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Research Opportunities for Undergraduates ### Hyperbolic geometry and Riemann surfaces, and Systolic geometry

Mentor: Bjoern Muetzel

If you are an undergraduate interested in a reading course, independent study or working on a research project, feel free to contact me. I am particularly interested in the following topics.

The hyperbolic plane is a space of constant negative curvature minus one, where different rules than in Euclidean space apply for geodesics, the geometry of polygons and the area of disks. A hyperbolic surface can be seen as a polygon in the hyperbolic plane with identified sides. We call such a surface a Riemann surface. Many questions about Riemann surfaces are still open or under study. Hyperbolic geometry is used in the theory of special relativity, particularly Minkowski spacetime.

A systole of a surface is a shortest non-contractible loop on a surface. Every surface has a genus $$g$$, where informally $$g$$ denotes the number of holes. Surprisingly given any surface of fixed genus $$g$$ and area one, the systole can not take a value larger than $$c \cdot \frac{\log(g)}{ \sqrt{g}}$$, where $$c$$ is a constant. A large number of families of short curves on surfaces satisfy this upper bound and example surfaces can be found among the hyperbolic Riemann surfaces.

### Research in Algebraic Combinatorics

I have a number of projects accessible to undergraduate students in Combinatorics, Algebra and Graph Theory. These projects can lead to a senior thesis for honors or high honors. The ideal student should have taken math 24 and preferably (although not required) Math 28, 31 (71), 38 and have some programming skills. For more details schedule an appointment.

### Explicit methods in number theory

Classical unsolved problems often serve as the genesis for the formulation of a rich and unified mathematical fabric. Diophantus of Alexandria first sought solutions to algebraic equations in integers almost two thousand years ago. For instance, he stated that if a two numbers can be written as the sum of squares, then their product is also a sum of two squares: since 5=2^2+1^2 and 13=3^2+2^2, then also 13*5=65 can be written as the sum of two squares, indeed 65=8^2+1^2. Equations in which only integer solutions are sought are now called Diophantine equations in his honor.

Diophantine equations may seem perfectly innocuous, but in fact within them can be found the deep and wonderously complex universe of number theory. Pierre de Fermat, a seventeenth century French lawyer and mathematician, famously wrote in his copy of Diophantus’ treatise “Arithmetica” that “it is impossible to separate a power higher than two into two like powers”, i.e., if n>2 then the equation x^n+y^n=z^n has no solution in integers x,y,z>= 1; provocatively, that he had “discovered a truly marvelous proof of this, which this margin is too narrow to contain.” This deceptively simple mathematical statement known as “Fermat’s last ‘theorem’” remained without proof until the pioneering work of Andrew Wiles, who in 1995 (building on the work of many others) used the full machinery of modern algebra to exhibit a complete proof. Over 300 years, attempts to prove there are no solutions to this innocent equation gave birth to some of the great riches of modern number theory.

Even before the work of Wiles, mathematicians recognized that geometric properties often govern the behavior of arithmetic objects. For example, Diophantus may have asked if there is a cube which is one more than a square, i.e., is there a solution in integers x,y to the equation E : x^3-y^2=1? This equation describes a curve in the plane called an elliptic curve, and a property of elliptic curves known as modularity was the central point in Wiles’s proof. One sees visibly the solution (x,y)=(1,0) to the equation—but are there any others? What happens if 1 is replaced by 2 or another number?

Computational tools provide a means to test conjectures and can sometimes furnish partial solutions; for example, one can check in a fraction of a second on a desktop computer that there is no integral point on E other than (1,0) with the coordinates x,y at most a million. Although this experiment does not furnish a proof, it is strongly suggestive. (Indeed, one can prove there are no other solutions following an argument of Leonhard Euler.) At the same time, theoretical advances fuel dramatic improvements in computation, allowing us to probe further into the Diophantine realm.

My research falls into this area of computational arithmetic geometry: I am concerned with algorithmic aspects of the problem of finding rational and integral solutions to polynomial equations, and I investigate the arithmetic of moduli spaces and elliptic curves. My work blends number theory with the explicit methods in algebra, analysis, and geometry in the exciting context of modern computation. This research is primarily theoretical, but it has potential applications in the areas of cryptography and coding theory. The foundation of modern cryptography relies upon the apparent difficulty of certain computational problems in number theory, in particular, the factorization of integers (in RSA) or the discrete logarithm problem (in elliptic curve cryptography).

I have several problems in the area of computational and explicit methods in number theory suitable for experimentation and possible resolution by motivated students. These problems can be tailored to the student based on interests, background, and personality, so there is little need to present the details here; but they all will feature a explicit mathematical approach and, very likely, some computational aspects. Mathematical maturity and curiosity is essential; some background (at the level of MATH 71) is desirable.

[Past projects]