# Jeremy Booher

I'm a second year mathematics graduate students at Stanford University. Before this, I completed my undergraduate studies in mathematics at Harvard University and spent a year doing Part III of Cambridge's Mathematical Tripos.

My email is jbooher [AT] math.stanford.edu

My office is 381-H in the math department (building 380).

## Seminars

Peter Hintz, Li-Cheng Tsai and I are organizing the KIDDIE Colloquium this year.

## Expository Notes and Articles

These include my senior thesis, Part III essay, and notes for many of the talks I have given.

Some of these come from the summers I spent as a counselor at PROMYS, and so use non-standard notation: taking a quotient of a ring by a principal ideal is denoted by a subscript. In particular, Zp is the integers modulo p, not the p-adic integers. Furthermore, the group of units in Zp is denoted by Up.

• The Brauer Group: Talk 1, Talk 2 (2013) Notes for CRAG talks about the Brauer group.
• Reciprocity Laws (2013) Notes for a KIDDIE talk about quadratic reciprocity and generalizations. Links the proof using Gauss sums with the standard algebraic number theory proof using the Frobenius, and then discusses cubic reciprocity, Eisenstein reciprocity, and the connection of quadratic reciprocity with class field theory.
• The Mordell-Weil Theorem for Elliptic Curves (2012) A proof of the Mordell-Weil theorem and an example of complete 2-descent, inspired by Tim Dokchitser's proof given in the Part III Elliptic Curves Course.
• Constructions with Fractional Ideals (2012) Notes for a talk in the 2011-2012 number theory learning seminar leading up to the proof of the Main Theorem of Complex Multiplication for Abelian varieties.
• Transcendental Numbers (2011) Notes for a KIDDIE talk explaining the techniques, going back to Hermite, used to show numbers like e are transcendental, and how a reasonable mathematician would discover them.
• Cubic Reciprocity (2011) Notes for a PROMYS talk explaining cubic reciprocity, which is the mathematics behind the picture on the 2011 PROMYS t-shirt. It gives the standard proof using Gauss and Jacobi sums assuming only the most elementary number theory.
• Representations of the Symmetric Group through Young Tableau (2011) Notes for a PROMYS counselor seminar about representation theory. It illustrates all of the general theory previously discussed in the case of the symmetric group, constructing all the irreducible representations of the symmetric group, showing how to combinatorially evaluate their characters and how to induce and restrict representations.
• Constructing the Integers: Ordinal Numbers and Transfinite Arithmetic (2011) Notes for a PROMYS talk explaining how to use set theory to construct a model for the integers, and at the same time introducing the more general arithmetic of ordinal numbers.
• The Limits of Computation (2011) Notes for a PROMYS talk explaining the limits of computation with deterministic finite automata and with Turing machines.
• Elementary Problems in Number Theory (2011) A collection of number theory problems I've given to students or heard from other counselors at the PROMYS program. They are of wildly varying difficulty.
• The Class Number One Problem for Imaginary Quadratic Fields (2011) Part III essay proving there are 9 imaginary quadratic fields with class number one. It gives two approaches, one following Heegner's original proof using modular functions, the second by finding rational points on a modular curve of level 24, and finally explaining a comment of Serre that these are essentially the same argument.
• Continued Fractions (2010) An explanation of the elementary theory of continued fractions, plus connections to Pell's equation and real quadratic fields along with some less common applications. This grew out of my notes for a review of continued fractions at PROMYS.
• The Isoperimetric Inequality (2010) Notes for a PROMYS talk proving the isoperimetric inequality, that of all closed curves of a given length, the one enclosing the largest area is the circle. There are two proofs given, the first uses the concept of action and is due to Hurwitz, the second uses Minkowski sums.
• Intersection Theory in Algebraic Geometry (2010) Notes for a PROMYS talk introducing basic notions in algebraic geometry and intersection theory. It presents the calculations on the Grassmanian of lines in three space to show there are two lines intersecting four general lines in projective three space, and ten lines in projective three space secant to each of two general twisted cubic curves.
• The Spirit of Moonshine: Connections Between the Mathieu Groups and Modular Forms (2010) Harvard undergraduate senior thesis, advised by Benedict Gross. An explanation of the connection between cycle shapes of elements of the Mathieu group M24 and modular forms that are products of the Dedekind eta function and for which the coefficients of the q-expansion are multiplicative. It also constructs an infinite dimensional graded virtual representation that in some sense explains this connection, analogously with the moonshine module connected with the monster group.
• The Circle Method, j Function, and Partitions (2010) Final paper for Harvard's Math 229, analytic number theory. An explanation of Rademacher's application of the circle method to obtain explicit formula for the q-expansion of the Dedekind eta function and the j function.
• Evaluation of Cubic Twisted Kloosterman Sheaf Sums REU project with Anastassia Etropolski and Amanda Hittson. In International Journal of Number Theory, 6 (2010), pages 1349-1365.
• The PoincarĂ©-Birkhoff-Witt Theorem (2009) Notes for a talk from Harvard's Math 222, Lie groups and Lie algebras. A standard and unenlightening proof of the PoincarĂ©-Birkhoff-Witt theorem.
• Number Theory in Cryptography (2008) Notes for a PROMYS talk about cryptography, in particular Rabin Encryption, Paillier Encryption, secret sharing, and zero knowledge proofs.
• Computability (2008) Notes for a PROMYS talk about Turing machines, the halting problem, and the arithmetic hierarchy.