The Hasse-Minkowski Theorem

The Hasse-Minkowski theorem concerns the classification of quadratic forms over global fields (i.e., finite extensions of Q and rational function fields with a finite constant field). Hasse proved the theorem over the rational numbers in his Ph.D. thesis in 1921. He extended the research of his thesis to quadratic forms over all number fields in 1924. Historically, the Hasse-Minkowski theorem was the first notable application of p-adic fields that caught the attention of a wide mathematical audience. The goal of this thesis is to discuss the Hasse-Minkowski theorem over the rational numbers and over the rational function fields with a finite constant field of odd characteristic. Our treatments of quadratic forms and local fields, though, are more general than what is strictly necessary for our proofs of the Hasse-Minkowski theorem over Q and its analogue over rational function fields (of odd characteristic). Our discussion concludes with some applications of the Hasse-Minkowski theorem.

A Median Voter Theorem for Postelection Politics

We analyze a model of 'postelection politics', in which (unlike in the more common Downsian models of 'preelection politics') politicians cannot make binding commitments prior to elections. The game begins with an incumbent politician in office, and voters adopt reelection strategies that are contingent on the policies implemented by the incumbent. We generalize previous models of this type by introducing heterogeneity in voters' ideological preferences, and analyze how voters' reelection strategies constrain the policies chosen by a rent-maximizing incumbent. We first show that virtually any policy (and any feasible level of rent for the incumbent) can be sustained in a Nash equilibrium. Then, we derive a 'median voter theorem': the ideal point of the median voter, and the minimum feasible level of rent, are the unique outcomes in any strong Nash equilibrium. We then introduce alternative refinements that are less restrictive. In particular, Ideologically Loyal Coalition-proof equilibrium also leads uniquely to the median outcome.

Automatic Stabilizer Feature of Fixed Exchange Rate Regimes in Emerging Markets

This paper shows that countries characterized by a financial accelerator mechanism may reverse the usual finding of the literature -- flexible exchange rate regimes do a worse job of insulating open economies from external shocks. I obtain this result with a calibrated small open economy model that endogenizes foreign interest rates by linking them to the banking sector's foreign currency leverage. This relationship renders exchange rate policy more important compared to the usual exogeneity assumption. I find empirical support for this prediction using the Local Projections method. Finally, 2nd order approximation to the model finds larger welfare losses under flexible regimes.

DeMoivre's Theorem

DeMoivre's theorem is of great utility in some parts of physical chemistry and is re-introduced here.

Ehrenfest's Theorem

A one dimensional presentation of Ehrenfest's theorem is presented.