Fact-checked by Grok 2 weeks ago

Holly Krieger

Holly Krieger is an American mathematician specializing in and , currently serving as a professor of at the , where she holds the position of Corfield Fellow at Murray Edwards College. Born and raised near , she completed her undergraduate studies in the mathematics honors program at the University of at Urbana-Champaign before earning her and PhD from the University of at in 2013, with a thesis on supervised by Laura DeMarco and Ramin Takloo-Bighash. Following her doctorate, Krieger held an NSF Postdoctoral Fellowship at the under , during which she transitioned toward research. In 2016, she joined the as the Corfield Lecturer and Fellow at Murray Edwards College, advancing to full professor in 2022. Her research focuses on the intersections of arithmetic geometry, , and unlikely intersections in dynamical systems, including studies of periodic points on Hénon maps, torsion points on elliptic curves, and arithmetic height functions related to problems like integer solutions to equations. Notable contributions include work on the uniform Mordell-Lang , resolved in a 2021 series of papers, and recent advancements in the Mordell , for which she delivered an invited at the 2024 Joint Mathematics Meetings. Krieger has received several prestigious awards for her work, including the 2023 Philip Leverhulme Prize, the 2020 Whitehead Prize from the London Mathematical Society, the 2020 Alexanderson Award from the American Institute of Mathematics, and the 2019 Mahler Lectureship from the Australian Mathematical Society. She was also a 2021–2022 Sally Starling Seaver Fellow at the Harvard Radcliffe Institute, where she explored connections between complex dynamics and arithmetic geometry. Beyond academia, Krieger is known for her public outreach efforts, including regular contributions to the Numberphile YouTube channel, making advanced mathematical concepts accessible to a broad audience.

Early life and education

Early life

Holly Krieger was born in , a Midwestern city approximately 130 miles south of . She was raised in the Champaign area during her early years. Krieger grew up in a supportive family environment that fostered and , with her mother pursuing singing and her uncle working as a professional player, though further specifics on family professions remain sparsely documented in available sources. As a child, she displayed an outgoing personality and interests in performance arts and , initially aspiring to become an or singer rather than focusing on . Her interest in mathematics began to emerge during high school in Champaign, where she excelled in the subject and briefly joined the math team, though she ultimately stepped away to prioritize social activities. This early exposure was shaped by the robust educational resources available in the Midwest, including proximity to institutions like the . She later transitioned to formal undergraduate studies at the .

Education

Krieger completed her undergraduate studies in the mathematics honors program at the University of Illinois at Urbana-Champaign, earning a Bachelor of Science degree in 2006. Her early interest in mathematics, influenced by her upbringing near Champaign, Illinois, led her to pursue this rigorous program focused on advanced coursework and research preparation. She then moved to the University of Illinois at Chicago, where she obtained a Master of Science degree in mathematics in 2008. Continuing at the same institution, Krieger earned her Ph.D. in mathematics in 2013. Her dissertation, titled Primitive Prime Divisors for Unicritical Polynomials, was supervised by Laura DeMarco and Ramin Takloo-Bighash. During her doctoral work, Krieger concentrated on introductory concepts in polynomial dynamics within the field of , employing primitive prime divisors to investigate properties of periodic points and orbits in unicritical polynomials. This approach allowed her to prove finiteness results for Zsigmondy sets associated with critical orbits, providing tools for analyzing divisibility patterns in dynamical systems over .

Professional career

Postdoctoral research

Following her Ph.D. in from the University of Illinois at Chicago in 2013, where she studied polynomial dynamics, Holly Krieger held a three-year (NSF) Mathematical Sciences Postdoctoral Research Fellowship at the (MIT). This prestigious fellowship supported her transition to independent research, providing resources to explore advanced topics in . At , Krieger worked under the supervision of , a leading expert in arithmetic geometry and . Her research during this period, titled "Arithmetic and Geometry in Algebraic Dynamics," focused on initial explorations in , examining the interplay between iterative processes and number-theoretic structures. This work marked the beginning of her efforts to bridge dynamical systems—rooted in her doctoral background—with arithmetic geometry and . From approximately 2013 to 2016, Krieger's postdoctoral investigations laid foundational insights into how algebraic maps over number fields exhibit arithmetic properties, influencing subsequent developments in the field. Her time at not only honed her expertise but also facilitated collaborations that shaped her long-term research trajectory in these interdisciplinary areas.

Positions at the University of Cambridge

In 2016, following her NSF Postdoctoral Fellowship at , Holly Krieger was appointed as a Lecturer in Mathematics in the Department of and at the . Concurrently, she became the Corfield Fellow at Murray Edwards College, a position she has held since 2016, and serves as Director of Studies in Mathematics there. In recognition of her contributions, Krieger was promoted to Professor of , effective 1 2022. These roles involve teaching and supervision in the Faculty of as well as administrative responsibilities within the college, supporting undergraduate and graduate education in .

Research

Arithmetic aspects of dynamical systems

Holly Krieger's research primarily investigates the arithmetic and algebraic properties of complex dynamical systems, with a particular emphasis on the behavior of periodic points under rational maps and the arithmetic structure of their orbits. In this context, she employs tools from , such as height functions and , to analyze families of maps like polynomials of the form f_c(z) = z^d + c, where c is a complex parameter. Her work highlights how arithmetic invariants, including canonical heights, reveal patterns in the distribution and density of periodic points, which are fixed under of the map. A central theme in Krieger's contributions is the application of primitive prime divisors to polynomial dynamics, which aids in understanding the arithmetic distribution of periodic points. Primitive prime divisors are primes that divide a specific iterate in an but not earlier ones, analogous to Zsigmondy primes in linear recurrence sequences. In her paper, Krieger proves the finiteness of the Zsigmondy set—the collection of integers n for which the nth term in the critical of f_c(z) = z^d + c (with rational c) lacks a primitive prime divisor—for rational parameters c. She establishes an effective bound on the size of this set, using dynamical height bounds for non-recurrent critical orbits and Thue-style for degrees d > 2, or complex-analytic methods for d = 2. This result constrains the possible arithmetic progressions in periodic point distributions, providing insights into the sparsity of torsion-like points in dynamical families. Krieger has also made significant advances in exploring unicritical polynomials—those with a single critical point—and their relation to the dynamical , which posits that exceptional loci (subvarieties where post-critically finite maps accumulate) in moduli spaces of dynamical systems have a specific . The , formulated by and DeMarco, draws parallels to the classical in Shimura varieties. In a 2016 paper, Krieger and collaborators prove a case of this for parametrized by polynomials, showing that if infinitely many points (a, b) on such a yield post-critically finite unicritical maps z^d + a and z^d + b, then the parametrization must be linear of a specific form involving roots of unity. Her 2017 paper in the Duke Mathematical Journal provides a detailed resolution for unicritical polynomial families, establishing the equidistribution of post-critically finite parameters with respect to the measure on the complement of the M_d (for degree d \geq 2). Using algebraic correspondences and combinatorial analysis, the work classifies all curves containing Zariski-dense sets of such pairs (a, b), confirming that exceptional loci are precisely the "arithmetic progressions" in parameter space. This proves the full dynamical André–Oort conjecture for these families, marking the first complete case and leveraging equidistribution theorems to bound the dimension of special subvarieties. These results underscore the arithmetic rigidity of dynamical systems, where infinite collections of periodic or post-critical finiteness impose strong algebraic constraints on parameter families. In 2025, Krieger co-authored a paper establishing a dynamical Shafarevich theorem for endomorphisms of infinite-dimensional projective space \mathbb{P}^\mathbb{N}, providing finiteness results for certain dynamical systems on this space.

Contributions to algebraic geometry

Krieger's research in algebraic geometry centers on the interplay between dynamical systems and geometric structures, particularly families of curves and polynomial maps on varieties. She employs tools from unlikely intersections to analyze torsion points and periodic behaviors in these settings, providing quantitative bounds that bridge arithmetic and complex geometry. This work extends classical conjectures, such as Manin-Mumford, to higher-genus contexts and dynamical families, revealing uniform constraints on exceptional points. A key contribution is her collaboration with Laura DeMarco and Hexi Ye on uniform versions of the Manin-Mumford conjecture for families of curves. In their 2020 paper, they establish quantitative bounds on the number of common torsion points for pairs of elliptic curves arising from a two-dimensional family of 2 curves, proving both a Manin-Mumford theorem over \mathbb{C} and a Bogomolov-type bound over \overline{\mathbb{Q}}. This resolves a conjecture by Bogomolov, Fu, and Tschinkel regarding torsion points on Legendre curves and introduces a general strategy for height-zero intersections in dynamical systems. Their approach leverages equidistribution techniques and unlikely intersection frameworks to achieve uniformity across the family, with the main stating that the intersection of torsion cosets is finite and bounded independently of parameters. Krieger has also advanced the study of rational periodic points on higher-dimensional varieties through her work on s. In a collaboration with Hyeonggeun Kim, Mara-Ioana Postolache, and Vivian Szeto, they construct, for each odd d \geq 3, a degree-d over \mathbb{Q} possessing at least (d-4)^2 integral periodic points of various s. This yields a lower bound on the number of rational periodic points, challenging conjectural uniform bounds in and highlighting the abundance of such points on algebraic surfaces. Unlike constructions, these maps exhibit cycles of arbitrarily large , with implications for the of periodic loci in \mathbb{P}^2. These efforts underscore Krieger's focus on torsion points and in higher-genus settings, such as 2 curves, where she applies unlikely intersections to constrain exceptional configurations in families of varieties. For instance, her results on bounded heights for unlikely intersections in algebraic dynamical systems provide foundational estimates for torsion anomalies, enabling applications to moduli spaces and uniform finiteness theorems beyond elliptic curves.

Recognition

Awards and prizes

In 2020, Holly Krieger received the from the London Mathematical Society, which recognizes early-career mathematicians for outstanding contributions to the field. The award specifically highlighted her work in , including advancements in equidistribution and bifurcation loci in parameter spaces. That same year, Krieger was awarded the Alexanderson Award by the American Institute of Mathematics, shared with collaborators Laura DeMarco and Hexi Ye for their joint paper on the uniform Manin-Mumford for a family of genus 2 curves. This prize honors exceptional collaborative in , emphasizing the team's contributions to and dynamical systems. In 2023, Krieger was granted the Philip Leverhulme Prize by the , which provides £100,000 to support mid-career researchers demonstrating exceptional promise in their discipline. The award acknowledged her innovative at the intersection of and dynamics, underscoring its potential for significant future impact in .

Invited lectureships

In 2019, Krieger served as the Mahler Lecturer for the Australian Mathematical Society, organized in collaboration with the Australian Mathematical Sciences Institute (AMSI), where she delivered a series of specialist lectures on topics in , including and dynamical systems, across multiple Australian universities. These lectures, part of an annual tour honoring contributions related to Kurt Mahler's work, also included public talks titled "The Mathematics of Life," aimed at broader audiences to explore connections between mathematics and biological systems. Krieger has received other notable invitations to speak at major international conferences. At the 63rd Annual Meeting of the Australian Mathematical Society in 2019, she delivered a plenary lecture, highlighting her expertise in dynamical systems. In 2023, she presented the London Mathematical Society (LMS)/ Lecture, titled "The Mathematical Vision of ," discussing the life and contributions of the Fields Medalist in a public forum. Additionally, at the 2024 Joint Mathematics Meetings (JMM) in , she gave a Current Events Bulletin Address entitled "Uniformity When Arithmetic Meets Geometry," addressing recent advances in arithmetic geometry. These invited lectureships underscore Krieger's role in advancing the visibility of and within the global mathematical community, providing platforms for her to share cutting-edge research while mentoring emerging scholars through accessible explanations and interactive sessions.

References

  1. [1]
    Holly Krieger — Cambridge
    Mathematical bio: Born and raised near Chicago, Professor Holly Krieger completed the undergraduate mathematics honors program at University of Illinois at ...Missing: biography | Show results with:biography<|control11|><|separator|>
  2. [2]
    Holly Krieger | Radcliffe Institute for Advanced Study at Harvard ...
    Krieger earned her bachelor's degree in mathematics at the University of Illinois at Urbana-Champaign and her master's and PhD from the University of ...Missing: biography | Show results with:biography
  3. [3]
    Professor Holly Krieger | Gresham College
    Born and raised near Chicago, Professor Holly Krieger completed the undergraduate mathematics honors program at University of Illinois at Urbana-Champaign.Missing: biography | Show results with:biography
  4. [4]
    Professor Holly Krieger | Faculty of Mathematics
    Professor Holly Krieger, Professor, Research Interests: Arithmetic and Complex Dynamics, Publications: Hénon maps with many rational periodic points.
  5. [5]
    A Century-Old Question Is Still Revealing Answers in Fundamental ...
    Oct 9, 2024 · Mathematicians have made lots of recent progress on a question called the Mordell conjecture, which was posed a century ago.
  6. [6]
    Champaign Mathematician (with Holly Krieger) - Numberphile Podcast
    Dec 13, 2019 · From Illinois to Cambridgeshire, Holly Krieger's path to mathematics was an unusual one.Missing: early family
  7. [7]
    Holly Krieger - AMSI Research and Higher Education
    Born and raised near Chicago, Dr Holly Krieger completed the undergraduate mathematics honors program at University of Illinois at Urbana-Champaign. She ...
  8. [8]
    [PDF] Holly Krieger - vita
    Holly Krieger. Department of Pure Mathematics and Mathematical Statistics. University of Cambridge. Wilberforce Road. Cambridge, UK CB3 0WB. Email: hkrieger ...
  9. [9]
    Professor Holly Krieger | Murray Edwards College
    Born and raised near Chicago, Dr Holly Krieger spent the career-building years of her life in Chicago, Berkeley, and Cambridge USA.Missing: high school
  10. [10]
    Primitive Prime Divisors for Unicritical Polynomials - UIC Indigo
    thesis. posted on 2015-11-01, 00:00 authored by Holly C. Krieger. We prove the finiteness of the Zsigmondy set associated to critical orbits of polynomials ...
  11. [11]
    PostDoctoral Research Fellowship - NASA ADS
    The title of the project for this fellowship to Holly Krieger is "Arithmetic and Geometry in Algebraic Dynamics. ... National Aeronautics and Space Administration ...Missing: MIT | Show results with:MIT
  12. [12]
    Dr Holly Krieger challenges the status quo - Support Cambridge
    Nov 28, 2019 · Appointed in 2016 as the Corfield Lecturer in Mathematics and the Corfield Fellow at Murray Edwards College, Holly's appointment in the Faculty ...Missing: professor | Show results with:professor
  13. [13]
    Women of mathematics: Holly Krieger | plus.maths.org
    Apr 12, 2017 · This article, the video and the podcast accompany the Women of Mathematics photo exhibition. You can download Holly Krieger's exhibition ...Missing: high | Show results with:high
  14. [14]
    Academic Staff | Department of Pure Mathematics and ... - DPMMS
    ... cam.ac.uk · Professor Holly Krieger. Algebraic Geometry. E1.19. 01223 337970. hkrieger@dpmms.cam.ac.uk · Professor Imre Leader. Combinatorics. C2.02. 01223 ...
  15. [15]
    Holly Krieger — Cambridge - DPMMS
    Professor of Mathematics, Department of Pure Mathematics and Mathematical Statistics. Corfield Fellow, Murray Edwards College. Research: I study the arithmetic ...Missing: thesis | Show results with:thesis
  16. [16]
    Primitive prime divisors in the critical orbit of z^d+c
    ### Summary of Abstract and Key Results
  17. [17]
    https://arxiv.org/abs/1203.2555
  18. [18]
    The Dynamical Andre-Oort Conjecture: Unicritical Polynomials - arXiv
    May 6, 2015 · Title:The Dynamical Andre-Oort Conjecture: Unicritical Polynomials. Authors:Dragos Ghioca, Holly Krieger, Khoa Nguyen, Hexi Ye. View a PDF of ...
  19. [19]
    [1901.09945] Uniform Manin-Mumford for a family of genus 2 curves
    Jan 28, 2019 · Title:Uniform Manin-Mumford for a family of genus 2 curves. Authors:Laura DeMarco, Holly Krieger, Hexi Ye. View a PDF of the paper titled ...
  20. [20]
    Uniform Manin-Mumford for a family of genus 2 curves
    Abstract. We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of ...Missing: contributions algebraic geometry
  21. [21]
    [2412.01668] Hénon maps with many rational periodic points - arXiv
    Dec 2, 2024 · Hénon maps with many rational periodic points. Authors:Hyeonggeun Kim, Holly Krieger, Mara-Ioana Postolache, Vivian Szeto.
  22. [22]
    https://annals.math.princeton.edu/2020/191-3/p05
  23. [23]
    LMS Prize Winners 2020 | London Mathematical Society
    Jun 26, 2020 · Dr Holly Krieger, of the University of Cambridge, is awarded a Whitehead Prize for her deep contributions to arithmetic dynamics, to ...
  24. [24]
    Alexanderson Award 2020 | American Inst. of Mathematics
    They study a family of surfaces in genus 2 that map to pairs of elliptic curves (surfaces of genus 1), and they give a positive answer to Mazur's question for ...
  25. [25]
    https://www.lms.ac.uk/news-entry/26062020-1657/lms-prize-winners-2020
  26. [26]
    Mahler Lecture Tour 2019: The Mathematics of Life
    Nov 25, 2019 · From 22 November to 19 December, Dr Holly Krieger will deliver two public lectures in Melbourne and Queensland and 11 specialist lectures to ...
  27. [27]
    Mahler Lecture: Dr Holly Krieger | School of Mathematics and Statistics
    Nov 22, 2019 · She went on to a master's degree and a Ph.D. from the University of Illinois at Chicago. Her initial research interests during graduate school ...
  28. [28]
    LMS/Gresham Lecture 2023 | London Mathematical Society
    May 24, 2023 · This lecture will be delivered by Holly Krieger, who is the Corfield Lecturer in Mathematics and the Corfield Fellow at Murray Edwards College, ...
  29. [29]
    Invited Speakers - A Closer Look - Joint Mathematics Meetings 2024
    ... algebraic geometry and, quite recently, through a connection with mirror symmetry. ... picture of Holly Krieger. Holly Krieger University of Cambridge ...