Welcome to Jonathan Hanke's Webpage

Welcome

Papers

Talks

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Software

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Personal

Thanks for stopping by my webpage. For a more personalized welcome, please select the options below that most interest you. Have fun! =)

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• My research interests are in better understanding the relationship between classical arithmetic (counting) problems in number theory, and features in the land of automorphic forms.
• Recently I have been working on the connection between understanding which numbers are represented by a quadratic form, and modular forms (via their theta functions). I have been particularly interested in quadratic forms in 3 and 4 variables, which correspond to modular forms of weights 3/2 and 2 (respectively).
• My major accomplishments along these lines are:
  • Explicit descriptions of the local densities of a quadratic form (representing a number) at all primes.
  • A global proof of the representation behavior of ternary quadratic forms by using the Shimura lift (to weight 2) and analyzing the associated Galois representations.
  • Computer software which efficiently computes all numbers represented by an integer-valued quaternary quadratic form. (This problem was previously unsolved for many "small" forms, e.g. x²+3y²+5z²+7w².)
  • A proof of the 290-Theorem (with M. Bhargava).
• To learn more about what I am interested in, please check out the professional documents below, as well as my Papers and Notes.

• If you are interested in learning about some aspect of Number Theory, or have questions relating to Algebra/Linear Algebra for your qualifying exams, feel free to stop by and ask about it.
• If you are curious about a topic you think I might know about, and think others might be interested too, feel free to contact me about giving a Graduate/Faculty Seminar Talk about it. In Fall 2005, I gave a series of two introductory talks on p-adic numbers entitled "The p-adic way of life", and another in Fall 2006 on "What numbers are a sum of 4 squares?"
• For writing mathematical papers, I use LaTeX through the friendly editor Kile.
• There is some useful math/computer reference information and macros available in the Useful Resources section. (My favorites are the Emacs/TeX/LaTeX reference information at the bottom, and the Math Geneology Project at the top.)

• If you are taking a course that I have taught, you might be interested in looking at past exams and review sheets for them, posted on my courses webpage.
• I like to teach, and really like to help people understand the idea behind the concepts we discuss. Math doesn't just fall from the sky -- it usually comes from an attempt to answer some very natural question. Once we can understand these motivating questions, their answers make a lot more sense.
• Because I try to get students to ask questions for themselves, my classes tend to be pretty challenging. Some of the most important questions I try to ask (and answer) are:
  • What's the main idea?
  • What is it good for?
  • What do you expect?
  • How do we know that?
• When approaching a problem, it's important to have some expectation or intuition about what's going on. Your intuition may not always be correct, but it's a very good place to start. Here the main questions you need to ask are always:
  • What do we know?
  • What do we want?
• My main (research) interest is in number theory, which basically means I really like whole numbers and prime numbers. Some good examples of questions in number theory are:
  • How many Pythagorean triples are there, and can we describe them?
  • How many primes are there, and how often do they occur in the number line?
  • Which numbers are a sum of 2 square numbers?
  Any homogeneous polynomial where every term has exactly 2 variables in it (i.e. degree 2) is called a quadratic form. A good example of this is the form corresponding to a sum of 4 squares: x²+y²+z²+w². I am interested in the values that come out of (i.e. are represented by) quadratic forms when we put in whole numbers.
• If you are interested in doing an independent study or project with me, you might like to look at this list of "final projects" from my number theory class. Also, I have had several independent study students, one of whom wrote a senior thesis with me. They are:
  • Mayank Varia -- Readings in Modular forms, Elliptic Curves, and Algebraic Geometry (Fall '04, Spring '05)
    His senior thesis is: Explicit computation of the L-function of a Kummer Surface
  • Mandy Frese -- Readings in Cryptography (Fall '04)
    Discussed the mathematics and implementation of several public-key protocols.
  • Mandy Frese -- Readings in Cryptography (Spring '06)
    Discussed the mathematics and implementation of Velu's formula for computing isogenies between elliptic curves, and its application to a hash funtion on the graph of supersingular curves. The associated SAGE source code is here. (Project suggested by Kristen Lauter.)
  • Andrew Lang -- The Prime Number Theorem (Spring '06)
    Discussed properties of the Riemann Zeta function, and its applications in number theory (the Prime Number Theorem, Special Values via Bernoulli Numbers, and Dirichlet's Theorem).
  • Catherine Law and Leah Yates -- Topics in Combinatorics (Summer '06)
    Reading course in Combinatorics, discussing binomial coefficients, generating functions, and graph theory.

• I like sharing, and try to make the as much of what I do available to anyone interested in mathematics. I have given several talks about quadratic forms, and written the following expository articles about quadratic forms.
• If you like computers and want to play with some math for yourself, you can try out the freely available SAGE computer algebra package together with some of the ideas here.

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• The (C++) source code for the SAGE quadratic forms routines are available in the "Software" section, as well as the SAGE website maintained by William Stein. These C++ routines are them wrapped to make them callable in Python.
• For some useful C++/Python references, as well as some other useful mathematics software of a similar style, check out the "Useful Resources" page.
• DejaVu file format and viewer -- script for converting PDF -> DJVU. The source for the DJVU viewer can be found here.

• You like my webpage, and maybe want to learn how to make yours do similar things. You might want to look at the CSS and GraphViz sections of the "Useful Resources" page for this kind of information.
• Cool things about this page:
  • CSS -- See the zen page and other tutorials to do the same...
  • Graphs -- These were made with GraphViz -- see the dot documentation to do it for yourself! =) They show the logical dependency/evolution of my papers, and give a reasonable idea of what I'm thinking about and how I work.
  • Client side image maps -- Used to make a clickable graph of my papers.
  • Javascript -- Handles some of the hilighting behavior and and makes various things appear/disappear. I found this book really helpful for learning how to use scripts to make neat things happen.

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Professional Information

Cover Sheet/Letters [AMS, Research, Teaching]	Curriculum Vitae [PS, DVI, PDF]	Research Summary [PS, DVI, PDF]
Research Statement [PS, DVI, PDF]	Teaching Statement [PS, DVI, PDF]	Publication List [PS, DVI, PDF]

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This page is still under active development, so if you have any comments about the site, please feel free to e-mail me or to leave a message in the guestbook.