statement o purpose

As an undergraduate I worked for my advisor's research group which
focused on designing magnetic crystal lattices, predicting their magnetic behavior,
and measuring their correlation to experimental data. The bulk of my work
consisted of synthesizing crystals, obtaining a suitable sample, recording its
magnetic moment inside of a magnetometer, and analyzing its fit to a curve derived
from the relevant model. When necessary, I obtained these curves by running
simulations [1] of finite rectangular spin lattices and fitting a rational function to the
generated data.
      While working in the laboratory I became interested in the two-dimensional
Ising model [2].  In the case of zero external field the model's partition function and
spontaneous magnetization can be solved using algebraic [3] or combinatorial [4]
methods, which required me to study modern algebra [5] and representation theory
[6], as well as graph theory [7] and path integrals [8].  The algebraic approach is a
very long and direct calculation, and in a sense relies on the symmetries of a 2L by 2L
transfer matrix, where each element (i,j) consists of terms of the partition function
contributed by two adjacent rows with spin configuration i and j. We are able to
generate a Clifford algebra using direct products of Pauli matrices which we use to
represent the transfer matrix. Many of the steps remain in the realm of linear
algebra, but an overview here would be impossible due to the sheer length of the
solution [9]. Conceptually the combinatorial approach is more elegant and recasts
the partition function as a sum over closed graphs of bonds between spins on the
lattice. Bonds must be properly weighted so that non-closed graphs and duplicates
do not contribute to the sum, and this weight is given by a nontrivial topological
theorem [10]. A Fourier transform of the path amplitudes Un, Dn, Ln, Rn reduces our
infinite sum to a geometric series in powers of a 4 by 4 matrix [8].
      The deep relationship between statistical physics and quantum field theory is
well known. The path integral formulation of QFT allows us to write correlation
functions and the partition function of a system of particles as a sum of amplitudes
(involving products of Green's functions of the Lagrangian) over Feynman diagrams
[7], which has been a useful method in many areas of theoretical physics. While the
Ising problem is very different, the combinatorial approach similarly reduces the
evaluation of large matrices to counting over graphs.
      The applicability of the Ising model relies on the fact that its spontaneous
magnetization emerges by a continuous phase transition, where the system is scale
invariant and the magnetization curve of the system is described by a small number
of parameters defined as critical exponents [11]. What is surprising is that these
exponents describe a universality class to which very different systems can belong.
The phase transition between a liquid and gas, for example, is in the same class as
the 3d Ising model. This is a very active area of research and recently there has been
much progress establishing a relationship between conformal field theories and the
parameters of continuous phase transitions [12]. I find the prominence of CFT in
modern physics fascinating, both in particle physics and statistical mechanics, and I
would like to actively pursue it as a student.  Academically, I am barely at a stepping
off point and I look forward to being challenged, expanding my knowledge of
physics and mathematics, and hopefully adding a significant contribution to the field
in the future.

References
[1]   S. Trebst, ALPS Documentation Wiki, Retrieved from The Algorithms and
Libraries for Physics Simulations Project: http://alps.comp-
phys.org/mediawiki/index.php/Main_Page, 2008
[2]   L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-
Disorder Transition, Physical Review, 65 (1944), 117-149.
[3]   B. Kaufman, Crystal Statistics. II. Partition Function Evaluated by Spinor
Analysis, Physical Review, 76 (1949), 1232-1243.
[4]   M. Kac, J.C. Ward, A Combinatorial Solution of the Two-Dimensional Ising
Model, Physical Review, 88 (1952), 1332-1337.
[5]   P. Girard, Quaternions, Clifford Algebras and Relativistic Physics, Birkhäuser
Verlag AG, Basel, 2007.
[6]   R. J. Baxter, Algebraic Reduction of the Ising Model, Journal of Statistical Physics,
132 (2008), 959-982.
[7]   P. Etingof, 18.238 Geometry and Quantum Field Theory, Fall 2002, Retrieved
from MIT OpenCourseWare: http://ocw.mit.edu
[8]   R. P. Feynman, Statistical Mechanics: A Set of Lectures, W. A. Benjamin, Inc.,
Reading, 1972.
[9]   R. J. Baxter, Onsager and Kaufman's Calculation of the Spontaneous
Magnetization of the Ising Model, Journal of Statistical Physics, 145 (2011),
518-548.
[10] S. Sherman, Combinatorial Aspects of the Ising Model for Ferromagnetism. I. A
Conjecture of Feynman on Paths and Graphs, Journal of Mathematical Physics,
1 (1960), 202-217.
[11] J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity,
Oxford University Press, Oxford, 2006.
[12] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi,
Solving the 3D Ising model with the conformal bootstrap, Physical Review D,
86 (2012), 025022.