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Research
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.
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 $\textrm{Gal}(K/F) \simeq \mathbb{Z}/p^n\mathbb{Z}$.
More recently I have begun a collaboration with Ben Brubaker to study the six-vertex model from statistical mechanics and its connection to Lie-theoretic phenomena.
Research Papers
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Download: PDF 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. -
Download: Available on the ArXiv 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 $F$. -
Download: Available at the ArXiv 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
- $2(x_m-x_1) \leq y_m-x_1$.
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Download: Available on the ArXiv 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. -
Download: Available on the ArXiv 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 $\mathbb{F}_p[G]$. -
Download: Available on the ArXiv 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. -
Download: Available on the ArXiv 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 $n$. -
Download: Available on the ArXiv 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 $k_m(E)$. -
Download: Available on the ArXiv 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 $p^i+1$. -
Download: Available at the ArXiv 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,
$\displaystyle \sum_{m \equiv j \mod{p}}(-1)^m \binom{n}{m} \equiv 0 \mod{p^{\left\lfloor\frac{n-1}{p-1}\right\rfloor}}.$
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Download: Available at the ArXiv 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. -
Download: Available at the ArXiv 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. -
Download: Available at the ArXiv 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. -
Download: Available at the ArXiv 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. -
Download: Available at the ArXiv Author(s): B.Brubaker and A.Schultz Status: Appeared in Journal of Combinatorial Theory, Series A. 55 (2018), 100---121. 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. -
Download: Available at the ArXiv Author(s): J.Mináč, A.Schultz, and J.Swallow Status: Appeared in Journal of Algebra 598 (2022), 194--235. Abstract: A powerful new perspective in the analysis of absolute Galois groups has recently emerged from the study of Galois modules related to classical parameterizing spaces of certain Galois extensions. The recurring trend in these decompositions is their stunning simplicity: almost all summands are free over some quotient ring. The non-free summands which appear are exceptional not only because they are different in form, but because they play the key role in controlling arithmetic conditions that allow the remaining summands to be easily described. In this way, these exceptional summands are the lynchpin for a bevy of new properties of absolute Galois groups that have been gleaned from these surprising decompositions.
In one such recent decomposition, a remarkable new exceptional summand was discovered which exhibited interesting properties that have not been seen before. The exceptional summand is drawn from a particular finite family that has not yet been investigated. The main goal of this paper is to introduce this family of modules and verify their indecomposability. We believe this module will be of interest to people working in Galois theory, representation theory, combinatorics, and general algebra. The analysis of these modules includes some interesting new tools, including analogs of $p$-adic expansions. -
Download: Available at the ArXiv Author(s): J.Mináč, A.Schultz, and J.Swallow Status: To appear in manuscripta mathematica. Abstract: Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{F}_p[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $\text{Gal}(K/F)$. For fixed values of $m$ and $n$, there are only finitely many possible isomorphism classes for the non-free indecomposable summand.
This result seems to be a significant breakthrough in the study of Galois modules. For one, these Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation should provide key insight into absolute Galois groups. More immediately, however, these results show that Galois cohomology is a context in which seemingly impossible module decompositions can practically be achieved: when $m,n>1$ the modular representation theory allows for an infinite number of indecomposable summands, with no known classification of indecomposable types. This paper marks the first time a Galois module decomposition has been computed under these circumstances over an infinite family of modules. -
Download: Available at the ArXiv Author(s): J.Mináč, A.Schultz, and J.Swallow Status: Appeared in Journal of Number Theory 242 (2023), 669--708. Abstract: The power classes of a field are well-known for their ability to parameterize elementary $p$-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple.
Suppose $m \in \mathbb{N}$ and $p$ is prime. In a recent paper, a surprising and elegant decomposition for $p^m$th power classes has been computed. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when $p=2$, the structure of this exceptional summand was determined by a vector $\mathbf{a}\in \{-\infty,0,\dots,n\}^m$ and a natural number $d$. In this paper we give field-theoretic interpretations for $\mathbf{a}$ and $d$, showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to $K/F$. -
Download: Available at the ArXiv or open access at Canadian Journal of Mathematics Author(s): F.Chemotti, J.Mináč, A.Schultz, and J.Swallow Status: Appeared in Canadian Journal of Mathematics 75, no. 3 (2023), 804--827. Abstract: For a Galois extension $K/F$ with $\text{char}(K)\neq 2$ and $\text{Gal}(K/F) \simeq \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, we determine the $\mathbb{F}_2[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times 2}$. Although there are an infinite number of (pairwise non-isomorphic) indecomposable $\mathbb{F}_2[\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}]$-modules, our decomposition includes at most $9$ indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types. -
Download: Available at the ArXiv Author(s): F.Chemotti, J.Mináč, T.T.Nguyen, A.Schultz, J.Swallow, and N.D.Tan Status: Accepted to Israel Journal of Mathematics Abstract: Recently the Galois module structure of square power classes of a field $K$ has been computed under the action of $\text{Gal}(K/F)$ in the case where $\text{Gal}(K/F)$ is the Klein $4$-group. Despite the fact that the modular representation theory over this group ring includes an infinite number of non-isomorphic indecomposable types, the decomposition for square power classes includes at most $9$ distinct summand types. In this paper we determine the multiplicity of each summand type in terms of a particular subspace of $\text{Br}(F)$, and show that all ``unexceptional" summand types are possible. -
Download: Available at the ArXiv Author(s): L.Heller, J.Mináč, T.T.Nguyen, A.Schultz, J.Swallow, and N.D.Tan Status: Accepted to Israel Journal of Mathematics Abstract: We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and that $F$ contains a primitive $p$th root of unity if $\text{char}(F)\neq p$.
Other Papers
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Download: PDF 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. -
Download: Available on the ArXiv 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. -
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, Princeton University Press, Princeton, NJ, 2011, and was reprinted in Notices of the AMS 69, no.1 (2022), 55--58 Abstract: This is an expository article I wrote to encourage graduate students to become active in the mathematical social network. -
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. -
Download: Available at ams.org Author(s): S.Chang, A.Schultz, and I.Volić Status: Appeared in Notices of the AMS 68, no.11 (2021), 1990--1993 Abstract: A communication to advertise the Institute for Mathematics and Democracy, an initiative founded by S.Chang and I.Volić to promote a deeper understanding of the mathematics that underlies many socioeconomic forces and political processes so that citizens can make informed political decisions and effect meaningful change. It supports research, teaching, and outreach in mathematics and democracy.
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.
During the academic year 2021-2022, I supervised Amy Tao's thesis project. Amy's project aimed to understand the $\mathbb{Z}/3\mathbb{Z}[\text{Gal}(K/F)]$-module structure of cube power classes of a field $K$ in the case where $\text{Gal}(K/F) \simeq \mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}$. This project is a natural follow-up to the work on square power classes over biquadratic extensions studied by F.Chemotti, J.Mináč, J.Swallow and myself. Moving to the case $p=3$, however, presents a whole slew of new complications.
During the academic year 2018-2019, I supervised Sappha O'Meara's thesis project. Sappha worked on sharpness results related to a $q$-analog of Fleck's congruence originally investigated with Robert Walker. This project is a follow-up to similar sharpness investigations conducted in collaboration with Madeleine Barowsky and Farrah Yhee.
During the academic year 2016-2017, I supervised Lauren Heller's thesis project. Lauren's thesis provided a module decomposition for Artin-Schreier classes (i.e., elements of $K/\wp(K)$) as a module over $\text{Gal}(K/F)$ in the case that $K/F$ is a rank $2$ elementary $p$-abelian extension. Her work is currently being generalized to give a module decomposition of $K/\wp(K)$ when $K/F$ is an arbitrary elementary $p$-abelian extension.
During the academic year 2015-2016, I supervised Phyllis Ju's thesis project. Phyllis's thesis works to provide enumerations for certain generalizations of alternating sign matrices which show up in the work of Brubaker, Bump and Friedberg, as well as subsequent collaborations between Brubaker and myself. During this same year I also explored some sharpness results related to Fleck's congruence with Madeleine Barowsky.
During the academic year 2012-2013, I supervised Ran Ji's thesis project. Ran's thesis gave some results which are part of a larger program to enumerate certain classes half-turn symmetric alternating sign matrices. Her methodology followed that pioneered by Greg Kuperberg in his paper Symmetry classes of alternating sign matrices under one roof.
During this same year I also worked with Farrah Yhee on some sharpness conjectures related to work I previously did with Robert Walker on a $q$-analog of Fleck's Congruence.
During the academic year 2011-2012 I'm supervised Melinda Lanius' thesis project. Melinda worked on constructing universal cycles for $k$-sets of $\{1,2,\cdots,n\}$ during a summer REU, and her thesis has her continuing with some of the ideas she had from the summer but wasn't able to complete.
In the summer of 2010, I supervised work by Robert Walker in the realm of q-binomial coefficients. Robert has presented this work at several conferences and has gone on to win a number of accolades, including an Outstanding Presentation award at the Young Mathematicians' Conference at OSU, the Outstanding Oral Presentation Award from the SROP, and an Exemplary Summer Research Citation from the National Center for Institutional Diversity. Our work culminated in the joint paper "A generalization of the Gaussian formula and a $q$-analog of Fleck's congruence" (for more on this paper, see above).
In the fall of 2009, Joe Nance and I started a project to investigate the convergence of certain sequences that Joe had been studying. The project has connections to infinite continued fractions of square roots of integers, particularly in certain "2 x 2" examples. It appears that linear algebra is the key to unlocking the convergence properties of these sequences. Joe is currently writing up his findings and will be starting a Master's program at UIUC in the fall of 2011.
In the fall of 2009, Jen Berg and I started a project in which we explore generalization to Artin-Schreier theory of some Galois modules (and their implications) related to Kummer theory. Having completed the "dictionary" between cyclic Artin-Schreier Galois modules and their corresponding extensions, we're now focused on studying the entire Artin-Schreier module as a whole We hope to write up our findings during the academic year 2010-2011. Jen is now a graduate student at the University of Texas at Austin. Our work culminated in the joint paper "$p$-groups have unbounded realization multiplicity" (for more on this paper, see above).
- In the fall of 2008, Greg Hurst came to me interested in studying recursive algorithms for computing Bell numbers. His initial idea turned into the collaborative paper "An elementary (number theory) proof of Touchard's congruence" (for more on this paper, see above).