I am an assistant professor at
the University of Notre Dame.

University of Notre
Dame

Department of Mathematics

275 Hurley Hall

Notre Dame, IN 46556

Department of Mathematics

275 Hurley Hall

Notre Dame, IN 46556

*On symmetric power L-invariants of Iwahori level Hilbert modular forms*(pdf, ), with Rob Harron. To appear in the American Journal of Mathematics.We compute L-invariants for symmetric powers of non-CM Iwahori level Hilbert modular forms in terms of logarithmic derivatives of Hecke eigenvalues on eigenvarieties.*p-adic Families and Galois Representations for GSp(4) and GL(2)*(pdf, ), Math. Research Letters 19 (2012), no 05, 1-10.We prove local-global compatibility for Iwahori level Siegel modular forms by combining a previous result (up to a quadratic twist) with p-adic families. We deduce information at p and l for two dimensional Galois representations on quadratic imaginary fields.*Lagrangian hyperplanes in holomorphic symplectic varieties*(arXiv, appendix, code). With Benjamin Bakker. To appear in CEJM.*Galois representations for holomorphic Siegel modular forms*(pdf, ), DOI 10.1007/s00208-012-0811-3, Mathematische Annalen: Volume 355, Issue 1 (2013), Page 381-400.We prove many cases of local-global compatibility (up to a quadratic twist) for holomorphic Siegel modular forms.*Crystalline representations for $\operatorname{GL}(2)$ over quadratic imaginary fields*(pdf, ), my thesis.If $\pi$ is an irreducible admissible regular algebraic cuspidal representation of $\textrm{GL}(2)$ over a quadratic imaginary field and $v$ is an unramified place of $K$ where $\pi_v$ and $\pi_{v^c}$ are unramified principal series with distinct Satake parameters, we show that the Galois representation associated to $\pi$ is crystalline at $v$.*Higher rank stable pairs on K3 surfaces*(arXiv, ). Communications in Number Theory and Physics, Volume 6, Number 4 (2012). With Benjamin Bakker.Virtual curve counts have been defined for threefolds by integration against virtual classes on moduli spaces of stable maps (Gromov-Witten theory), ideal sheaves (Donaldson-Thomas theory), and stable pairs (Pandharipande-Thomas theory). The first two theories are proven to be equivalent for toric threefolds, and all three are conjecturally equivalent for arbitrary threefolds. One may ask whether there is such a correspondence for surfaces. In particular, the Gromov-Witten theory of $K3$ surfaces has recently been computed by Maulik, Pandharipande, and Thomas; it is governed by quasimodular forms and is closely related to invariants obtained from the moduli spaces of rank $r = 0$ stable pairs with $n = 1$ sections. We compute the Hodge polynomials of the moduli spaces of stable pairs for higher rank $r \geq 0$ and level $n \geq 1$, and explore the modularity properties and relationship to Gromov-Witten theory.*Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves*(pdf), Math. Comp.**78**(2009), no. 268, 2397--2425. With G. Grigorov, S. Patrikis, W. Stein, C. Tarniţǎ.

- Applications of Global Class Field Theory, notes for Math 160c, Caltech, Spring 2013
- Introduction to p-adic Hodge theory, notes for Math 162b, Caltech, Winter 2012

- Math 20550, Calculus 3, Notre Dame, Spring 2016.

- Math 40520, Theory of Numbers, Notre Dame, Fall 2015.
- Math 60220, Basic Algebra 2: Graduate Algebra, Notre Dame, Spring 2015.
- Math 60210, Basic Algebra 1: Graduate Algebra, Notre Dame, Fall 2014.
- Math 10560, Calculus 2: Integration, series, differential equations, Notre Dame, Fall 2014.
- Math 80220, Topics in Algebra 2: Introduction to Algebraic Number Theory, Notre Dame, Spring 2014.
- Math 20550, Calculus 3, Notre Dame, Spring 2014.
- Math 10350, Calculus A for Life Sciences, Notre Dame, Fall 2013.
- Math 5c, Galois Theory and Representations of Finite Groups, Caltech, Spring 2013.
- Math 160c, Applications of Global Class Field Theory, Caltech, Spring 2013. Lecture notes
- Math 1a (Section 1), Freshman Mathematics, Caltech, Fall 2011.
- Math 160b, Local Class Field Theory, Caltech, Winter 2012.
- Math 162b, p-adic Galois Representations, Caltech, Winter 2012. Lecture notes
- Math 5c, Galois Theory and Representations of Finite Groups, Caltech, Spring 2011.
- Math 160b, Local Class Field Theory, Caltech, Winter 2011
- Math 203, Multivariate Calculus, Princeton, Fall 2009
- Math 103, Calculus, Princeton, Fall 2007

- Math 217, Honors Linear Algebra, Princeton, Spring 2010
- Math 453, Analytic Number Theory, Princeton, Spring 2009
- Math 322, Galois Theory, Princeton, Fall 2008
- Math 202, Linear Algebra, Princeton, Spring 2008
- Math 217, Honors Linear Algebra, Princeton, Spring 2007
- Math 215, Honors Real Analysis, Princeton, Fall 2006
- Math 129, Algebraic Number Theory, Harvard, Spring 2005
- Math 250, Graduate Algebra, Harvard, Fall 2004
- Math 130, Topology, Harvard, Spring 2004
- Math 112, Real Analysis, Harvard, Fall 2003
- Math 55b, Honors Linear Algebra and Analysis, Harvard, Spring 2003
- Math 55a, Honors Linear Algebra and Analysis, Harvard, Fall 2002