Topics in Asymptotic Geometric Analysis and Applications
Topics in Asymptotic Geometric Analysis and Applications
Joscha Prochno
(ORCID: 0000-0002-0750-2850)
Disciplines
Mathematics (100%)
Keywords
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Asymptotic Geometric Analysis,
Tensor products,
Large Deviations,
Schatten classes,
Random matrix,
Central Limit Theorem
The theory of Asymptotic Geometric Analysis is a comparably young and lively mathematical discipline that emerged from the local theory of Banach spaces. It bridges, in essence, three areas of mathematics: functional analysis, convex and discrete geometry, and probability theory, studying systems that involve a huge amount of parameters. The theory demonstrates new and unexpected phenomena characteristic of high dimensions. Considering that high- dimensional systems are very frequent in mathematics and applied sciences today, understanding of those phenomena is becoming increasingly important. The research carried out within this project reflects the broad spectrum of the theory in several ways. Some problems have an analytic flavor, while others are more geometric or probabilistic in nature. However, they all share a lively interplay of geometric, analytic and probabilistic ideas as is typical for the theory. We shall study central limit theorems and large deviations for the volume of random projections of lp-balls, large deviations for random projections of product measures, the geometry of tensor products of lp-spaces, entropy and Gelfand numbers for embeddings of Schatten classes, operator norms of structured random matrices, and look at new applications of asymptotic geometric analysis to information-based complexity, especially minimal dispersion of point sets.
The theory of Asymptotic Geometric Analysis is a comparably young and lively mathematical discipline that emerged from the local theory of Banach spaces. It bridges, in essence, three areas of mathematics: functional analysis, convex and discrete geometry, and probability theory, studying systems that involve a huge amount of parameters. The theory demonstrates new and unexpected phenomena characteristic of high dimensions. Considering that high-dimensional systems are very frequent in mathematics and applied sciences today, understanding of those phenomena is becoming increasingly important. The research carried out within this project reflects the broad spectrum of the theory in several ways. Some problems that have been solved have an analytic flavor, while others are more geometric or probabilistic in nature. However, they all share a lively interplay of geometric, analytic and probabilistic ideas as is typical for the theory. We studied central limit theorems and large deviations for the volume of random projections of lp-balls, large deviations for random projections of product measures, the geometry of tensor products of lp-spaces, entropy and Gelfand numbers for embeddings of Schatten classes, operator norms of structured random matrices, and looked at new applications of asymptotic geometric analysis to information-based complexity, especially minimal dispersion of point sets. Last but not least, we made a number of contributions to the probabilistic theory of lacunary systems.
Research institution(s)
- Universität Passau - 100%
International project participants
- Alexander Litvak, University of Alberta - Canada
- Jan Vybiral, Czech Technical University in Prague - Czechia
- Olivier Guedon, Universite Paris-Est - Marne-la-Vallee - France
- Andreas Defant, Universität Oldenburg - Germany
- Carsten Schütt, Universität Kiel - Germany
- Zakhar Kabluchko, Universität Münster - Germany
- Christoph Thäle, Ruhr-Universität Bochum - Germany
- Erich Novak, Friedrich Schiller Universität Jena - Germany
- Nina Gantert, Technische Universität München - Germany
- Radoslaw Adamczak, University of Warsaw - Poland
- Rafal Latala, University of Warsaw - Poland
- David Alonso-Gutierrez, University of Zaragoza - Spain
- Ramon Van Handel, Princeton University - USA
- Kavita Ramanan, Brown University - USA
- Elisabeth Werner, Case Western Reserve University - USA
- Konstantin Tikhomirov, Georgia Institute of Technology - USA
Research Output
- 123 Citations
- 73 Publications
- 2 Scientific Awards
Publications
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2023
Title Random planar trees and the Jacobian conjecture DOI 10.48550/arxiv.2301.08221 Type Preprint Author Bisi E Link Publication -
2023
Title A probabilistic approach to Lorentz balls DOI 10.48550/arxiv.2303.04728 Type Other Author Kabluchko Z Link Publication -
2022
Title Large deviation principles for lacunary sums DOI 10.1090/tran/8788 Type Journal Article Author Aistleitner C Journal Transactions of the American Mathematical Society Pages 507-553 Link Publication -
2022
Title The large deviation behavior of lacunary sums DOI 10.1007/s00605-022-01733-x Type Journal Article Author Frühwirth L Journal Monatshefte für Mathematik Pages 113-133 Link Publication -
2022
Title Limit theorems for mixed-norm sequence spaces with applications to volume distribution DOI 10.48550/arxiv.2209.08937 Type Preprint Author Juhos M -
2022
Title Hölder's inequality and its reverse-a probabilistic point of view DOI 10.48550/arxiv.2209.13442 Type Preprint Author Frühwirth L -
2024
Title The large and moderate deviations approach in geometric functional analysis DOI 10.48550/arxiv.2403.03940 Type Preprint Author Prochno J Link Publication -
2024
Title Asymptotic theory of Schatten classes DOI 10.48550/arxiv.2404.07145 Type Preprint Author Juhos M Link Publication -
2024
Title Norms of structured random matrices. DOI 10.1007/s00208-023-02599-6 Type Journal Article Author Adamczak R Journal Mathematische annalen Pages 3463-3527 -
2024
Title Brunn-Minkowski Inequality for -Convolution Bodies via Ball's Bodies. DOI 10.1007/s12220-023-01508-2 Type Journal Article Author Alonso-Gutiérrez D Journal Journal of geometric analysis Pages 58 -
2024
Title The Minimal Spherical Dispersion. DOI 10.1007/s12220-023-01527-z Type Journal Article Author Prochno J Journal Journal of geometric analysis Pages 85 -
2020
Title Exact asymptotic volume and volume ratio of Schatten unit balls DOI 10.1016/j.jat.2020.105457 Type Journal Article Author Kabluchko Z Journal Journal of Approximation Theory Pages 105457 Link Publication -
2023
Title Moderate deviation principles and Mod-Gaussian convergence for lacunary trigonometric sums DOI 10.48550/arxiv.2312.09137 Type Preprint Author Prochno J Link Publication -
2023
Title A note on critical intersections of classical and Schatten $p$-balls DOI 10.48550/arxiv.2308.10635 Type Preprint Author Sonnleitner M Link Publication -
2023
Title The Maclaurin inequality through the probabilistic lens DOI 10.48550/arxiv.2312.12134 Type Other Author Frühwirth L Link Publication -
2023
Title Hölder's inequality and its reverse-A probabilistic point of view DOI 10.1002/mana.202200411 Type Journal Article Author Frühwirth L Journal Mathematische Nachrichten -
2023
Title On the power of iid information for linear approximation DOI 10.48550/arxiv.2310.12740 Type Other Author Sonnleitner M Link Publication -
2023
Title Diophantine conditions in the law of the iterated logarithm for lacunary systems DOI 10.48550/arxiv.2310.20257 Type Preprint Author Aistleitner C Link Publication -
2020
Title Limit theorems for random points in a simplex DOI 10.48550/arxiv.2005.04911 Type Preprint Author Baci A -
2020
Title Zur Irrationalität in der Schule DOI 10.48550/arxiv.2005.12238 Type Preprint Author Leobacher G -
2020
Title The maximum entropy principle and volumetric properties of Orlicz balls DOI 10.48550/arxiv.2007.05247 Type Preprint Author Kabluchko Z -
2020
Title Tractability properties of the discrepancy in Orlicz norms DOI 10.1016/j.jco.2020.101468 Type Journal Article Author Dick J Journal Journal of Complexity Pages 101468 Link Publication -
2024
Title Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions DOI 10.1214/22-aihp1340 Type Journal Article Author Kabluchko Z Journal Annales de l'Institut Henri Poincaré, Probabilités et Statistiques -
2024
Title Limit theorems for mixed-norm sequence spaces with applications to volume distribution DOI 10.1214/24-ejp1158 Type Journal Article Author Juhos M Journal Electronic Journal of Probability -
2024
Title Sharp concentration phenomena in high-dimensional Orlicz balls DOI 10.48550/arxiv.2407.15579 Type Preprint Author Frühwirth L Link Publication -
2021
Title The maximum entropy principle and volumetric properties of Orlicz balls DOI 10.1016/j.jmaa.2020.124687 Type Journal Article Author Kabluchko Z Journal Journal of Mathematical Analysis and Applications Pages 124687 Link Publication -
2021
Title Large deviations, moderate deviations, and the KLS conjecture DOI 10.1016/j.jfa.2020.108779 Type Journal Article Author Alonso-Gutiérrez D Journal Journal of Functional Analysis Pages 108779 Link Publication -
2021
Title Sharp concentration for the largest and smallest fragment in a $k$-regular self-similar fragmentation DOI 10.48550/arxiv.2102.08935 Type Preprint Author Dyszewski P -
2021
Title Best and random approximation of a convex body by a polytope DOI 10.48550/arxiv.2111.07306 Type Preprint Author Prochno J -
2021
Title Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions DOI 10.48550/arxiv.2110.12977 Type Preprint Author Kabluchko Z -
2021
Title Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls DOI 10.48550/arxiv.2111.04691 Type Preprint Author Fruehwirth L -
2021
Title Norms of structured random matrices DOI 10.48550/arxiv.2112.14413 Type Preprint Author Adamczak R -
2021
Title The large deviation behavior of lacunary sums DOI 10.48550/arxiv.2107.12860 Type Preprint Author Frühwirth L -
2021
Title Spectral flatness and the volume of intersections of $p$-ellipsoids DOI 10.48550/arxiv.2107.01097 Type Preprint Author Juhos M -
2021
Title Random sections of $\ell_p$-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators DOI 10.48550/arxiv.2109.14504 Type Preprint Author Hinrichs A -
2022
Title A probabilistic way to discover the rainbow DOI 10.1017/mag.2022.15 Type Journal Article Author Prochno J Journal The Mathematical Gazette Pages 103-115 Link Publication -
2022
Title Best and random approximation of a convex body by a polytope DOI 10.1016/j.jco.2022.101652 Type Journal Article Author Prochno J Journal Journal of Complexity Pages 101652 Link Publication -
2022
Title Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings DOI 10.1016/j.jat.2022.105736 Type Journal Article Author Prochno J Journal Journal of Approximation Theory Pages 105736 Link Publication -
2022
Title Thin-shell theory for rotationally invariant random simplices DOI 10.1214/21-ejp734 Type Journal Article Author Heiny J Journal Electronic Journal of Probability Pages 1-41 Link Publication -
2022
Title Projections of the uniform distribution on the cube: a large deviation perspective DOI 10.4064/sm210413-16-9 Type Journal Article Author Johnston S Journal Studia Mathematica Pages 103-119 Link Publication -
2022
Title Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures DOI 10.1016/j.jfa.2021.109291 Type Journal Article Author Alonso-Gutiérrez D Journal Journal of Functional Analysis Pages 109291 Link Publication -
2022
Title Spectral flatness and the volume of intersections of p-ellipsoids DOI 10.1016/j.jco.2021.101617 Type Journal Article Author Juhos M Journal Journal of Complexity Pages 101617 Link Publication -
2022
Title Faà di Bruno's formula and inversion of power series DOI 10.1016/j.aim.2021.108080 Type Journal Article Author Johnston S Journal Advances in Mathematics Pages 108080 Link Publication -
2022
Title Limit theorems for random points in a simplex DOI 10.1017/jpr.2021.77 Type Journal Article Author Baci A Journal Journal of Applied Probability Pages 685-701 Link Publication -
2022
Title Sharp concentration for the largest and smallest fragment in a k-regular self-similar fragmentation DOI 10.1214/21-aop1556 Type Journal Article Author Dyszewski P Journal The Annals of Probability Link Publication -
2023
Title A Maxwell principle for generalized Orlicz balls DOI 10.1214/22-aihp1298 Type Journal Article Author Johnston S Journal Annales de l'Institut Henri Poincaré, Probabilités et Statistiques -
2023
Title On the power of iid information for linear approximation DOI 10.30970/ana.2023.1.88 Type Journal Article Author Sonnleitner M Journal Journal of Applied and Numerical Analysis -
2020
Title Large deviations, moderate deviations, and the KLS conjecture DOI 10.48550/arxiv.2003.11442 Type Preprint Author Alonso-Gutiérrez D -
2020
Title Gelfand numbers of embeddings of Schatten classes DOI 10.48550/arxiv.2011.06554 Type Preprint Author Hinrichs A -
2020
Title Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures DOI 10.48550/arxiv.2011.07523 Type Preprint Author Alonso-Gutiérrez D -
2020
Title A probabilistic way to discover the rainbow DOI 10.48550/arxiv.2011.01030 Type Preprint Author Prochno J -
2020
Title A Maxwell principle for generalized Orlicz balls DOI 10.48550/arxiv.2012.11568 Type Preprint Author Johnston S -
2020
Title Large Deviation Principles for Lacunary Sums DOI 10.48550/arxiv.2012.05281 Type Preprint Author Aistleitner C -
2020
Title Intersection of unit balls in classical matrix ensembles DOI 10.1007/s11856-020-2052-6 Type Journal Article Author Kabluchko Z Journal Israel Journal of Mathematics Pages 129-172 -
2020
Title Statistical independence in mathematics–the key to a Gaussian law DOI 10.1007/s00591-020-00287-z Type Journal Article Author Leobacher G Journal Mathematische Semesterberichte Pages 69-104 Link Publication -
2020
Title On the relation of the spectral test to isotropic discrepancy and $L_q$-approximation in Sobolev spaces DOI 10.48550/arxiv.2010.04522 Type Preprint Author Sonnleitner M -
2019
Title High-dimensional limit theorems for random vectors in lpn-balls. II DOI 10.1142/s0219199719500731 Type Journal Article Author Kabluchko Z Journal Communications in Contemporary Mathematics Pages 1950073 -
2022
Title Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies DOI 10.48550/arxiv.2211.17069 Type Preprint Author Alonso-Gutiérrez D -
2021
Title On the relation of the spectral test to isotropic discrepancy and L q -approximation in Sobolev spaces DOI 10.1016/j.jco.2021.101576 Type Journal Article Author Sonnleitner M Journal Journal of Complexity Pages 101576 Link Publication -
2021
Title Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings DOI 10.48550/arxiv.2103.13050 Type Preprint Author Prochno J -
2021
Title The minimal spherical dispersion DOI 10.48550/arxiv.2103.11701 Type Preprint Author Prochno J -
2021
Title Thin-shell theory for rotationally invariant random simplices DOI 10.48550/arxiv.2103.11872 Type Preprint Author Heiny J -
2021
Title Projections of the uniform distribution on the cube -- a large deviation perspective DOI 10.48550/arxiv.2103.16430 Type Preprint Author Johnston S -
2021
Title A new look at random projections of the cube and general product measures DOI 10.3150/20-bej1303 Type Journal Article Author Kabluchko Z Journal Bernoulli Link Publication -
2021
Title Gelfand numbers of embeddings of Schatten classes DOI 10.1007/s00208-021-02203-9 Type Journal Article Author Hinrichs A Journal Mathematische Annalen Pages 1563-1593 -
2021
Title Random sections of ellipsoids and the power of random information DOI 10.1090/tran/8502 Type Journal Article Author Hinrichs A Journal Transactions of the American Mathematical Society Pages 8691-8713 Link Publication -
2025
Title A general functional version of Grünbaum's inequality DOI 10.1016/j.jmaa.2024.129065 Type Journal Article Author Alonso-Gutiérrez D Journal Journal of Mathematical Analysis and Applications -
2019
Title Independence in Mathematics -- the key to a Gaussian law DOI 10.48550/arxiv.1911.07733 Type Preprint Author Leobacher G -
2019
Title Faà di Bruno's formula and inversion of power series DOI 10.48550/arxiv.1911.07458 Type Preprint Author Johnston S -
2019
Title High-dimensional limit theorems for random vectors in $\ell_p^n$-balls. II DOI 10.48550/arxiv.1906.03599 Type Preprint Author Kabluchko Z -
2019
Title A new look at random projections of the cube and general product measures DOI 10.48550/arxiv.1910.02676 Type Preprint Author Kabluchko Z -
2019
Title Berry-Esseen bounds for random projections of $\ell_p^n$-balls DOI 10.48550/arxiv.1911.00695 Type Preprint Author Johnston S -
2019
Title Tractability properties of the discrepancy in Orlicz norms DOI 10.48550/arxiv.1910.12571 Type Preprint Author Dick J
Scientific Awards
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2022
Title Joseph F. Traub Information-Based Complexity Young Researcher Award Type Research prize Level of Recognition Continental/International -
2021
Title Prize of the Austrian Mathematical Society Type Research prize Level of Recognition National (any country)