About

Theresa Anderson is an Associate Professor of Mathematical Sciences at Carnegie Mellon University. The provided page text does not include additional details about her research focus, background, or key contributions.

Research topics

• Mathematics
• Physics
• Pure mathematics
• Computer Science
• Mathematical analysis
• Combinatorics
• Quantum mechanics
• Discrete mathematics

Selected publications

• Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials

  arXiv (Cornell University) · 2026-03-16

  preprintOpen access1st authorCorresponding

  We study the Galois group $G_f$ of a random polynomial $f$ of height at most $H$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$, which arise in a wide variety of applications. Our main result is a theorem of van der Waerden-Bhargava type: the probability that $G_f$ is not the full hyperoctahedral group $S_2 \wr S_n$ is $Θ(H^{-1}\log H)$, independent of $b$, with the leading-order group $G_1$ being of index $2$. This paper is a companion to a recent paper by the authors and Bertelli addressing reciprocal polynomials (i.e. the case $b = 1$).

  PublisherDOI

• Galois groups of reciprocal polynomials and thevan der Waerden–Bhargava theorem

  Algebra & Number Theory · 2026-03-24

  preprintOpen access1st authorCorresponding

  We study the Galois groups G f of degree 2n reciprocal (a.k.a.palindromic) polynomials f of height at most H , finding that G f falls short of the maximal possible group S 2 S n for a proportion of all f bounded above and below by constant multiples of H -1 log H , whether or not f is required to be monic.This answers a 1998 question of Davis, Duke and Sun and extends Bhargava's 2023 resolution of van der Waerden's 1936 conjecture on the corresponding question for general polynomials.Unlike in that setting, the dominant contribution comes not from reducible polynomials but from those f for which (-1) n f (1) f (-1) is a square, causing G f to lie in an index-2 subgroup.

  PublisherOA PDFDOI

• Infinite intersections of doubling measures, weights, and function classes

  Research in the Mathematical Sciences · 2026-03-31

  articleOpen access1st authorCorresponding

  Abstract A series of long-standing questions in harmonic analysis ask whether the intersection of all prime “ p -adic versions” of an object, such as a doubling measure or a Muckenhoupt or reverse Hölder weight, recovers the full object. Investigation into these questions was reinvigorated in 2019 by work of Boylan–Mills–Ward, culminating in showing that this recovery fails for a finite intersection in work of Anderson–Bellah–Markman–Pollard–Zeitlin. Via generalizing a new number theoretic construction therein, we answer these questions.

  PublisherOA PDFDOI

• Galois groups of reciprocal polynomials II: Twisted reciprocal polynomials

  ArXiv.org · 2026-03-16

  articleOpen access1st authorCorresponding

  We study the Galois group $G_f$ of a random polynomial $f$ of height at most $H$ in the family of polynomials of degree $2n$ satisfying the twisted reciprocal relation $f(x) = x^{2n}/b^n \cdot f(b/x)$, which arise in a wide variety of applications. Our main result is a theorem of van der Waerden-Bhargava type: the probability that $G_f$ is not the full hyperoctahedral group $S_2 \wr S_n$ is $Θ(H^{-1}\log H)$, independent of $b$, with the leading-order group $G_1$ being of index $2$. This paper is a companion to a recent paper by the authors and Bertelli addressing reciprocal polynomials (i.e. the case $b = 1$).

  PublisherOA PDF

• The structure of the double discriminant

  ArXiv.org · 2025-07-22

  preprintOpen access1st authorCorresponding

  For a polynomial $f(x) = \sum_{i=0}^n a_i x^i$, we study the double discriminant $DD_{n,k} = \operatorname{disc}_{a_k} \operatorname{disc}_x f(x)$. This object has been well studied in algebraic geometry, but has been brought to recent prominence in number theory by its key role in the proof of the Bhargava--van der Waerden theorem. We bridge the knowledge gap for this object by proving an explicit factorization: $DD_{n,k}$ is the product of a square, a cube, and possibly a linear monomial. Our proof is entirely algebraic. We also investigate other aspects of this factorization.

  PublisherOA PDFDOI

• Characterizing the support of semiclassical measures for higher-dimensional cat maps

  arXiv (Cornell University) · 2024-10-17

  preprintOpen access

  Quantum cat maps are toy models in quantum chaos associated to hyperbolic symplectic matrices $A\in \operatorname{Sp}(2n,\mathbb{Z})$. The macroscopic limits of sequences of eigenfunctions of a quantum cat map are characterized by semiclassical measures on the torus $\mathbb{R}^{2n}/\mathbb{Z}^{2n}$. We show that if the characteristic polynomial of every power $A^k$ is irreducible over the rationals, then every semiclassical measure has full support. The proof uses an earlier strategy of Dyatlov-Jézéquel [arXiv:2108.10463] and the higher-dimensional fractal uncertainty principle of Cohen [arXiv:2305.05022]. Our irreducibility condition is generically true, in fact we show that asymptotically for $100\%$ of matrices $A$, the Galois group of the characteristic polynomial of $A$ is $S_2 \wr S_n$. When the irreducibility condition does not hold, we show that a semiclassical measure cannot be supported on a finite union of parallel non-coisotropic subtori. On the other hand, we give examples of semiclassical measures supported on the union of two transversal symplectic subtori for $n=2$, inspired by the work of Faure-Nonnenmacher-De Bièvre [arXiv:nlin/0207060] in the case $n=1$. This is complementary to the examples by Kelmer [arXiv:math-ph/0510079] of semiclassical measures supported on a single coisotropic subtorus.

  PublisherOA PDFDOI

• Bounds on 10th moments of (𝑥,𝑥³) for ellipsephic sets

  Contemporary mathematics - American Mathematical Society · 2024-01-01

  other1st authorCorresponding

  Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an ellipsephic set which satisfies digital restrictions in a given base. Using the method developed by Hughes and Wooley, we bound the number of integer solutions to the system of equations <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column Blank 2nd Column a m p semicolon sigma-summation Underscript i equals 1 Overscript 2 Endscripts left-parenthesis x Subscript i Superscript 3 Baseline minus y Subscript i Superscript 3 Baseline right-parenthesis equals sigma-summation Underscript i equals 3 Overscript 5 Endscripts left-parenthesis x Subscript i Superscript 3 Baseline minus y Subscript i Superscript 3 Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column a m p semicolon sigma-summation Underscript i equals 1 Overscript 2 Endscripts left-parenthesis x Subscript i Baseline minus y Subscript i Baseline right-parenthesis equals sigma-summation Underscript i equals 3 Overscript 5 Endscripts left-parenthesis x Subscript i Baseline minus y Subscript i Baseline right-parenthesis comma EndLayout"> <mml:semantics> <mml:mtable columnalign="right left" rowspacing="3pt" columnspacing="0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd/> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:munderover> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo> − </mml:mo> <mml:msubsup> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:mn>5</mml:mn> </mml:munderover> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo> − </mml:mo> <mml:msubsup> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd/> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:munderover> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:mn>5</mml:mn> </mml:munderover> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>y</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{equation*} \begin {split} &amp; \sum _{i=1}^2 \left (x_i^3-y_i^3 \right )=\sum _{i=3}^5 \left (x_i^3-y_i^3 \right )\\ &amp; \sum _{i=1}^2 (x_i-y_i)=\sum _{i=3}^5 (x_i-y_i), \end{split} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold x bold comma bold y element-of script upper A Superscript 5"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">x</mml:mi> <mml:mo mathvariant="bold">,</mml:mo> <mml:mi mathvariant="bold">y</mml:mi> </mml:mrow> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mn>5</mml:mn> </mml:msup>

  PublisherDOI

• Infinite intersections of doubling measures, weights, and function classes

  ArXiv.org · 2024-09-26

  preprintOpen access1st authorCorresponding

  A series of longstanding questions in harmonic analysis ask if the intersection of all prime ``$p$-adic versions" of an object, such as a doubling measure, or a Muckenhoupt or reverse Hölder weight, recovers the full object. Investigation into these questions was reinvigorated in 2019 by work of Boylan-Mills-Ward, culminating in showing that this recovery fails for a finite intersection in work of Anderson-Bellah-Markman-Pollard-Zeitlin. Via generalizing a new number-theoretic construction therein, we answer these questions.

  PublisherOA PDFDOI

• Weakly porous sets and Muckenhoupt A distance functions

  Journal of Functional Analysis · 2024-07-03 · 7 citations

  articleOpen access1st author

  We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight w(x)=dist(x,E)−α belongs to the Muckenhoupt class A1, for some α>0, if and only if E⊂Rn is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of E. When E is weakly porous, we obtain a similar quantitative characterization of w∈Ap, for 1<p<∞, as well. At the end of the paper, we give an example of a set E⊂R which is not weakly porous but for which w∈Ap∖A1 for every 0<α<1 and 1<p<∞.

  PublisherDOI

• Arbitrary finite intersections of doubling measures and applications

  Journal of Functional Analysis · 2024-07-04 · 1 citations

  articleOpen access1st author

  We make major progress on a folkloric conjecture in analysis by constructing a measure on the real line which is doubling on all n-adic intervals for any finite list of n∈N, yet not doubling overall. In particular, we extend previous results in the area, including those of Boylan-Mills-Ward and Anderson-Hu, by using a wide array of substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, Ap weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.

  PublisherDOI

Recent grants

• Questions at the Interface of Analysis and Number Theory

  NSF · $98k · 2022–2025

• PostDoctoral Research Fellowship

  NSF · $150k · 2015–2019

• CAREER: Building bridges between number theory and harmonic analysis

  NSF · $328k · 2023–2028

Frequent coauthors

• Bingyang Hu

  17 shared

• Eyvindur A. Palsson

  8 shared

• Angel Kumchev

  Towson University

  7 shared

• Kevin Hughes

  6 shared

• Kabe Moen

  5 shared

• Robert J. Lemke Oliver

  4 shared

• Gloria Marı́ Beffa

  University of Wisconsin–Madison

  4 shared

• Olli Tapiola

  Universitat Autònoma de Barcelona

  4 shared

Education

• Ph.D.

  Brown University