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Properties of High Rank Subvarieties of Affine Spaces 3
PROPERTIES OF HIGH RANK SUBVARIETIES OF AFFINE SPACES DAVID KAZHDAN AND TAMAR ZIEGLER Abstract. We use tools of additive combinatorics for the study of subvarieties defined by high rank families of polynomials in high dimensional Fq-vector spaces. In the first, analytic part of the paper we prove a number properties of high rank systems of polynomials. In the second, we use these properties to deduce results in Algebraic Geometry , such as an effective Stillman conjecture over algebraically closed fields, an analogue of Nullstellensatz for varieties over finite fields, and a strengthening of a recent result of [5]. We also show that for k-varieties X ⊂ An of high rank any weakly polynomial function on a set X(k) ⊂ kn extends to a polynomial. 1. Introduction Let k be a field. For an algebraic k-variety X we write X(k) := X(k). To simplify notations we often write X instead of X(k). In particular we write V := V(k) when V is a vector space and write kN for AN (k). For a k-vector space V we denote by Pd(V) the algebraic variety of polynomials on V of degree ≤ d and by Pd(V ) the set of polynomials functions P : V → k of degree ≤ d. We always assume that d< |k|, so the restriction map Pd(V)(k) → Pd(V ) is a bijection. For a family P¯ = {Pi} of polynomials on V we denote by X V P¯ ⊂ the subscheme defined by the ideal generated by {Pi} and by XP¯ the set XP (k) ⊂ V . We will not distinguish between the set of affine k-subspaces of V and the set of affine subspaces of V since for an affine k-subspace W ⊂ V, the map W → W(k) is a bijection. -
Arxiv:1910.04675V1 [Math.DS] 10 Oct 2019 UNIAIEDSONNS FNILFLOWS of DISJOINTNESS QUANTITATIVE Ekts,Cmie Ihtesrcua Hoesfrnlosby Ziegler
QUANTITATIVE DISJOINTNESS OF NILFLOWS FROM HOROSPHERICAL FLOWS ASAF KATZ Abstract. We prove a quantitative variant of a disjointness the- orem of nilflows from horospherical flows following a technique of Venkatesh, combined with the structural theorems for nilflows by Green, Tao and Ziegler. arXiv:1910.04675v1 [math.DS] 10 Oct 2019 1 2 1. Introduction In a landmark paper [10], H. Furstenberg introduced the notion of joinings of two dynamical systems and the concept of disjoint dynam- ical systems. Ever since, this property has played a major role in the field of dynamics, leading to many fundamental results. In his paper, Furstenberg proved the following characterization of a weakly-mixing dynamical system: 1.1. Theorem (Furstenberg [10], Theorem 1.4). A dynamical system (X,β,µ,T ) is weakly-mixing if and only if (X, T ) is disjoint from any Kronecker system. Namely, given any compact abelian group A, equipped with the ac- tion of A on itself by left-translation Ra, the only joining between (X, T ) and (A, Ra) is the trivial joining given by the product mea- sure on the product dynamical system (X A, T R ). The Wiener- × × a Wintner ergodic theorem readily follows from Furstenberg’s theorem. In [2], J. Bourgain derived a strengthening of Furstenberg’s disjoint- ness theorem, which amounts to the uniform Wiener-Wintner theorem. The proof is along the lines of Furstenberg’s but utilizing the Van-der- Corput trick to show uniformity over various Kronecker systems. In a recent work [41], A. Venkatesh gave a quantitative statement of the uniform Wiener-Wintner theorem for the case of the horocyclic flow on compact homogeneous spaces of SL2(R) (and in principal, his proof works also for non-compact spaces as well, by allowing the decay rate to depend on Diophantine properties of the origin point of the orbit). -
Arxiv:1006.0205V4 [Math.NT]
AN INVERSE THEOREM FOR THE GOWERS U s+1[N]-NORM BEN GREEN, TERENCE TAO, AND TAMAR ZIEGLER Abstract. This is an announcement of the proof of the inverse conjecture for the Gowers U s+1[N]-norm for all s > 3; this is new for s > 4, the cases s = 1, 2, 3 having been previously established. More precisely we outline a proof that if f : [N] → [−1, 1] is a function with kfkUs+1[N] > δ then there is a bounded-complexity s-step nilsequence F (g(n)Γ) which correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. In particular, one obtains an asymptotic formula for the number of k-term arithmetic progres- sions p1 <p2 < ··· <pk 6 N of primes, for every k > 3. 1. Introduction This is an announcement and summary of our much longer paper [20], the pur- pose of which is to establish the general case of the Inverse Conjecture for the Gowers norms, conjectured by the first two authors in [15, Conjecture 8.3]. If N is a (typically large) positive integer then we write [N] := {1,...,N}. Throughout the paper we write D = {z ∈ C : |z| 6 1}. For each integer s > 1 the inverse conjec- ture GI(s), whose statement we recall shortly, describes the structure of functions st f :[N] → D whose (s + 1) Gowers norm kfkU s+1[N] is large. -
Jakub M. Konieczny Curriculum Vitae
Jakub M. Konieczny Curriculum Vitae E-mail: [email protected] CONTACT Einstein Institute of Mathematics Webpages: INFORMATION Edmond J. Safra Campus The Hebrew University of Jerusalem mathematics.huji.ac.il/people/jakub-konieczny Givat Ram. Jerusalem, 9190401, Israel jakubmichalkonieczny.wordpress.com RESEARCH Ergodic theory and dynamical systems. Applications to combinatorial number theory. Sym- INTERESTS bolic dynamics. Fourier analysis. Higher order Fourier analysis. Dynamics on nilmani- folds. Generalised polynomials. Automatic sequences. q-multiplicative sequences. Ultrafil- ters. Ramsey theory. Extremal combinatorics. CURRENT Postdoctoral Fellow, October 2017 to present ACADEMIC Einstein Institute of Mathematics, The Hebrew University of Jerusalem APPOINTMENTS under ERC grant: Ergodic Theory and Additive Combinatorics Advisor: Tamar Ziegler EDUCATION University of Oxford Doctor of Philosophy in Mathematics, 1 August 2017 • Supervisor: Ben Green • Area of study: Ergodic theory/Additive combinatorics • Thesis title: On combinatorial properties of nil-Bohr sets of integers and related problems Jagiellonian University, Kraków and Vrije Universiteit, Amsterdam (joint degree) Master of Science in Mathematics, 18 July 2013 (UJ) | 30 August 2013 (VU) • Supervisor: P. Niemiec (UJ) and T. Eisner-Lobova (VU) and A. J. Homburg (VU) • Area of study: Ergodic theory/Combinatorial number theory • Thesis title: Applications of ultrafilters in ergodic theory and combinatorial number theory Jagiellonian University, Kraków Bachelor of Science in Mathematics, 15 July 2011 • Individualised Studies in Mathematics and Natural Sciences (SMP) programme • Extended curriculum incorporating material from Computer Science and Physics JOURNAL PAPERS [1] J. Byszewski, J. Konieczny, and E. Krawczyk. Substitutive systems and a finitary version (PUBLISHED OR of Cobham’s theorem. To appear in Combinatorica, available at arXiv: 1908.11244 ACCEPTED FOR [math.CO] PUBLICATION) [2] J. -
Issue 118 ISSN 1027-488X
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY S E European M M Mathematical E S Society December 2020 Issue 118 ISSN 1027-488X Obituary Sir Vaughan Jones Interviews Hillel Furstenberg Gregory Margulis Discussion Women in Editorial Boards Books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Recent books in the EMS Monographs in Mathematics series Massimiliano Berti (SISSA, Trieste, Italy) and Philippe Bolle (Avignon Université, France) Quasi-Periodic Solutions of Nonlinear Wave Equations on the d-Dimensional Torus 978-3-03719-211-5. 2020. 374 pages. Hardcover. 16.5 x 23.5 cm. 69.00 Euro Many partial differential equations (PDEs) arising in physics, such as the nonlinear wave equation and the Schrödinger equation, can be viewed as infinite-dimensional Hamiltonian systems. In the last thirty years, several existence results of time quasi-periodic solutions have been proved adopting a “dynamical systems” point of view. Most of them deal with equations in one space dimension, whereas for multidimensional PDEs a satisfactory picture is still under construction. An updated introduction to the now rich subject of KAM theory for PDEs is provided in the first part of this research monograph. We then focus on the nonlinear wave equation, endowed with periodic boundary conditions. The main result of the monograph proves the bifurcation of small amplitude finite-dimensional invariant tori for this equation, in any space dimension. -
Additive and Analytic Combinatorics
IMA Workshops September 29-October 3, 2014 Additive and Analytic Combinatorics ORGANIZERS SPEAKERS David Conlon, University of Oxford Tim Austin, Courant Institute of Mathematical Sciences Ernie Croot, Georgia Institute of Technology Vitaly Bergelson, The Ohio State Van Vu, Yale University University Tamar Ziegler, Hebrew University Emmanuel Breuillard, Université de Paris XI (Paris-Sud) Boris Bukh, Carnegie Mellon University Mei-Chu Chang, University of California, Additive combinatorics is the theory of counting Riverside additive structures in sets. This theory has seen David Conlon, University of Oxford exciting developments and dramatic changes in Ernie Croot, Georgia Institute of direction in recent years thanks to its connections Technology with areas such as harmonic analysis, ergodic Jacob Fox, Massachusetts Institute of Technology theory, and representation theory. As it turns out, Bob Guralnick, University of Southern many combinatorial ideas that have existed in the California combinatorics community for quite some time can Akos Magyar, University of British be used to attack notorious problems in other areas Columbia of mathematics. A typical example is the Green- Hamed Hatami, McGill University Tao theorem on the existence of long arithmetic Frederick Manners, University of Oxford progressions in primes, which uses a famous Lilian Matthiesen, Institut de theorem of Szemerédi on arithmetic progressions Mathematiques de Jussieu in dense sets as a key component. The field is also Jozsef Solymosi, University of British of great interest to computer scientists; a number of Columbia the techniques and theorems have seen application Terence Tao, University of California, Los Angeles in, for example, communication complexity, Van Vu, Yale University property testing, and the design of randomness Melanie Wood, University of Wisconsin, extractors. -
An Inverse Theorem for the Gowers U4-Norm
Glasgow Math. J. 53 (2011) 1–50. C Glasgow Mathematical Journal Trust 2010. doi:10.1017/S0017089510000546. AN INVERSE THEOREM FOR THE GOWERS U4-NORM BEN GREEN Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK e-mail: [email protected] TERENCE TAO UCLA Department of Mathematics, Los Angeles, CA 90095-1555, USA e-mail: [email protected] and TAMAR ZIEGLER Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel e-mail: [email protected] (Received 30 November 2009; accepted 18 June 2010; first published online 25 August 2010) Abstract. We prove the so-called inverse conjecture for the Gowers U s+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f :[N] → ރ is a function with |f (n)| 1 for all n and f U4 δ then there is a bounded complexity 3-step nilsequence F(g(n)) which correlates with f . The approach seems to generalise so as to prove the inverse conjecture for s 4as well, and a longer paper will follow concerning this. By combining the main result of the present paper with several previous results of the first two authors one obtains the generalised Hardy–Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p1 < p2 < p3 < p4 < p5 N of primes. 2010 Mathematics Subject Classification. Primary: 11B30; Secondary: 37A17. NOTATION . By a 1-bounded function on a set X we mean a function f : X → ރ with |f (x)| 1 for all x∈ X. -
Introducing the 2021 Class of Karen EDGE Fellows Emille Davie
Introducing the 2021 Class of Karen EDGE Fellows The EDGE Foundation is delighted to announce the 2021 Class of Karen EDGE Fellows. The Karen EDGE Fellowship Program was established with a generous gift from Karen Uhlenbeck on the occasion of her 2019 Abel Prize. The Fellowships are designed to support and enhance the research programs and collaborations of mid- career mathematicians who are members of an underrepresented minority group. The 2021 Fellows were selected on the basis of their excellent research programs and their plans to use the funds for enhancing those programs through collaboration and travel. The Karen EDGE Fellows for 2021 are Emille Davie Lawrence, University of San Francisco, and Manuel Rivera, Purdue University. Emille Davie Lawrence Manuel Rivera Emille Davie Lawrence received her Ph.D. in Mathematics from the University of Georgia in 2007, under the direction of Will Kazez and Clint McCrory. She was an undergraduate at Spelman College. She was a postdoctoral fellow at the Univer- sity of California, Santa Barbara, taught at California State Polytechnic University, Pomona, and subsequently joined the faculty of University of San Francisco, where she has taught since 2011. She is currently Term Associate Professor and serves as Department Chair. Emille's research is in spatial graph theory, a branch of geometric topology in the intersection of knot theory and graph theory. Specifically, spatial graph theory is the study of embeddings of graphs in manifolds, with the manifold S3 being of particular interest. Her most recent research centers around classifying all groups which can occur as the topological symmetry group for some embedding of an abstract graph or family of graphs. -
Joint Journals Catalogue EMS / MSP 2020
Joint Journals Cataloguemsp EMS / MSP 1 2020 Super package deal inside! msp 1 EEuropean Mathematical Society Mathematical Science Publishers msp 1 The EMS Publishing House is a not-for-profit Mathematical Sciences Publishers is a California organization dedicated to the publication of high- nonprofit corporation based in Berkeley. MSP quality peer-reviewed journals and high-quality honors the best traditions of quality publishing books, on all academic levels and in all fields of while moving with the cutting edge of information pure and applied mathematics. The proceeds from technology. We publish more than 16,000 pages the sale of our publications will be used to keep per year, produce and distribute scientific and the Publishing House on a sound financial footing; research literature of the highest caliber at the any excess funds will be spent in compliance lowest sustainable prices, and provide the top with the purposes of the European Mathematical quality of mathematically literate copyediting and Society. The prices of our products will be set as typesetting in the industry. low as is practicable in the light of our mission and We believe scientific publishing should be an market conditions. industry that helps rather than hinders scholarly activity. High-quality research demands high- Contact addresses quality communication – widely, rapidly and easily European Mathematical Society Publishing House accessible to all – and MSP works to facilitate it. Technische Universität Berlin, Mathematikgebäude Straße des 17. Juni 136, 10623 Berlin, Germany Contact addresses Email: [email protected] Mathematical Sciences Publishers Web: www.ems-ph.org 798 Evans Hall #3840 c/o University of California Managing Director: Berkeley, CA 94720-3840, USA Dr. -
Mathematics People
NEWS Mathematics People currently holds a Packard Fellowship and an NSF CAREER Wood Receives NSF grant. In high school, Wood won silver medals at the 1998 and 1999 International Mathematical Olympiad as the first Waterman Award US female in the competition. In 2001, she was awarded the Melanie Matchett Wood of Harvard Elizabeth Lowell Putnam Prize in the Putnam Mathemati- University has been named a co- cal Competition; in 2002 she became a Putnam Fellow. Her recipient of the 2021 National Science honors include the Alice T. Schafer Prize of the Association Foundation (NSF) Alan T. Waterman for Women in Mathematics (2002), the AMS–MAA–SIAM Award for “fundamental contribu- Morgan Prize (2003), a Clay Mathematics Institute Liftoff tions in number theory, algebraic Fellowship (2009), and the AWM–Microsoft Research Prize geometry, topology, and probability.” in Algebra and Number Theory (2018). She was named an She is the first woman mathemati- Inaugural Fellow of the AMS in 2012. cian to receive the honor. The Waterman Award annually recognizes an outstand- ing young researcher in any field of science or engineering Melanie Matchett The NSF statement reads in part: Wood “Throughout her scientific career, supported by NSF. Researchers forty years of age or younger, Wood has worked on some of the or up to ten years post-PhD, are eligible. Awardees receive most difficult problems in mathematics and has developed US$1 million distributed over five years. methods that combine ideas from across mathematics. Wood studies the statistical behavior of how factorization —Elaine Kehoe, from NSF and Harvard University announcements works in different systems of numbers, aiming to answer questions posed 200 years ago by Gauss.” Her work is aimed at understanding the distribution of number fields Akhtari Awarded and their fundamental structures, including class groups, p-class tower groups, and the Galois groups of their max- Michler Memorial Prize imal unramified extensions. -
Ewm Newsletter23.Pdf
Newsletter 23 Edited by Sara Munday (University of Bristol, UK) and Elena Resmerita (University of Klagenfurt, Austria) 2013/2 In this issue This issue is dedicated to the EWM general meeting which took place in September 2013 at the Hausdorff Centre for Mathematics, Bonn. A report and the minutes of the meeting are included below, followed by interesting data related to gender in Math publications provided by colleagues from Zentralblatt Math (presented also at the meeting). The interviews focus on the change of convenor: Marie-Francoise Roy (former convenor) and Susanna Terracini (new convenor), and on Tamar Ziegler, a guest speaker at Bonn who has been the European Mathematics Society 2013 lecturer. The series of articles on the STEM situation in different European countries continues here with Germany and Italy. An article published a few years ago in the magazine "Role Model Book" of the Nagoya University, Japan is included courtesy to its author: Yukari Ito. Prestigious prizes for research awarded to woman mathematicians this year in Europe are highlighted. A series of announcements on Math events and miscellanea are pointed out at the end of this issue. The 2013 EWM general meeting Report from the meeting The meeting was opened by the EWM convenor Marie-Francoise Roy and by Dr. Michael Meier, Chief Administrator of HCM Bonn, who talked about the mission and activities of HCM, as well as the personality of the mathematician Felix Hausdorff , who on top of his contributions to topology and many other topics in mathematics was also a philosopher and playwrighter under the pseudonym Paul Mongré. -
2007-2008 Catalog Until the Catalog Expires (Six Years)
Oakland FINAL 5/3/07 4:22 PM Page 1 OAKLAND UNIVERSITY 2007–2008 UNDERGRADUATE CATALOG May 2007 Volume XLVII Published by Oakland University, Rochester, Michigan All data in this catalog reflect information as it was available at the publication date. Oakland University reserves the right to revise all announcements contained in this publication at its discretion and to make reasonable changes in requirements to improve or upgrade academic and non-academic programs. The academic requirements described in this catalog are in effect fall semester 2007 through summer session 2014. Undergraduate students admitted to a degree-granting program may use provisions in this catalog to meet requirements within that time frame. Oakland FINAL 5/3/07 4:22 PM Page 2 Oakland University is a legally autonomous state institution of higher learning. Legislation creating Oakland University as an independent institution, separate from Michigan State University, was established under Act No. 35, Public Acts of 1970. The university is governed by an eight member board of trustees appointed by the governor with the advice and consent of the Michigan Senate. As an equal opportunity and affirmative action institution, Oakland University is committed to compliance with federal and state laws prohibiting discrimination, including Title VI of the Civil Rights Act of 1964, Title IX of the Education Amendments of 1972, Section 504 of the Rehabilitation Act of 1973 and the Americans with Disabilities Act. It is the policy of Oakland University that there shall be no unlawful discrimination against any person on the basis of race, sex, sexual orientation, color, religion, creed, national origin or ancestry, age, height, weight, mar- ital status, handicap, familial status, veteran status or other prohibited factors in employment, admissions, educational programs or activities.