Me, John Swallow, and Ján Mináč at an AMS conference in Davidson

Research

As a graduate student my advisor was Ravi Vakil, and my project was to extend some results of Mináč and Swallow concerning the module structure of certain Galois cohomology groups, particularly the $\mathbb{F}_p[\textrm{Gal}(K/F)]$-structure of $H^i(K,\mu_p)$ when $\textrm{Gal}(K/F) \simeq \mathbb{Z}/p^n\mathbb{Z}$.

I had the privilege to work with John Swallow while I was an undergrad at Davidson college, and through John I was introduced to Ján Mináč and their Galois module work. Since then I have worked on several projects which aim to extend their initial results concerning pth power classes of field extensions of degree p to more general settings.

I have recently begun a collaboration with Ben Brubaker to study the six-vertex model from statistical mechanics and its connection to Lie-theoretic phenomena.

Research Papers

1. A five color zero-sum generalization
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Author(s): D.Grynkiewicz and A. Schultz
Status: Appeared in Graphs and Combinatorics 22 (2006), 351 -- 360
Abstract: Several theorems of Ramsey-type have been generalized by considering $\mathbb{Z}/m\mathbb{Z}$-colorings and zero-sum configurations rather than $2$-colorings and monochromatic configurations; such generalizations are called EGZ. There is also a notion of EGZ generalizations of $r$-colorings for $r>2$. Recently in a sequence of three papers the first author showed that a certain Ramsey-type theorem with $4$-colors admitted an EGZ generalization. In this paper we show that a Ramsey-type problem with $5$-colors considered in a paper of the second author admits an EGZ generalization. This is the only EGZ generalization in $5$ colors which known to the authors at the time of this paper.
2. Galois module structure of $p$th-power classes of cyclic extensions of degree $p^n$
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Author(s): J.Mináč, A.Schultz, and J.Swallow
Status: Appeared in Proceedings of the London Mathematical Society 92 (2006), 307--341
Abstract: In the mid-1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of $p$th-power classes of cyclic extensions $K/F$ of $p$th-power degree. They determined the structure of these modules in the case when $F$ is a local field. In this paper we determine these Galois modules for all base fields $F$.
3. On a Modification of a Problem of Bialostocki, Erdös, and Lefmann
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Author(s): A.Schultz
Status: Appeared in Discrete Mathematics 306 (2006), 244--253
Abstract: For positive integers $m$ and $r$, one can easily show there exist integers $N$ such that for every map $D:\{1,2,...,N\} \to \{1,2,...,r\}$ there exist $2m$ integers $x_1 < ... < x_m < y_1 < ... < y_m$ which satisfy:
  • $D(x_1) = ... = D(x_m)$,
  • $D(y_1) = ... = D(y_m)$, and
  • $2(x_m-x_1) \leq y_m-x_1$.
In this paper we investigate the minimal such integer, which we call $g(m,r)$. We compute $g(m,2)$ for $m \geq 2$; $g(m,3)$ for $m \geq 4$; and $g(m,4)$ for $m \geq 3$. Furthermore, we consider $g(m,r)$ for general $r$. Along with results that bound $g(m,r)$, we compute $g(m,r)$ exactly for the following infinite families of $r$: $\{f_{2n+3}\}, \{2f_{2n+3}\}, \{18f_{2n}-7f_{2n-2}\}$, and $\{23f_{2n}-9f_{2n-2}\}$, where here $f_i$ is the $i$th Fibonacci number defined by $f_0 = 0$ and $f_1=1$.
4. Hilbert 90 for biquadratic extensions
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Author(s): R.Dwilewicz, J.Mináč, A.Schultz, and J.Swallow
Status: Appeared in American Mathematical Monthly 114 (2007), no. 7, 577--587
Abstract: Hilbert's Theorem 90 is a classical result in the theory of cyclic extensions. The quadratic case of Hilbert 90, however, generalizes in noncyclic directions as well. Informed by a poem of Richard Wilbur, the article explores several generalizations, discerning connections among multiplicative groups of fields, values of binary quadratic forms, a bit of module theory over group rings, and even Galois cohomology.
5. Automatic realizations of Galois groups with cyclic quotient of order $p^n$
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Author(s): J.Mináč, A.Schultz, and J.Swallow
Status: Appeared in Journal de Théorie des Nombres de Bordeaux 20 (2008), 419--430 (2008)
Abstract: We establish automatic realizations of Galois groups among groups $M \rtimes G$, where $G$ is a cyclic group of order $p^n$ for a prime $p$ and $M$ is a quotient of the group ring $\mathbb{F}_p[G]$.
6. Galois module structure of Milnor $K$-theory mod $p^s$ in characteristic $p$
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Author(s): J.Mináč, A.Schultz, and J.Swallow
Status: Appeared in New York Journal of Mathematics 14 (2008), 225--233
Abstract: Let $E$ be a cyclic extension of $p$th-power degree of a field $F$ of characteristic $p$. For all $m, s \in \mathbb{N}$, we determine $K_mE/p^sK_mE$ as a $(\mathbb{Z}/p^s\mathbb{Z})[\textrm{Gal}(E/F)]$-module. We also provide examples of extensions for which all of the possible nonzero summands in the decomposition are indeed nonzero.
7. Hilbert 90 for Galois cohomology
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Author(s): N.Lemire, J.Mináč, A.Schultz, and J.Swallow
Status: Appeared in Communications in Algebra 38 (2010), 361--372
Abstract: Assuming the Bloch-Kato Conjecture, we determine precise conditions under which Hilbert 90 is valid for Milnor $k$-theory and Galois cohomology. In particular, Hilbert 90 holds for degree $n$ when the cohomological dimension of the Galois group of the maximal $p$-extension of $F$ is at most $n$.
8. Galois module structure of Galois cohomology for embeddable cyclic extensions of degree $p^n$
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Author(s): N.Lemire, J.Mináč, A.Schultz, and J.Swallow
Status: Appeared in Journal of the London Mathematical Society 81 (2010), no. 3, 525--543
Abstract: In the case that $E/F$ is a cyclic, degree $p^n$ extension which embeds in a cyclic, degree $p^{n+1}$ extension $L/F$, we give the Galois module structure of the reduced Milnor $k$-groups $k_m(E)$.
9. Cyclic algebras, Schur indices, norms, and Galois modules
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Author(s): J.Mináč, A.Schultz, and J.Swallow
Status: Appeared in Annales des Sciences mathematiques du Quebec 35 (2011), no.1, 123--136
Abstract: Let $p$ be a prime and suppose that $K/F$ is a cyclic extension of degree $p^n$ with group $G$. Let $J=\mathbb{F}_p[G]$-module $K^{\times}/K^{\times p}$ of $p$th-power classes. In our previous paper we established precise conditions for $J$ to contain an indecomposable direct summand of dimension not a power of $p$. At most one such summand exists, and its dimension must be $p^i+1$ for some $0\leq i < n$. We show that for all primes $p$ and all $0 \leq i< n$, there exists a field extension $K/F$ with a summand of dimension $p^i+1$.
10. A generalization of the Gaussian formula and a $q$-analog of Fleck's congruence
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Author(s): A.Schultz and R.Walker
Status: Appeared in Journal of Number Theory 133 (2013), 3717-3738
Abstract:

The $q$-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row $n \in \mathbb{Z}^+$ of Pascal's triangle is captured by the so-called Gaussian Formula, which states that

$\sum_{m=0}^{n}(-1)^m \binom{n}{m}_q=\left\{\begin{array}{ll}\prod_{k~odd} (1-q^k),&\mbox{ if }n\mbox{ is even}\\ 0,&\mbox{ if }n\mbox{ is odd}.\end{array}\right.$

In this paper, we find a $q$-binomial congruence which synthesizes this result and Fleck's congruence for binomial coefficients, which asserts that for $n , p \in \mathbb{Z}^+$, with $p$ a prime,

$\displaystyle \sum_{m \equiv j \mod{p}}(-1)^m \binom{n}{m} \equiv 0 \mod{p^{\left\lfloor\frac{n-1}{p-1}\right\rfloor}}.$

11. $p$-groups have unbounded realization multiplicity
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Author(s): J.Berg and A.Schultz
Status: Appeared in Proceedings of the AMS 142 (2014), no.~7, 2281--2290
Abstract: In this paper we interpret the solutions to a particular Galois embedding problem over an extension $K/F$ satisfying $\textrm{Gal}(K/F) \simeq \mathbb{Z}/p^n\mathbb{Z}$ in terms of certain Galois submodules within the parameterizing space of elementary $p$-abelian extensions of $K$; here $p$ is a prime. Combined with some basic facts about the module structure of this parameterizing space, this allows us to exhibit a class of $p$-groups whose realization multiplicity is unbounded.
12. Parameterizing solutions to any Galois embedding problem over $\mathbb{Z}/p^n\mathbb{Z}$ with elementary $p$-abelian kernel
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Author(s): A.Schultz
Status: Appeared in Journal of Algebra 411 (2014), 50--91
Abstract: In this paper we use the Galois module structure for the classical parameterizing spaces for elementary $p$-abelian extensions of a field $K$ to give necessary and sufficient conditions for the solvability of any embedding problem which is an extension of $\mathbb{Z}/p^n\mathbb{Z}$ with elementary $p$-abelian kernel. This allows us to count the total number of solutions to a given embedding problem when the appropriate modules are finite, and leads to some nontrivial automatic realization and realization multiplicity results for Galois groups.
13. The 6-vertex model and deformations of the Weyl character formula
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Author(s): B.Brubaker and A.Schultz
Status: Appeared in Journal of Algebraic Combinatorics 42 (2015), no.~4, 917--958
Abstract: We use statistical mechanics -- variants of the six-vertex model in the plane studied by means of the Yang-Baxter equation -- to give new deformations of Weyl's character formula for classical groups of Cartan type $B$, $C$, and $D$. In each case, the corresponding Boltzmann weights are associated to the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types $A$ and $C$ by Tokuyama and Hamel-King, respectively. A special case recovers deformations of the Weyl denominator formula due to Okada.
14. Galois $p$-groups and Galois modules
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Author(s): S.Chebolu, J.Mináč, and A.Schultz
Status: Appeared in Rocky Mountain Journal of Mathematics. 46 (2016), 1405--1446.
Abstract: The smallest non-abelian $p$-groups play a fundamental role in the theory of Galois $p$-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups --- as well as other closely related, larger $p$-groups --- occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.
15. On Hamiltonians for six-vertex models
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Author(s): B.Brubaker and A.Schultz
Status: Preprint
Abstract: In this paper, we explain a connection between a family of free-fermionic six-vertex models and a discrete time evolution operator on one-dimensional Fermionic Fock space. The family of ice models generalize those with domain wall boundary, and we focus on two sets of Boltzmann weights whose partition functions were previously shown to generalize a generating function identity of Tokuyama. We produce associated Hamiltonians that recover these Boltzmann weights, and furthermore calculate the partition functions using commutation relations and elementary combinatorics. We give an expression for these partition functions as determinants, akin to the Jacobi-Trudi identity for Schur polynomials.

Other Papers

16. Galois module structure of Galois cohomology (thesis)
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Author(s): A.Schultz
Status: Final Draft
Abstract: My thesis, in which I explore the structure of certain Galois cohomology groups in the vein of Mináč and Swallow. This includes generalizations to $p$-adic extensions, the first result on pro-$p$ groups in this family of projects.
17. An elementary (number theory) proof of Touchard's congruence
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Author(s): G.Hurst and A.Schultz
Abstract: We use well known recurrence relations for Bell and Stirling numbers to develop a new recurrence relation on Bell numbers. This allows us to give a new proof of Touchard's congruence which uses some techniques approachable by undergraduates.
18. Meta-morphism: From Graduate Student to Networked Mathematician
, Princeton University Press, Princeton, NJ, 2011.
Download: Available at ams.org
Author(s): A.Schultz
Status: Appeared in Notices of the AMS 57 (2010), no. 9, 1132--1135. Also appeared in Best Writing in Mathematics 2011
Abstract: This is an expository article I wrote to encourage graduate students to become active in the mathematical social network.
19. An inquiry-based learning approach to enhancing proof writing skills via cross-institutional peer review.
Author(s): D.Ernst, A.Hodge, A.Schultz
Status: To appear in Problems, Resources and Issues in Mathematics Undergraduate Studies.
Abstract: In the spring of 2011, D.C.~Ernst and A.~Schultz taught number theory courses at their respective institutions. Twice during the semester, students in each class submitted proofs of 2--3 theorems to be peer reviewed by students in the other class. Each student wrote anonymous and formal referee reports of the submitted theorems which were then returned to the original student author. In this paper we detail the peer review process, discuss a survey that gauged student perception of peer review, and reflect on the strengths and weaknesses of this pedagogical method.

Work with Undergraduates

I've had the benefit of working with many excellent undergraduate students. Below is a list of research projects I have pursued with some of the most enthusiastic of these students.