feel free to use the materials in your own classrooms as you see fit (I only ask that you give proper credit/references).
I was born in Knoxville, Tennessee in the summer of the year the World's Fair came to town. Being a clumsy and double crowned catastrophe of a child, I found solace in working Rubik's cubes and memorizing various nations' flags. I also played soccer; however, in the 2nd scrimmage of my 8th grade season, I was slide-tackled from behind, which action snapped both bones in my left leg directly above the ankle. I didn't play much soccer after that. I'm pretty sure that something happened in high school, but I refuse to remember it. [scene missing]. Next, it was on to washing dishes at a BBQ restaurant and then to stocking auto parts for a warehouse in the heart of K-town's 'Mechanicsville'. This is where I realized the need for college. I came from humble beginnings, and my parents sacrificed everything to ensure I had as much freedom and support as one could ever need. Also, I discovered the tantalizing Siren song of number theoretic conjectures (often so simple to state, yet so incredibly difficult to solve). Previously, I had planned on charging astronomical fees for mediocre, surrealistic artwork, but armed with a new love of puzzles I decided to pursue mathematics instead. I spent some time at a community college in the boondocks to get my feet wet and also found time to play in a rock band once described by a friend as "southern fried chaos". In the spring of 2006, I received my Bachelor of Science (summa cum laude) from the University of Tennessee. In the fall of that same year I came to the University of Arizona in Tucson to earn my PhD.
Without question, my interests in this deductive science lie squarely in the area of number theory. In particular, I've done research in algebraic number theory, Iwasawa theory, and elliptic curves. Naturally, I also have great inquisitiveness in other topics of number theory, algebra, and algebraic geometry, like continued fractions, K-theory, Dedekind schemes, function field arithmetic, and transcendental numbers. Moreover, I plan on doing research in the connections between number theory and phyics as well as continuing research in the mathematics of social justice (i.e., quantifying social and environmental issues related to the unequal distribution of resources: see The Rincon Blog for more details).
My PhD advisor William McCallum and I ran a research group for second year graduate students in the fall of 2011. Many aspects of this project could be adapted for undergraduates at various levels. The project revolves around the following observation: Heron triangles (i.e., triangles with rational sides) with the same area and perimeter not need be similar. For example, the (3, 4, 5) right triangle has area 6 and perimeter 12, while the (41/15, 101/21, 156/35) non-right triangle also has area 6 and perimeter 12. In fact, it turns out that we can generate infinitely many more such Heron triangles with area 6 and perimeter 12 by using the group law on an elliptic curve. In general, consider a triangle with side lengths a, b, c and incircle of radius r as in the figure below.
Then we have the angle sum (α + β + γ)/2 = π, so if x = tan(α/2), y = tan(β/2), z = tan(γ/2), then the perimeter P satisfies
P/(2r) = x + y + z = xyz
by a trigonometric identity. Fixing the perimeter P and area A = rP/2 means that k := P/(2r) is constant. Eliminating z and simplifying in the above equation yields
x2y + xy2 = k(xy - 1).
This equation defines an elliptic curve for relevant values of k, and students at various levels can analyze these curves in a variety of ways. This could serve as an introduction to the Method of Descent and Nagell-Lutz Theorem trying to find large ranks and compute torsion of these curves. For the students with background in complex analysis, one can study L-functions of these curves, or the j-invariant, which turns out to be the rational function of k
j = k2(k2 - 24)3/(k2 - 27).
In particular, different values of k can give isomorphic curves: is there a nice geometric interpretation in terms of dual triangles or symmetries which explains this? Students interested in algebraic number theory could investigate which of these curves have complex multiplication. Furthermore, this could serve as a wonderful opportunity to introduce students to aspects of mathematical software and programming; for instance, students could create animations and perform complicated arithmetic computations with computer software like Cinderella2, GeoGebra, SAGE or Maple.
Here is a link to my senior undergraduate thesis from UTK. It examines Lehmer's totient problem in PIDs by generalizing the notions of Euler's phi function and Carmichael numbers. I also had the occasion as an undergraduate to write some nifty code in Fortran. In particular, here is a link to my final project for a computer literacy course. This employs Miller's primality test (as translated from pseudo-code found in Crandall and Pomerance's Prime Numbers: A Computational Perspective), finding full reptend primes, and computing approximations to Aritn's constant (although to do the last, one would need to remove the exclamation point from the appropriate line).
I worked through many interesting homeworks sets at the University Arizona. A couple of my favorites include this assignmnet for group theory in which I wrote some GAP code to search through groups of small order and check to see if their commutator subgroups consisted entirely of commutators. Of course, there are finite groups whose sets of commutators are not equal to their derived subgroups; the smallest such group has order 96. A year later, Victor Piercey and myself were two of the only students in our homological algebra course to complete this technical homework problem set from Hilton and Stammbach's text. It was a great test of diagram chasing and xy-mtrix skills.
I wrote a project for the 2007 integration workshop (the Arizona Math Department's "welcome to the neighborhood") on periodic continued fractions. It covers the basic properties of convergents and leads the reader through the proof of the characterization of the continued fraction expansions of quadratic irrationals. I also wrote a project for the 2008 integration workshop on quadratic reciprocity. It outlines Eisenstein's beautiful proof for this classical theorem of Gauss.
I first spoke at the department's ANTS in the fall of 2007. Here is a link to the slides on Artin L-functions from that talk. The main references used included a group theory text by Arizona's own late Larry Grove and an article by Heilbronn. I also spoke at ANTS in the fall of 2008; here are the slides on a Riemann-Hurwitz formula for number fields from that talk. In the fall of 2009, I gave another (two-part) talk at ANTS on the Iwasawa theory of elliptic curves and an application to the rank zero case of the Birch and Swinnerton-Dyer conjecture; here is a link to the slides which are primarily based off of Ralph Greenberg's work on the subject. I spoke at ANTS yet again in the spring of 2010 on arithmetic dynamics; here are the slides. The slides are ;arge;y dranw from the first three chapters in Joseph Silverman's The Arithmetic of Dynmaical Systems. I gave another ANTS talk in February 2011 on the Brumer-Stark Conjecture, but I did not make slides for that occasion.
Here is my RTG paper on Bernoulli numbers and Fermat's Last Theorem. The paper begins with elementary results, congruences, and the von Staudt-Clausen theorem. Then a glance at functional values of Dirichlet series is given. Finally, an outline of the proofs of Herbrand's theorem and Ribet's converse are included. In the fall of 2011, my advisor Bill McCallum and I worked with some second year students on their RTG project on the family of elliptic curves which parametrize Heronian triangles having the same perimeter and area.
This is my final project for homological algebra. This beast was done with three other co-authors and contains an introduction to sheaves and sheaf cohomology. I wrote the chapter on Cech cohomology. Here is a link to my final project for commutative algebra on K-theory. I only discuss the lower algebraic K-groups, but it should serve as a nice introduction to these functors inasmuch as the paper condenses the first few chapters of Milnor. Here is a link to a term paper on Hilbert functions done for the first part of a two semester sequence in algebraic geometry taught by Douglas Ulmer.
I spent the 08-09 academic year finishing all of my non-thesis requirements. For one of my distribution credits, I took General Relativity; here is a link to a paper on the Møller energy of wormholes I wrote for that course. For my other "outside the dept" excursion, I took Game Theory; here is a link to the poster I made for that course. I studied algebraic number theory, Iwasawa theory, homological algebra, and algebraic geometry, for my comprehensive exam. The written component of my comp exam (linked here as a pdf) investigated Hurwitz-type formuls for number fields, namely Kida's formula for CM-fields and Iwasawa's formula for Zp-fields.
Here is my CV, a comprehensive academic résumé. The CV includes lists of my grants and awards, talks given, conferences attended, and education history.
My love affair with art and music is deep and life-long. I had shown some talent in mathematics through K-12 schooling, but I was certain that I would spend my days doing something creative. Little did I know growing up that math was absolutely compatible with that...
In 2010, I attended the annual Daniel Bartlett Memorial Lecture with my office mate Victor. The talk combined two of my most favorite things: mathematics and surrealism. Tom Banchoff of Brown University was speaking about his correspondence with the eccentric artist Salvador Dali.
Banchoff and Dali
Both men were interested in visually representing four dimensional objects. By this I mean a fourth spatial dimension, not a dimension for time. One work of art mentioned in that evening dealt with precisely that. Seen below, this painting depicts a four dimensional "hypercube" unfolded in three dimensions.
"Corpus Hypercubicus" (1954) |
Folding a cross (2D) into a cube (3D) 
Folding a hypercross (3D) into a hypercube (4D) |
Victor and I were particularly taken with the painting since we both simultaneously realized that this "hypercross" arrangment of cubes was, in fact, the solution to a problem in algebraic topology that we were given in our of first year of graduate school. Let me explain. We were assigned the following standard exercise from Hatcher:
The problem itself is not so difficult. One builds up a 1-skeleton as a corner of a cube and then attaches faces (two-cells) according to the 1-quarter twists. Assuming the attaching was done carefully, one then uses a corollary of van Kampen to get a description of the fundamental group with generators and relations. The group is recognized to be the quaternion group Q8 and everyone's happy. However, we were also told to identify all the path connected covers of this space. D'oh! Since the size of the fundamental group was eight, we knew that the universal cover had to have eight sheets. This means we had to stack together eight cubes and wrap them up in a nice way. At the time, we didn't know which arrangements of cubes would give us the right cover and which would give us nonsense. As it turns out, there are exactly 261 ways of stacking the cubes correctly, and the most symmetric one seems to be Dali's hypercross.
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| Fig. 1: Fellow with protruding ocular tube and a relief map hat. | Fig. 2: A bust of Euclid laced with a Penrose tiling. |
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| Fig. 3: die Guitarre | Fig. 4: Joe crashing the hi-hat |
