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Pieter Allaart

Professor

University of North Texas

Department of Mathematics

(940) 369-7313

Email: allaart@unt.edu

Education

  • PhD, Vrije Universiteit Amsterdam, 1998
    Major: Mathematics & Computer Science
    Dissertation: Ranges of vector measures and optimal-partitioning inequalities
  • MA, Free University, 1994
    Major: Mathematics; Master's thesis: A constant-sum characterization of Benford's Law

Professional Positions

  • Associate Professor, University of North Texas. University of North Texas. (2007 - Present).
  • GCOE Foreign Researcher, Kyoto University. Kyoto University. (2009 - 2009).
  • Assistant Professor, University of North Texas. University of North Texas. (2001 - 2007).
  • Visiting Assistant Professor, University of North Texas. University of North Texas. (1998 - 2001).

Teaching

Teaching Experience

    University of North Texas

  • MATH 1680 - Elementary Probability and Statistics, 2 courses.
  • MATH 1710 - Calculus I, 2 courses.
  • MATH 1720 - Calculus II, 2 courses.
  • MATH 1780 - Probability Models, 2 courses.
  • MATH 2510 - Real Analysis I, 2 courses.
  • MATH 2700 - Linear Algebra and Vector Geometry, 9 courses.
  • MATH 2730 - Multivariable Calculus, 7 courses.
  • MATH 3000 - Real Analysis I, 4 courses.
  • MATH 3010 - Seminar in Problem Solving Techniques, 8 courses.
  • MATH 3180 - Probability for Engineers, 2 courses.
  • MATH 3410 - Differential Equations I, 7 courses.
  • MATH 3510 - Abstract Algebra I, 3 courses.
  • MATH 3680 - Applied Statistics, 4 courses.
  • MATH 4610 - Probability, 10 courses.
  • MATH 4650 - Statistics, 2 courses.
  • MATH 4900 - Special Problems, 5 courses.
  • MATH 5110 - Introduction to Analysis, 2 courses.
  • MATH 5120 - Introduction to Analysis, 2 courses.
  • MATH 5310 - Real Analysis, 5 courses.
  • MATH 5320 - Real Analysis, 7 courses.
  • MATH 5350 - Markov Processes, 1 course.
  • MATH 5810 - Probability and Statistics, 5 courses.
  • MATH 5820 - Probability and Statistics, 1 course.
  • MATH 5900 - Special Problems, 25 courses.
  • MATH 5910 - Special Problems, 1 course.
  • MATH 6810 - Probability., 6 courses.
  • MATH 6900 - Special Problems, 10 courses.
  • MATH 6940 - Individual Research, 1 course.
  • MATH 6950 - Doctoral Dissertation, 27 courses.

Directed Student Learning

  • Master's Thesis Committee Chair, Mathematics. (February 2025).
  • Doctoral Advisory Committee Member, Mathematics. (January 2025).
  • Directed Individual/Independent Study, Mathematics. (August 2024 - December 2024).
  • Directed Individual/Independent Study, Mathematics. (August 2024 - December 2024).
  • Master's Thesis Committee Chair, Mathematics. (January 2024 - June 2024).
  • Directed Individual/Independent Study, Mathematics. (January 2024 - May 2024).
  • Directed Individual/Independent Study, Mathematics. (August 2023 - December 2023).
  • Other (Helped a new graduate student prepare for an NSF Graduate Research Fellowship application), Mathematics. (October 2023).
  • Doctoral Advisory Committee Chair, "Dimensions of statistically self-affine functions and random Cantor sets," Mathematics. (May 2023).
  • Directed Individual/Independent Study, Mathematics. (August 2022 - December 2022).
  • Directed Individual/Independent Study, "Groups as unions of proper subgroups," Mathematics. (September 2021 - November 2021).
  • Directed Individual/Independent Study, Mathematics. (January 2021 - August 2021).
  • Doctoral Advisory Committee Chair, "Optimal look-ahead stopping rules for simple random walk," Mathematics. (June 2021).
  • Directed Individual/Independent Study, "N/A," Mathematics. (January 2021 - May 2021).
  • Directed Individual/Independent Study, Mathematics. (July 2020 - December 2020).
  • Other (Dissertation Research Hours), Mathematics. (August 2019 - December 2020).
  • Directed Individual/Independent Study, Mathematics. (January 2020 - May 2020).
  • Doctoral Advisory Committee Chair, "Prophet inequalities for multivariate random variables with cost for observations," Mathematics. (May 2019).
  • Other (Dissertation Research Hours), Mathematics. (January 2019 - May 2019).
  • Directed Individual/Independent Study, Mathematics. (August 2018 - December 2018).
  • Other (Dissertation Research Hours), Mathematics. (August 2018 - December 2018).
  • Doctoral Advisory Committee Chair, "A random walk version of Robbins' problem," Mathematics. (October 2018).
  • Other (Dissertation Research Hours), Mathematics. (January 2018 - January 2018).
  • Other (Dissertation Research Hours), Mathematics. (August 2017 - December 2017).
  • Dissertation Committee Member, Mathematics. (June 2017).
  • Directed Individual/Independent Study, Mathematics. (January 2017 - May 2017).
  • Other (Dissertation Research Hours), Mathematics. (January 2017 - May 2017).
  • Doctoral Advisory Committee Chair, Mathematics. (December 2015).
  • Major Professor, Mathematics. (2015).
  • Major Professor, Mathematics. (2014).
  • Major Professor, "Project title: The optimal stopping time for the American call/put option using a multiplicative random walk," Mathematics. (2008 - 2008).

Research

Published Intellectual Contributions

    Journal Article

  • Allaart, P.C., Kong, D. (2023). Critical values for the beta-transformation with a hole at 0. Ergodic Theory and Dynamical Systems. 43 (6) 1785-1828.
  • Allaart, P.C., Jackson, S.C., Jones, R.T., Lambert, D. (2023). On the existence of numbers with matching continued fraction and decimal expansions. Monatshefte für Mathematik. 202 (1) 1-30.
  • Allaart, P., Jones, R.T. (2023). Box-counting dimension and differentiability of box-like statistically self-affine functions. Journal of Mathematical Analysis and Applications. 521 (2) Article 126909, 20 pages.
  • Allaart, P.C., Kong, D. (2022). Density spectrum of Cantor measure. Nonlinearity. 35 (12) 6453–6484.
  • Allaart, P.C., Kong, D. (2021). Relative bifurcation sets and the local dimension of univoque bases. Ergodic Theory and Dynamical Systems. 41 (8) 2241-2273.
  • Allaart, P.C., Kong, D. (2021). On the smallest base in which a number has a unique expansion. Transactions of the American Mathematical Society. 374 (9) 6201-6249.
  • Norouzi Kandalan, R., Allaart, P.C., Namuduri, K. (2020). Mobility Accelerates Consensus-Building in Sensor Networks.. IEEE Sensors Letters. 4 (12) 1-4.
  • Allaart, P.C. (2020). The pointwise Holder spectrum of general self-affine functions on an interval.. Journal of Mathematical Analysis and Applications. 488 (2) Article 124096, 35 pages. Elsevier.
  • Allaart, P., Allen, A. (2019). A random walk version of Robbins' problem: small horizon. Mathematica Applicanda. 47 (2) 293-312.
  • Allaart, P. (2019). An algebraic approach to entropy plateaus in non-integer base expansions. Discrete and Continuous Dynamical Systems. 39 (11) 6507-6522.
  • Allaart, P., Kong, D. (2019). On the continuity of the Hausdorff dimension of the univoque set. Advances in Mathematics. 354 24 pages.
  • Allaart, P., Kong, D., Baker, S. (2019). Bifurcation sets arising from non-integer base expansions. Journal of Fractal Geometry. 6 (4) 301-341.
  • Allaart, P. (2018). Differentiability and Holder spectra of a class of self-affine functions. Advances in Mathematics. 328 1-39.
  • Allaart, P. (2017). Differentiability of a two-parameter family of self-affine functions. Journal of Mathematical Analysis and Applications. 450 954-968.
  • Allaart, P. (2017). On univoque and strongly univoque sets. Advances in Mathematics. 308 575-598.
  • Allaart, P., Islas, J.A. (2016). A sharp lower bound for choosing the maximum of an independent sequence. Journal of Applied Probability. 53 (4) 1041-1051.
  • Allaart, P. (2016). The infinite derivatives of Okamoto's function: an application of beta-expansions. Journal of Fractal Geometry. 3 (1) 1-31.
  • P. Allaart. (2014). Correction and strengthening of ``How large are the level sets of the Takagi function?".
  • P. Allaart. (2014). Digital sum inequalities and approximate convexity of Takagi-type functions.
  • P. Allaart. (2014). Hausdorff dimension of level sets of generalized Takagi functions.
  • Allaart, P. (2014). On the level sets of the Takagi-van der Waerden functions.
  • P. Allaart. (2013). Level sets of signed Takagi functions. Acta Mathematica Hungarica.
  • Allaart, P. (2012). How large are the level sets of the Takagi function?.
  • Allaart, P. (2012). Predicting the supremum: optimality of "stop at once or not at all".
  • Allaart, P. (2012). The finite cardinalities of level sets of the Takagi function.
  • with K. Kawamura. (2012). The Takagi function: a survey.
  • Allaart, P. (2011). An inequality for sums of binary digits, with application to Takagi functions http://arxiv.org/abs/1009.1308 Related Links: http://arxiv.org/abs/1009.1308.
  • P. Allaart and K. Kawamura. (2011). Takagi function: Survey. Denton,
  • Allaart, P. (2010). A general `bang-bang' principle for predicting the maximum of a random walk..
  • Allaart, P. (2010). How to stop near the top in a random walk.
  • Allaart, P. (2010). Optimal stopping rules for American and Russian options in a correlated random walk model..
  • with K. Kawamura. (2010). The improper infinite derivatives of Takagi's nowhere-differentiable function.
  • Allaart, P. (2009). A sharp ratio inequality for optimal stopping when only relative record times are observed.
  • with R. D. Mauldin. (2009). Injectivity of the Dubins-Freedman construction of random distributions.
  • Allaart, P. (2009). On a flexible class of continuous functions with uniform local structure.
  • Allaart, P. (2008). Distribution of the extrema of random Takagi functions.
  • with M. Monticino. (2008). Optimal buy/sell strategies for directionally reinforced processes..
  • with K. Kawamura. (2007). Dimensions of the coordinate functions of space-filling curves.
  • Allaart, P. (2007). Prophet inequalities for i.i.d. random variables with random arrival times.
  • with K. Kawamura. (2006). Extreme values of some continuous, nowhere differentiable functions.
  • Allaart, P. (2006). Prophet regions for discounted, uniformly bounded random variables.
  • with K. Kawamura. (2005). On the coordinate functions of L´evy's dragon curve.
  • Allaart, P. (2005). Prophet regions for independent [0, 1]-valued random variables with random dis- counting.
  • Allaart, P. (2004). An application of prophet regions to optimal stopping with a random number of observations.
  • Allaart, P. (2004). Optimal stopping rules for correlated random walks with a discount.
  • Allaart, P. (2004). Stopping the maximum of a correlated random walk, with cost for observation.
  • Allaart, P. (2003). Moments of the mean of Dubins-Freedman random probability distributions.
  • with M. Monticino. (2003). Pseudo-prophet inequalities in average-optimal stopping.
  • with M. Monticino. (2001). Optimal stopping rules for directionally reinforced processes.
  • Allaart, P. (2000). Inequalities relating maximal moments to other measures of dispersion.
  • Allaart, P. (1999). A sharp non-convexity bound for partition ranges of vector measures with atoms.
  • Allaart, P. (1999). Bounds on the non-convexity of ranges of vector measures with atoms.
  • Allaart, P. (1998). Minimax risk inequalities for the location-parameter classification problem.
  • Allaart, P. (1997). An invariant-sum characterization of Benford's law. Journal of Applied Probability. 34 288-291.

Presentations Given

  • Allaart, P. (Other), On the level sets of the Takagi function, (2011 - 2011).
  • Allaart, P. (Other), Optimal prediction of the maximum of a stochastic process, (2011 - 2011).
  • Allaart, P. (Other), The improper infinite derivatives of Takagi's nowhere-differentiable function, (2010 - 2010).
  • Allaart, P. (Other), General secretary-type problems for random walk, (2009 - 2009).
  • Allaart, P. (Other), Injectivity of the Dubins-Freedman construction of random distributions, (2009 - 2009).
  • Allaart, P. (Other), Maximum and minimum values of some continuous nowhere-differentiable functions, (2009 - 2009).
  • Allaart, P. (Other), On generalized Takagi functions, (2009 - 2009).
  • Allaart, P. (Other), Optimal exercise times for American and Russian options under a correlated random walk model, (2009 - 2009).
  • Allaart, P. (Other), Injectivity of the Dubins-Freedman construction of random distributions, (2008 - 2008).
  • Allaart, P. (Other), Injectivity of the Dubins-Freedman construction of random distributions, a followup, (2008 - 2008).
  • Allaart, P. (Other), Inequalities for optimal stopping with unequal information, (2007 - 2007).
  • Allaart, P. (Other), Dimension estimates for the coordinate functions of P´olya's space-filling curve, (2006 - 2006).
  • Allaart, P. (Other), Prophet inequalities for i.i.d. random variables with random arrival times, (2006 - 2006).
  • Allaart, P. (Other), Prophet inequalities for i.i.d. random variables with random arrival times, (2006 - 2006).
  • Allaart, P. (Other), Prophet regions for uniformly bounded random variables with random discounting, (2005 - 2005).
  • Allaart, P. (Other), The coordinate functions of L´evy dragon curve, (2005 - 2005).
  • Allaart, P. (Other), Stopping the maximum of a correlated random walk, with application to Russian options, (2004 - 2004).
  • Allaart, P. (Other), Introduction to optimal stopping, (2003 - 2003).
  • Allaart, P. (Other), Optimal stopping rules for correlated random walks with a discount, (2002 - 2002).
  • Allaart, P. (Other), Optimal stopping rules for directionally reinforced processes, (2001 - 2001).
  • Allaart, P. (Other), Random probability distributions and average-optimal stopping, (2001 - 2001).
  • Allaart, P. (Other), A generalization of Lyapounov's convexity theorem to measures with atoms, (1999 - 1999).
  • Allaart, P. (Other), Maximal moments as a measure of dispersion, (1999 - 1999).
  • Allaart, P. (Other), Some old and some new maximal moment inequalities, (1999 - 1999).
  • Allaart, P. (Other), Cake cutting, L´evy concentration and the location-parameter classification problem, (1997 - 1997).
  • Allaart, P. (Other), Two sharp inequalities related to measure-partitioning in statistics and fair division theory, (1997 - 1997).
  • Allaart, P. (Other), Lyapounov's convexity theorem and the classification problem, (1996 - 1996).
  • Allaart, P. (Other), The first digit problem, (1995 - 1995).

    • Invited Talk

  • Allaart, P. (Author & Presenter), Analysis Seminar, Random subsets of self-similar sets, University of St. Andrews, St. Andrews, United Kingdom. (2025).
  • Allaart, P. (Author & Presenter), Jones, T. (Author), AMS Central Sectional Meeting, Random subsets of Cantor sets generated by trees of coin flips, American Mathematical Society, San Antonio, TX, United States of America. (2024 - 2024).
  • Allaart, P. (Author & Presenter), AMS Southeastern Sectional Meeting, The beta-transformation with a hole at zero, American Mathematical Society, Mobile, AL, United States of America. (2023).
  • Allaart, P. (Author & Presenter), Chongqing University Dynamical Systems Seminar, Differentiability and Holder spectrum of a class of self-affine functions, Chongqing University, online, China. (2023 - 2023).
  • Allaart, P. (Author & Presenter), American Mathematical Society Western Sectional Meeting, Density spectrum of Cantor measure, American Mathematical Society, online, United States of America. (2022 - 2022).
  • Allaart, P.C. (Author & Presenter), One World Numeration Seminar (online), On the existence of Trott numbers relative to multiple bases, online, United States of America. (2021 - 2021).
  • Allaart, P.C. (Author & Presenter), One World Numeration Seminar, On the smallest base in which a number has a unique expansion, Online, France. (2020).
  • Allaart, P.C. (Author & Presenter), Probability Seminar, On univoque and strongly univoque sets, University of Waterloo, Waterloo, Ontario (virtual seminar), Canada. (2020).
  • Allaart, P.C. (Author & Presenter), American Mathematical Society Southeastern Sectional Meeting, The pointwise Holder spectrum of self-affine functions, American Mathematical Society, Gainesville, Florida, United States of America. (2019).
  • Allaart, P.C. (Author & Presenter), Kyoto University Dynamical Systems Seminar, The pointwise Holder spectrum of self-affine functions, Kyoto, Japan. (2019).
  • Allaart, P.C. (Author & Presenter), Tsukuba University Number Theory Seminar, On univoque and strongly univoque sets, Tsukuba, Japan. (2019).
  • Allaart, P. (Author & Presenter), 60 years of dynamics and number expansions, On the local dimension of the univoque set, Pisa, Italy. (2018).
  • Allaart, P. (Author & Presenter), Symposium on Optimal Stopping, Some difficult to solve optimal stopping problems, Rice University, Houston, TX, United States of America. (2018).
  • Allaart, P. (Author & Presenter), Ergodic Theory Seminar, On univoque and strongly univoque sets, Warwick University (U.K.), United Kingdom. (2018).
  • Allaart, P. (Author & Presenter), Ergodic Theory Seminar, On univoque and strongly univoque sets, Leiden University, Netherlands. (2018).
  • Allaart, P. (Author & Presenter), Workshop on Ergodic Theory and Fractals, Differentiability and Holder spectra of a class of self-affine functions, Utrecht, Netherlands. (2018).
  • Allaart, P. (Author & Presenter), AMS Central Sectional Meeting, Differentiability and Holder spectra of a class of self-affine functions, American Mathematical Society, Denton, TX, United States of America. (2017).
  • Allaart, P. (Author & Presenter), Real Analysis Symposium, Differentiability and Holder spectra of a class of self-affine functions, Wooster, Ohio, United States of America. (2017 - 2017).

    • Oral Presentation

  • Allaart, P. (Author & Presenter), Jones, T. (Author), Geometry and Fractals under the Midnight Sun, Random subsets of Cantor sets generated by trees of coin flips, University of Oulu, Oulu, Finland. (2024 - 2024).
  • Allaart, P. (Author & Presenter), Geometry of Deterministic and Random Fractals, Density spectrum of Cantor measure, Budapest University of Engineering and Economics, Budapest, Hungary. (2022 - 2022).
  • Allaart, P. (Author & Presenter), AMS Southeastern Sectional Meeting, The infinite derivatives of Okamoto's function and beta-expansions, American Mathematical Society, Memphis, TN, United States of America. (2015).
  • Allaart, P. (Author & Presenter), Kyoto University Dynamical Systems Seminar, The infinite derivatives of Okamoto's function and beta-expansions, Kyoto, Japan. (2015).
  • Allaart, P. (Author & Presenter), AMS Sectional Meeting, Zero sets and maximum sets of randomized Takagi functions, American Mathematical Society, Lubbock, TX, United States of America. (2014).
  • Allaart, P. (Author & Presenter), Real Analysis Symposium, Zero sets and maximum sets of randomized Takagi functions, Prague, Czech Republic. (2014).
  • Allaart, P. (Author & Presenter), AMS Southeastern Sectional Meeting, The Hausdorff dimension of level sets of generalized Takagi functions, American Mathematical Society, Oxford, MS, United States of America. (2013).
  • Allaart, P. (Author & Presenter), Kyoto University Mathematics Department Colloquium, A new look at the Takagi function, Kyoto, Japan. (2012).
  • Allaart, P. (Author & Presenter), RTG Conference on logic, dynamics and their interactions, with a celebration of the work of Dan Mauldin, The Hausdorff dimension of level sets of generalized Takagi functions, Denton, TX, United States of America. (2012).
  • Allaart, P. (Author & Presenter), XVIII SIMMAC (International Symposium on Mathematical Methods Applied to the Sciences), Optimal prediction of the maximum of a stochastic process, San Jose, Costa Rica, Costa Rica. (2012).

Contracts, Grants, Sponsored Research

    Grant - Research

  • Allaart, P.C. (Principal), "Non-integer base expansions and multifractal analysis," sponsored by Simons Foundation, Private, $42000 Funded. (2020 - 2025).
  • Allaart, P. (Principal), Lazebnik, K.Y. (Co-Principal), Kawamura, K. (Co-Principal), "Conference: Dynamical Systems and Fractal Geometry," sponsored by National Science Foundation, Federal, $32017 Funded. (2024 - 2025).
  • Allaart, P. (Principal), "Visitor grant for a 2.5 month research visit to Utrecht University," sponsored by NWO (Dutch National Science Foundation), International, $7000 Funded. (2018 - 2018).
  • Allaart, P. (Principal), Schmidt, R. (Principal), "Non-integer base expansions and multifractal analysis," sponsored by Simons Foundation, FOND, Funded. (2020 - 2025).
  • Allaart, P.C. (Principal), "Visitor grant for a 2.5 month research visit to Utrecht University," sponsored by NWO (Dutch National Science Foundation), International, Funded. (2018 - 2018).

Service

University Service

  • Committee Member, Probability and Statistics Qualifying Exam Committee. (June 2018 - Present).
  • Committee Member, Faculty Council. (September 2021 - August 2024).
  • Chairperson, Real Analysis Qualifying Exam Committee. (June 2023 - May 2024).
  • Committee Member, Executive Committee. (August 2022 - May 2024).
  • Member, Administrator Evaluation Committee. (October 2023 - December 2023).
  • Chairperson, Faculty Council. (August 2022 - June 2023).
  • Member, Chair Search Committee. (January 2023 - May 2023).
  • Committee Member, Real Analysis Qualifying Exam Committee. (June 2021 - May 2023).
  • Member, Statistics Search Committee. (January 2023 - March 2023).
  • Committee Member, Applied PDE search committee. (January 2022 - March 2022).
  • Committee Chair, Graduate Affairs Committee. (June 2020 - May 2021).
  • Committee Member, Executive Committee. (May 2019 - May 2021).
  • Committee Member, Faculty Senate Committee on Teaching Fellows and Teaching Assistants. (September 2015 - May 2021).
  • Committee Member, Graduate Affairs Committee. (June 2012 - June 2020).
  • Committee Chair, Ad-hoc grade appeal committee. (June 2019 - July 2019).
  • Committee Chair, Graduate Affairs Committee. (January 2019 - June 2019).
  • Committee Member, Applied PDE Search Committee. (September 2018 - May 2019).
  • Committee Member, Real Analysis Qualifying Exam Committee. (June 2017 - May 2018).
  • Committee Chair, Graduate Affairs Committee. (August 2017 - December 2017).
  • Committee Member, Probability and Statistics Qualifying Exam Committee. (June 2015 - May 2017).
  • Committee Member, Strategic Planning Committee. (June 2013 - May 2017).
  • Committee Chair, Real Analysis Qualifying Exam Committee. (June 2015 - May 2016).
  • Committee Chair, Graduate Affairs Committee. (2011 - 2013).
  • Other, Executive Committee. (2010 - 2012).
  • Committee Chair, Harmonic Analysis and Differential Equations Search Committee. (2007 - 2008).
  • Other, Undergraduate Curriculum Committee. (2006 - 2008).
  • Other, Undergraduate Affairs Committee. (2002 - 2008).
  • Other (Coordinator), UNT Stochastics Seminar. (2001 - 2005).

Professional Service

  • Reviewer, Journal Article, Refereeing. (2022 - 2024).
  • Workshop Organizer, Conference on Dynamical Systems and Fractal Geometry. Denton, TX. (June 2022 - May 2024).
  • Reviewer, Journal Article, American Mathematical Societry Math Reviews. (2005 - June 2020).
  • Reviewer, Book, American Mathematical Societry Math Reviews. (2018 - 2018).
  • Reviewer, Grant Proposal, Grant proposal to the Israel Science Foundation. (2007 - 2007).