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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. -
April 2010 Contents
International Association of Mathematical Physics News Bulletin April 2010 Contents International Association of Mathematical Physics News Bulletin, April 2010 Contents Reflections on the IAMP Geography 3 Aharonov-Bohm & Berry Phase Anniversaries 50/25 5 The 25th anniversary of the founding of HARL 8 An interview with Huzihiro Araki 10 Shing-Tung Yau the Wolf Prize laureate 2010 in Mathematics 14 News from the IAMP Executive Committee 22 A new associated member: PIMS 26 Bulletin editor Valentin Zagrebnov Editorial board Evans Harrell, Masao Hirokawa, David Krejˇciˇr´ık, Jan Philip Solovej Contacts [email protected] http://www.iamp.org Cover photo (courtesy of Professor A.Tonomura): From double-slit experiment to the Aharonov-Bohm effect. See a comment at the end of the page 7. The views expressed in this IAMP News Bulletin are those of the authors and do not necessary represent those of the IAMP Executive Committee, Editor or Editorial board. Any complete or partial performance or reproduction made without the consent of the author or of his successors in title or assigns shall be unlawful. All reproduction rights are henceforth reserved, mention of the IAMP News Bulletin is obligatory in the reference. (Art.L.122-4 of the Code of Intellectual Property). 2 IAMP News Bulletin, April 2010 Editorial Reflections on the IAMP Geography by Pavel Exner (IAMP President) The topic of today’s meditation was inspired by complaints of American colleagues about the shaky position our discipline enjoys in the U.S. True, such woes are ubiquitous since com- petition for resources in science was and will always be tough. -
Fifteenth Annual Commencement Exercises Benjamin N
Yeshiva University, Cardozo School of Law LARC @ Cardozo Law Pre-2019 Commencement Programs Graduation 6-6-1993 Fifteenth Annual Commencement Exercises Benjamin N. Cardozo School of Law Follow this and additional works at: https://larc.cardozo.yu.edu/pre-19-programs Part of the Law Commons Recommended Citation Benjamin N. Cardozo School of Law, "Fifteenth Annual Commencement Exercises" (1993). Pre-2019 Commencement Programs. 15. https://larc.cardozo.yu.edu/pre-19-programs/15 This Book is brought to you for free and open access by the Graduation at LARC @ Cardozo Law. It has been accepted for inclusion in Pre-2019 Commencement Programs by an authorized administrator of LARC @ Cardozo Law. For more information, please contact [email protected], [email protected]. - FIFTEENTI-1 ANNUAL COMMENCEMENT EXERCISES Sunday, June Sixth Nineteen Hundred and Ninety-Three at Two-Thirty in the Afternoon Avery Fisher Hall Lincoln Center New York City This program contains a listing of candidates for degrees and honors during the period July 1992-June 1993; it is not an official roster of graduates. ORDER of EXERCISES Processional HERBERT C. DOBRINSKY, Ed.D. Vice President for University Affairs Herald EGON BRENNER, D.E.E. Executive Vice President Chief Marshal Presiding NORMAN LAMM, Ph.D. President ISRAEL MILLER, D.D. Senior Vice President The National Anthem CANTOR IRA W. HELLER, M.A. Yeshiva College, 1983 Ferkauf Graduate School of Psychology, 1985 The Jewish Center, New York City Invocation MARK NELSON WILDES Class of 1993 Candidate for Semikhah, 1993 Rabbi Isaac Elchanan Theological Seminary Welcome FRANK J. MACCHIAROLA, LL.B., Ph.D. -
Sastra Prize 2011
UF SASTRA PRIZE Mathematics 2011 Research Courses Undergraduate Graduate News Resources People ROMAN HOLOWINSKY TO RECEIVE 2011 SASTRA RAMANUJAN PRIZE The 2011 SASTRA Ramanujan Prize will be awarded to Roman Holowinsky, who is now an Assistant Professor at the Department of Mathematics, Ohio State University, Columbus, Ohio, USA. This annual prize which was established in 2005, is for outstanding contributions by very young mathematicians to areas influenced by the genius Srinivasa Ramanujan. The age limit for the prize has been set at 32 because Ramanujan achieved so much in his brief life of 32 years. The $10,000 prize will be awarded at the International Conference on Number Theory, Ergodic Theory and Dynamics at SASTRA University in Kumbakonam, India (Ramanujan's hometown) on December 22, Ramanujan's birthday. Dr. Roman Holowinsky has made very significant contributions to areas which are at the interface of analytic number theory and the theory of modular forms. Along with Professor Kannan Soundararajan of Stanford University (winner of the SASTRA Ramanujan Prize in 2005), Dr. Holowinsky solved an important case of the famous Quantum Unique Ergodicity (QUE) Conjecture in 2008. This is a spectacular achievement. In 1991, Zeev Rudnick and Peter Sarnak formulated the QUE Conjecture which in its general form concerns the correspondence principle for quantizations of chaotic systems. One aspect of the problem is to understand how waves are influenced by the geometry of their enclosure. Rudnick and Sarnak conjectured that for sufficiently chaotic systems, if the surface has negative curvature, then the high frequency quantum wave functions are uniformly distributed within the domain. -
When Algebraic Entropy Vanishes Jim Propp (U. Wisconsin) Tufts
When algebraic entropy vanishes Jim Propp (U. Wisconsin) Tufts University October 31, 2003 1 I. Overview P robabilistic DYNAMICS ↑ COMBINATORICS ↑ Algebraic DYNAMICS 2 II. Rational maps Projective space: CPn = (Cn+1 \ (0, 0,..., 0)) / ∼, where u ∼ v iff v = cu for some c =6 0. We write the equivalence class of (x1, x2, . , xn+1) n in CP as (x1 : x2 : ... : xn+1). The standard imbedding of affine n-space into projective n-space is (x1, x2, . , xn) 7→ (x1 : x2 : ... : xn : 1). The “inverse map” is (x : ... : x : x ) 7→ ( x1 ,..., xn ). 1 n n+1 xn+1 xn+1 3 Geometrical version: CPn is the set of lines through the origin in (n + 1)-space. n The point (a1 : a2 : ... : an+1) in CP cor- responds to the line a1x1 = a2x2 = ··· = n+1 an+1xn+1 in C . The intersection of this line with the hyperplane xn+1 = 1 is the point a a a ( 1 , 2 ,..., n , 1) an+1 an+1 an+1 (as long as an+1 =6 0). We identity affine n-space with the hyperplane xn+1 = 1. Affine n-space is a Zariski-dense subset of pro- jective n-space. 4 A rational map is a function from (a Zariski-dense subset of) CPn to CPm given by m rational functions of the affine coordinate variables, or, the associated function from a Zariski-dense sub- set of Cn to CPm (e.g., the “function” x 7→ 1/x on affine 1-space, associated with the function (x : y) 7→ (y : x) on projective 1-space). -
From the Editor
Department of Mathematics University of Wisconsin From the Editor. This year’s biggest news is the awarding of the National Medal of Science to Carl de Boor, professor emeritus of mathematics and computer science. Accompanied by his family, Professor de Boor received the medal at a White House ceremony on March 14, 2005. Carl was one of 14 new National Medal of Science Lau- reates, the only one in the category of mathematics. The award, administered by the National Science Foundation originated with the 1959 Congress. It honors individuals for pioneering research that has led to a better under- standing of the world, as well as to innovations and tech- nologies that give the USA a global economic edge. Carl is an authority on the theory and application of splines, which play a central role in, among others, computer- aided design and manufacturing, applications of com- puter graphics, and signal and image processing. The new dean of the College of Letters and Science, Gary Sandefur, said “Carl de Boor’s selection for the na- tion’s highest scientific award reflects the significance of his work and the tradition of excellence among our mathematics and computer science faculty.” We in the Department of Mathematics are extremely proud of Carl de Boor. Carl retired from the University in 2003 as Steen- bock Professor of Mathematical Sciences and now lives in Washington State, although he keeps a small condo- minium in Madison. Last year’s newsletter contains in- Carl de Boor formation about Carl’s distinguished career and a 65th birthday conference held in his honor in Germany. -
Fall 2006 [Pdf]
Le Bulletin du CRM • www.crm.umontreal.ca • Automne/Fall 2006 | Volume 12 – No 2 | Le Centre de recherches mathématiques A Review of CRM’s 2005 – 2006 Thematic Programme An Exciting Year on Analysis in Number Theory by Chantal David (Concordia University) The thematic year “Analysis in Number The- tribution of integers, and level statistics), integer and rational ory” that was held at the CRM in 2005 – points on varieties (geometry of numbers, the circle method, 2006 consisted of two semesters with differ- homogeneous varieties via spectral theory and ergodic theory), ent foci, both exploring the fruitful interac- the André – Oort conjectures (equidistribution of CM-points tions between analysis and number theory. and Hecke points, and points of small height) and quantum The first semester focused on p-adic analy- ergodicity (quantum maps and modular surfaces) The main sis and arithmetic geometry, and the second speakers were Yuri Bilu (Bordeaux I), Bill Duke (UCLA), John semester on classical analysis and analytic number theory. In Friedlander (Toronto), Andrew Granville (Montréal), Roger both themes, several workshops, schools and focus periods Heath-Brown (Oxford), Elon Lindenstrauss (New York), Jens concentrated on the new and exciting developments of the re- Marklof (Bristol), Zeev Rudnick (Tel Aviv), Wolfgang Schmidt cent years that have emerged from the interplay between anal- (Colorado, Boulder and Vienna), K. Soundararajan (Michigan), ysis and number theory. The thematic year was funded by the Yuri Tschinkel (Göttingen), Emmanuel Ullmo (Paris-Sud), and CRM, NSF, NSERC, FQRNT, the Clay Institute, NATO, and Akshay Venkatesh (MIT). the Dimatia Institute from Prague. In addition to the partici- The workshop on “p-adic repre- pants of the six workshops and two schools held during the sentations,” organised by Henri thematic year, more than forty mathematicians visited Mon- Darmon (McGill) and Adrian tréal for periods varying from two weeks to six months. -
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). -
Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings Via Transfer-Impedances
LOCAL CHARACTERISTICS, ENTROPY AND LIMIT THEOREMS FOR SPANNING TREES AND DOMINO TILINGS VIA TRANSFER-IMPEDANCES Running Head: LOCAL BEHAVIOR OF SPANNING TREES Robert Burton 1 Robin Pemantle 2 3 Oregon State University ABSTRACT: Let G be a finite graph or an infinite graph on which ZZd acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, methods from [Pem] show that this still makes sense, producing a random essential spanning forest of G. A method for calculating local characteristics (i.e. finite-dimensional marginals) of T from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix (n-dimensional minors) are needed to compute small (n-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for G, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when G is an infinite graph admitting a Zd-action with finite quotient. The same computation also gives the entropy of the law of T. These results are applied to the problem of tiling certain lattices by dominos { the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous [Al2] on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. -
Superirreducibility of Polynomials, Binomial Coefficient Asymptotics
Superirreducibility of Polynomials, Binomial Coefficient Asymptotics and Stories From My Classroom by Lara Du A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2020 Doctoral Committee: Professor Mattias Jonsson, Co-Chair Professor Trevor Wooley, Co-Chair Professor Ratindranath Akhoury Professor Jeffrey Lagarias Dr Elaine Lande Lara Du [email protected] ORCID id: 0000-0001-6749-4867 c Lara Du 2020 ACKNOWLEDGEMENTS This work was partially supported by NSF grant DMS-1701577 and a Department of Mathematics Fellowship. I was extremely fortunate during my graduate school years to have had excellent mentors that helped me grow both as a mathematician and as a teacher. Mathematically, I benefitted from regular meetings with Professors Trevor Wooley, Mattias Jonsson and Jeff Lagarias, who have all advised me in some form during my PhD programme. Working with Trevor, I really found my place in the mathematical commu- nity: learning how to develop as an academic and have meaningful mathematical conver- sations while staying true to my own beliefs and roots. From Mattias, I learned to pursue the math that I found interesting, seeking out opportunities that were best for me and find- ing the conviction I needed to see them through. From Jeff, I learned to problem-solve, to adjust the research question I was asking if appropriate, to take things as slowly as I needed, but to never give up on seeking beautiful mathematical truths. Throughout graduate school, I’ve also benefitted enormously from collaboration with Gene Kopp, Jonathan Bober and Dan Fretwell and from the mathematical mentorship of Evangelia Gazaki. -
Michigan Section–MAA Newsletter U
Michigan Section – MAA PAID Permit No. 144 Non-Profit Org. U. S. POSTAGE Ann Arbor, Mich. Arbor, Ann Volume 34, Number 2 April 2008 Michigan Section Annual Meeting Grand Valley State University May 2–3, 2008 Coming to Grand Rapids … Anna Gilbert A Few Good Terms DATED MATERIAL DATED PLEASE EXPEDITE Gail Burrill The Mathematical Knowledge Needed for Teaching Jeffrey Lagarias Hilbert’s 18th Problem Carl Pomerance The Covering Congruences of Paul Newsletter Erdős 8604 Francis Wu Teaching Research: Encouraging Discoveries … and much more 8604 ALSO IN THIS ISSUE: 51st Annual Michigan Mathematics Prize Competition Norman Richert, Editor Michigan Section–MAA Mathematical Reviews University of Michigan Box P.O. MI 48107- Arbor, Ann www.michmaa.org Mathematical Association of America Michigan Section Newsletter Calendar of Events Volume 34, Number 2 May 2–3, 2008 Michigan Section Meeting, GVSU Contents Pages April 9–12, 2008 NCTM Annual Meeting, Salt Lake City Section Officers’ Reports .......................................................................... 1 July 31–August 2, 2008 MAA MathFest, Madison Remembering Janet Andersen ................................................................... 8 November 20–23, 2008 AMATYC Annual Meeting, Washington, D.C. Michigan NExT......................................................................................... 9 Webmaster’s Report ................................................................................ 10 January 5–8, 2009 MAA/AMS Annual Meeting, Washington, D.C. Teaching Award Nominations -
2014-2015 Annual Report
Institute for Computational and Experimental Research in Mathematics Annual Report August 1, 2014 – July 31, 2015 Jill Pipher, Director Jeffrey Hoffstein, Consulting Associate Director Bjorn Sandstede, Associate Director Sergei Tabachnikov, Deputy Director Homer Walker, Deputy Director Ulrica Wilson, Associate Director for Diversity and Outreach 1 Table of Contents Mission ................................................................................................................................................ 5 Core Programs and Events ................................................................................................................... 5 Virtual Institute of Mathematical and Statistical Sciences (VI-MSS) ...................................................... 7 Participant Summaries by Program Type .............................................................................................. 9 ICERM Funded Participants ................................................................................................................................................... 9 All Participants (ICERM funded and Non-ICERM funded) .................................................................................... 10 All Speakers (ICERM funded and Non-ICERM funded) .......................................................................................... 12 All Postdocs (ICERM funded and Non-ICERM funded) ........................................................................................... 14 All Graduate Students (ICERM funded