Viggo Brun Prize 2022

The Viggo Brun Prize for 2022 is awarded to:

Nina Holden

for her exceptionally deep and broad contributions to probability theory, especially for her work on random surfaces and quantum gravity in two dimensions.

Nina Holden (born 1986) is a Norwegian mathematician with MSc from the University of Oslo in 2010 and PhD from the Massachusetts Institute of Tech- nology in 2018. She is currently a Post-Doc at ETH Zu ̈rich and has accepted a position as Associate Professor at the Courant Institute of Mathematical Sciences, New York University, starting in fall 2022. In 2021 Nina Holden was honored with the Maryam Mirzakhani New Frontiers Prize.

Holden’s research field is probability theory. She has successfully pur- sued an impressive number of topics: Liouville Quantum Gravity (LQG), Schramm–Loewner evolutions (SLE), random planar maps, statistical physics, data reconstruction, graphons, fine properties of Brownian motion, the Schelling model, and more. Her results are of theoretical nature, but have important implications in theoretical physics (conformal field theory, string theory, the understanding of gravity), as well as real-world phenomena such as phase transitions for random systems. Her proofs involve a wide variety of different techniques and ideas.

Holden’s perhaps most remarkable achievement, for which she was awarded the Maryam Mirzakhani New Frontiers Prize, concerns the convergence of uniform random planar triangulations (URPT) to LQG. Physicists have be- lieved, since at least the 1980s, that if one takes an appropriate limit of large number of vertices of the URPT, one should obtain some form of convergence to LQG. In other words, just as discrete random walks converge to contin- uum random paths of Brownian motion, it should be the case that discrete random surfaces converge to continuum random surfaces.

In joint work with Xin Sun Holden establishes this fact in a very precise way. They prove that if uniformly random planar triangulations are embed- ded in the plane via the so called Cardy embedding, then the embedded map converges to LQG. This result and various ramifications span six publica- tions, with several co-authors, that are noteworthy in their own right and that have been published or accepted for publication in highest level journals such as Memoirs of the American Mathematical Society, Annales de l’Institut Henri Poincar ́e and Acta Mathematica.

In addition to the monumental results related to the Cardy embedding, Holden also has several other impressive achievements, in particular within the broad subject of conformal probability, from which we give some samples:

In A distance exponent for Liouville quantum gravity, with E. Gwynne and X. Sun, Probability Theory and Related Fields (2018), she has made some of the most important progress on the still somewhat mysterious problem of understanding distance functions on Liouville quantum gravity surfaces.

The paper SLE as a mating of trees in Euclidean geometry, with X. Sun, Communications in Mathematical Physics (2018), explains a fundamental property of the scaling limit of a uniformly random spanning tree, namely that it is in some sense determined by the structure of the graph itself.

The paper An almost sure KPZ relation for SLE and Brownian motion, with E. Gwynne and J. Miller, Annals of Probability (2020), provides a very general way to understand dimensions of random fractals arising in conformal probability, many related to statistical physics models.

In Brownian motion correlation in the peanosphere for κ > 8, with E. Gwynne, J. Miller and X. Sun, Annales de l’Institut Henri Poincar ́e (2017), an answer is given to a fundamental question asked by Miller and Sheffield in one of their papers.

The paper Conformal welding for critical Liouville quantum gravity, with E. Powell, Annales de l’Institut Henri Poincar ́e (2021), solves a problem about SLE and the conformal welding theory of LQG surfaces by extending the γ < 2 theory to the critical γ = 2 case.

In Dimension transformation formula for conformal maps into the com- plement of an SLE curve, with E. Gwynne and J. Miller, Probability Theory and Related Fields (2019), a formula is proved relating the Hausdorff dimen- sion of a deterministic Borel subset of R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an SLE curve.

Outside of conformal probability, Holden has solved, in How round are the complementary components of planar Brownian motion?, with S. Na ̧cu, Y. Peres,and T. S. Salisbury, Annales de l’Institut Henri Poincar ́e (2019), one of the more fundamental questions about the complementary compo- nents of planar Brownian motion, namely the extent to which they tend to be “round”, that is, have area on the same order of the square of their diameter. In the well-cited and recognized Sparse exchangeable graphs and their limits via graphon processes, with C. Borgs, J. T. Chayes and H. Cohn, Journal of Machine Learning Research (2018), so–called graphons are stud- ied, which constitute an approach to large dense graphs. In addition, Holden has obtained interesting results on trace reconstruction problems for random strings and deletion probabilities in Trace reconstruction with varying dele- tion probabilities, with L. Hartung and Y. Peres, Analytic Algorithmics and Combinatorics (ANALCO) (2018), and in Subpolynomial trace reconstruction for random strings and arbitrary deletion probability, with R. Pemantle and Y. Peres, Conference On Learning Theory (COLT) (2018). The problems in this area involve the extent to which one can recover information about a string of characters when some of the characters have been randomly re- moved.

In conclusion, Holden’s research record is truly astonishing, with papers of exceptional depth and breadth published in first class journals, and she has very rapidly established herself as one of the leading researchers in probability theory at highest international level.