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Curriculum Vitae
Curriculum Vitae Larry Guth Education B.S. Mathematics, Yale University, 2000 PhD Mathematics, MIT, 2005 Employment Samelson Fellow, Stanford University 2005-6 Szego Assistant Professor, Stanford University, 2006 - 2008 National Science Foundation Postdoctoral Fellow, 2006 - 2008 Tenure-stream Assistant Professor, University of Toronto, 2008 - 2011 Member, Institute for Advanced Study, 2010 - 2011 Professor, New York University, 2011 - 2012 Professor, MIT, 2012 - Fellowships National Science Foundation Graduate Fellowship, 2001 - 2003 National Science Foundation Postdoctoral Fellowship , 2006 - 2008 Alfred P. Sloan Research Fellowship, 2010 - 2014 Simons Investigator, 2014 - Honors Salem prize 2013 Invited talks Invited sectional speaker at the 2010 International Congress of Mathematics Marston Morse lectures, IAS, 2013 Namboodiri lectures, University of Chicago, 2015 1 Selected publications 1. Lipshitz maps from surfaces, Geom. Funct. Anal. 15 (2005), no. 5, 1052-109 2. The width-volume inequality, Geom. Funct. Anal. 17 (2007) no. 4, 1139-1179 3. Notes on Gromov's systolic estimate, Geom. Dedicata 123 (2006) 113-129 4. Symplectic embeddings of polydisks, Invent. Math. 172 (2008) no. 3, 477-489 5. Minimax problems related to cup powers and Steenrod squares, Geom. Funct. Anal. 18 (2009), no. 6, 1917{1987 6. The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture, Acta Math. 205 (2010), no. 2, 263-286. 7. Systolic inequalities and minimal hypersurfaces, Geom. Funct. Anal. 19 (2010), no. 6, 1688-1692. 8. Volumes of balls in large Riemannian manifolds, Ann. of Math. (2) 173 (2011), no. 1, 5176. 9. Area-expanding embeddings of rectangles, preprint 10. Algebraic methods in discrete analogs of the Kakeya problem (joint with Nets Katz), Adv. -
2017-2018 Annual Report
Institute for Computational and Experimental Research in Mathematics Annual Report May 1, 2017 – April 30, 2018 Brendan Hassett, Director Mathew Borton, IT Director Jeff Brock, Associate Director Ruth Crane, Assistant Director Sigal Gottlieb, Deputy Director Elisenda Grigsby, Deputy Director Jeffrey Hoffstein, Consulting Associate Director Caroline Klivans, Associate Director Jill Pipher, Consulting Associate Director Bjorn Sandstede, Associate Director Homer Walker, Consulting Associate Director Ulrica Wilson, Associate Director for Diversity and Outreach Table of Contents Mission ....................................................................................................................................... 6 Core Programs and Events ......................................................................................................... 6 Participant Summaries by Program Type ................................................................................... 9 ICERM Funded Participants ................................................................................................................. 9 All Participants (ICERM funded and Non-ICERM funded) .................................................................. 10 ICERM Funded Speakers ................................................................................................................... 11 All Speakers (ICERM funded and Non-ICERM funded) ...................................................................... 12 ICERM Funded Postdocs .................................................................................................................. -
The Extended Relativity Theory in Clifford Spaces
THE EXTENDED RELATIVITY THEORY IN CLIFFORD SPACES C. Castroa and M. Pav·si·cb May 21, 2004 aCenter for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta bJo·zef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia; Email: [email protected] Abstract A brief review of some of the most important features of the Extended Rela- tivity theory in Cli®ord-spaces (C-spaces) is presented whose " point" coordinates are noncommuting Cli®ord-valued quantities and which incorporate the lines, ar- eas, volumes,.... degrees of freedom associated with the collective particle, string, membrane,... dynamics of p-loops (closed p-branes) living in target D-dimensional spacetime backgrounds. C-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, noncommuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable di- mensions/signatures that allows to study the dynamics of all (closed) p-branes, for all values of p, on a uni¯ed footing. It resolves the ordering ambiguities in QFT and the problem of time in Cosmology. A discussion of the maximal-acceleration Rela- tivity principle in phase-spaces follows along with the study of the invariance group of symmetry transformations in phase-space that allows to show why Planck areas are invariant under acceleration-boosts transformations and which seems to suggest that a maximal-string tension principle may be operating in Nature. We continue by pointing out how the relativity of signatures of the underlying n-dimensional spacetime results from taking di®erent n-dimensional slices through C-space. The conformal group emerges as a natural subgroup of the Cli®ord group and Relativity in C-spaces involves natural scale changes in the sizes of physical objects without the introduction of forces nor Weyl's gauge ¯eld of dilations. -
Polynomial Methods in Combinatorics
UNIVERSITY LECTURE SERIES VOLUME 64 Polynomial Methods in Combinatorics Larry Guth American Mathematical Society https://doi.org/10.1090//ulect/064 Polynomial Methods in Combinatorics UNIVERSITY LECTURE SERIES VOLUME 64 Polynomial Methods in Combinatorics Larry Guth American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jordan S. Ellenberg Robert Guralnick William P. Minicozzi II (Chair) Tatiana Toro 2010 Mathematics Subject Classification. Primary 05D99. For additional information and updates on this book, visit www.ams.org/bookpages/ulect-64 Library of Congress Cataloging-in-Publication Data Names: Guth, Larry, 1977– Title: Polynomial methods in combinatorics / Larry Guth. Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Univer- sity lecture series ; volume 64 | Includes bibliographical references. Identifiers: LCCN 2016007729 | ISBN 9781470428907 (alk. paper) Subjects: LCSH: Combinatorial geometry. | Polynomials. | Geometry, Algebraic. | AMS: Combina- torics – Extremal combinatorics – None of the above, but in this section. msc Classification: LCC QA167 .G88 2016 | DDC 511/.66–dc23 LC record available at http://lccn.loc. gov/2016007729 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. -
Algebraic Foundations of Split Hypercomplex Nonlinear Adaptive
Mathematical Methods in the Research Article Applied Sciences Received XXXX (www.interscience.wiley.com) DOI: 10.1002/sim.0000 MOS subject classification: 60G35; 15A66 Algebraic foundations of split hypercomplex nonlinear adaptive filtering E. Hitzer∗ A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Copyright c 2011 John Wiley & Sons, Ltd. Keywords: Quaternionic adaptive filtering, Hypercomplex adaptive filtering, Nonlinear adaptive filtering, Hypercomplex Multilayer Perceptron, Clifford geometric algebra 1. Introduction Split quaternion nonlinear adaptive filtering has recently been treated by [23], who showed its superior performance for Saito’s Chaotic Signal and for wind forecasting. The quaternionic methods constitute a generalization of complex valued adaptive filters, treated in detail in [19]. A method of quaternionic least mean square algorithm for adaptive quaternionic has previously been developed in [22]. Additionally, [24] successfully proposes the usage of local analytic fully quaternionic functions in the Quaternion Nonlinear Gradient Descent (QNGD). Yet the unconditioned use of analytic fully quaternionic activation functions in neural networks faces problems with poles due to the Liouville theorem [3]. The quaternion algebra of Hamilton is a special case of the higher dimensional Clifford algebras [10]. The problem with poles in nonlinear analytic functions does not generally occur for hypercomplex activation functions in Clifford algebras, where the Dirac and the Cauchy-Riemann operators are not elliptic [21], as shown, e.g., for hyperbolic numbers in [17]. -
Mathematical Research in High School: the PRIMES Experience
Mathematical Research in High School: The PRIMES Experience The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Etingof, Pavel, Slava Gerovitch, and Tanya Khovanova. “Mathematical Research in High School: The PRIMES Experience.” Notices of the American Mathematical Society 62, no. 08 (September 1, 2015): 910–918. © American Mathematical Society (AMS) As Published http://dx.doi.org/10.1090/noti1270 Publisher American Mathematical Society (AMS) Version Final published version Citable link http://hdl.handle.net/1721.1/110267 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Mathematical Research in High School: The PRIMES Experience Pavel Etingof, Slava Gerovitch, and Tanya Khovanova Consider a finite set of lines in 3-space. A joint is a point where three of these lines (not lying in the same plane) intersect. If there are L lines, what is the largest possible number of joints? Well, let’s try our luck and randomly choose k planes. Any pair of planes produces a line, and any triple of planes, a joint. Thus, they produce L := k(k − 1)/2 lines and J :=p k(k − 1)(k − 2)/6 joints. If k is large, J is 2 3=2 about 3 L . For many years it was conjectured that one cannot do much better than that, in the 3=2 sense that if L is large, thenp J ≤ CL , where C is a constant (clearly, C ≥ 2 ). -
Hermitian Clifford Analysis on Superspace
Hermitian Clifford analysis on superspace Hennie De Schepper, Alí Guzmán Adán, Frank Sommen Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Galglaan 2, 9000 Ghent, Belgium. Abstract In this paper we first recall the proper algebraic framework, i.e. the radial algebra, needed to extend Hermitian Clifford analysis to the superspace setting. The fundamental objects for this extension then are introduced by means of an abstract complex structure on the Hermitian radial algebra. This leads to a natural representation of this Hermitian radial algebra on superspace. Keywords. Hemitian Clifford analysis, Superspace, Radial algebra Mathematics Subject Classification (2010). 1 Introduction Clifford analysis nowadays is a well established mathematical discipline constituting a natural refinement of harmonic analysis. In its most simple setting, it focusses on the null solutions of the Dirac operator Pm m @x = j=1 ej@xj , where the elements (e1; : : : ; em) form an orthonormal basis for the Euclidean space R leading to the construction of the real Clifford algebra R0;m. This setting is known as the orthogonal case since the fundamental group leaving the Dirac operator invariant is Spin(m) which is a double covering of SO(m). By taking the dimension even, say 2m, and introducing a so-called complex structure J 2 SO(2m), the fundamental elements of Hermitian Clifford analysis arise in a natural way from the orthogonal setting. The Hermitian case focusses on h-monogenic functions, h-monogenicity being expressed by means of two mutually adjoint Dirac operators which are invariant under the realization of the unitary group U(m) 1 in Spin(2m). -
Representations of the Lie Superalgebra Osp(1L2n)
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 031, 27 pages Representations of the Lie Superalgebra osp(1j2n) with Polynomial Bases Asmus K. BISBO a, Hendrik DE BIE b and Joris VAN DER JEUGT a a) Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium E-mail: [email protected], [email protected] b) Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Krijgslaan 281-S8, B-9000 Gent, Belgium E-mail: [email protected] Received June 30, 2020, in final form March 10, 2021; Published online March 25, 2021 https://doi.org/10.3842/SIGMA.2021.031 Abstract. We study a particular class of infinite-dimensional representations of osp(1j2n). These representations Ln(p) are characterized by a positive integer p, and are the lowest com- ponent in the p-fold tensor product of the metaplectic representation of osp(1j2n). We con- struct a new polynomial basis for Ln(p) arising from the embedding osp(1j2np) ⊃ osp(1j2n). The basis vectors of Ln(p) are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra valued polynomials with integer coefficients in np variables. Using com- binatorial properties of these tableau vectors it is deduced that they form indeed a basis. The computation of matrix elements of a set of generators of osp(1j2n) on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau. Key words: representation theory; Lie superalgebras; Young tableaux; Clifford analysis; parabosons 2020 Mathematics Subject Classification: 17B10; 05E10; 81R05; 15A66 1 Introduction In representation theory of Lie algebras, Lie superalgebras or their deformations, there are often three problems to be tackled. -
Linking Clifford Analysis and Combinatorics Through Bijective Methods
9th International Conference on Clifford Algebras and their Applications in Mathematical Physics K. Gurlebeck¨ (ed.) Weimar, Germany, 15–20 July 2011 LINKING CLIFFORD ANALYSIS AND COMBINATORICS THROUGH BIJECTIVE METHODS H. R. Maloneka;c and M. I. Falcao˜ b;c a Department of Mathematics, University of Aveiro, Portugal. b Department of Mathematics and Applications, University of Minho, Portugal. c Center for Research and Development in Mathematics and Applications, Portugal. Keywords: Monogenic polynomials, Special Functions, Appell sequences, combinatorial iden- tities Abstract. The application of Clifford Analysis methods in Combinatorics has some peculiari- ties due to the use of noncommutative algebras. But it seems natural to expect from here some new results different from those obtained by using approaches based on several complex vari- able. For instances, the fact that in Clifford Analysis the point-wise multiplication of monogenic functions as well as their composition are not algebraically closed in this class of generalized holomorphic functions causes serious problems. Indeed, this is one of the reasons why in poly- nomial approximation almost every problem needs the construction of specially adapted poly- nomial bases. Our aim is to show that the analysis and comparison of different representations of the same polynomial or entire function allow to link Clifford Analysis and Combinatorics by means of bijective methods. In this context we also stress the central role of the hypercomplex derivative for power series representations in connection with the concept of Appell sequences as analytic tools for establishing this link. 1 1 INTRODUCTION “Remember, it’s supposed to be fun” Richard Feynman, quoted in Some time with Feynman by L. -
A Theory of Neural Computation with Clifford Algebras
A Theory of Neural Computation with Clifford Algebras Dissertation zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften (Dr.–Ing.) der Technischen Fakult¨at der Christian-Albrechts-Universit¨at zu Kiel Sven Buchholz Kiel 2005 1. Gutachter Prof. Dr. Gerald Sommer (Kiel) 2. Gutachter Prof. Dr. Reinhold Schneider (Kiel) 3. Gutachter Prof. Dr. Thomas Martinetz (L ¨ubeck) Datumderm¨undlichenPr¨ufung: 16.03.2005 Abstract The present thesis introduces Clifford Algebra as a framework for neural compu- tation. Clifford Algebra subsumes, for example, the reals, complex numbers and quaternions. Neural computation with Clifford algebras is model–based. This principle is established by constructing Clifford algebras from quadratic spaces. Then the subspace grading inherent to any Clifford algebra is introduced, which allows the representation of different geometric entities like points, lines, and so on. The above features of Clifford algebras are then taken as motivation for introducing the Basic Clifford Neuron (BCN), which is solely based on the geometric product of the underlying Clifford algebra. Using BCNs the Linear Associator is generalized to the Clifford associator. As a second type of Clifford neuron the Spinor Clifford Neuron (SCN) is presented. The propagation function of a SCN is an orthogonal transformation. Examples of how Clifford neurons can be used advantageously are given, including the linear computation of M¨obius transformations by a SCN. A systematic basis for Clifford neural computation is provided by the important notions of isomorphic Clifford neurons and isomorphic representations. After the neuron level is established, the discussion continues with (Spinor) Clifford Multi- layer Perceptrons. The treatment is divided into two parts according to the type of activation function used. -
LARRY GUTH Curriculum Vitae Department Of
LARRY GUTH Curriculum Vitae Department of Mathematics 617-253-4326 Room 2-278 [email protected] 77 Massachusetts Avenue Cambridge, MA 02139 Degrees: B.S. Mathematics, Yale University, 2000 Ph.D. Mathematics, MIT, 2005 Employment: Samelson Fellow, Stanford University, 2005-2006 Szego Assistant Professor, Stanford University, 2006-2008 National Science Foundation Postoctoral Fellow, 2006-2008 Tenure-stream Assistant Professor, University of Toronto, 2008-2011 Member, Institute for Advanced Study, 2010-2011 Professor, New York University, 2011-2012 Professor, MIT, Department of Mathematics, 2012- Claude Shannon Professor of Mathematics, MIT, 2019- Fellowships: National Science Foundation Graduate Fellowship, 2001-2003 National Science Foundation Postdoctoral Fellowship, 2006-2008 Alfred P. Sloan Research Fellowship, 2010-2014 Simons Investigator, 2014- Honors: Sloan Research Fellowship, 2010 Salem Prize, 2013 Simons Investigator, Simons Fnd, 2014 Clay Research Award, 2015 School of Science Prize for Excellence in Graduate Teaching, 2015 New Horizons in Mathematics Breakthrough Prize, 2015 Fellow of the American Academy of Arts and Sciences, 2018 Best Paper Award, International Consortium of Chinese Math., 2018 Fellow of the American Mathematical Society, 2019 B^ocher Memorial Prize, 2020 Maryam Mirzakhani Prize in Mathematics, 2020 Margaret MacVicar Faculty Fellow, MIT, 2021 Member of the National Academy of Sciences, 2021 1 B.S. Student Thesis Supervised (University of Toronto): Lisha (Janet) Li: University of Toronto: Received a 2017 PhD in Statistics & Machine Learning, from U.C. Berkeley; currently at Amplify Partners, S.F. Bay Area Graduate Students Supervised: University of Toronto: Dominic Dotterrer: PhD, University of Toronto, 2013: Dickson Instr., University of Chicago, 2013-2016; currently: Senior Research Scientist, Goldman Sachs, 2019-. -
A Variant of Hörmander's L2 Theorem for Dirac Operator in Clifford Analysis
Noname manuscript No. (will be inserted by the editor) A variant of H¨ormander's L2 theorem for Dirac operator in Clifford analysis Yang Liu · Zhihua Chen · Yifei Pan Received: date / Accepted: date Abstract In this paper, we give the H¨ormander's L2 theorem for Dirac operator over open subset Ω 2 Rn+1 with Clifford algebra. Some sufficient conditions on the existence of weak solutions for Dirac operators have been found in the sense of Clifford analysis. In particular, if Ω is bounded, then we prove that for any f in L2 space with value in Clifford algebra, we can find a weak solution of Dirac operator such that Du = f with the solution u in the L2 space as well. The method is based on H¨ormander's L2 existence theorem in complex analysis and the L2 weighted space is utilised. Keywords H¨ormander's L2 theorem · Clifford analysis · weak solution · Dirac operator Mathematics Subject Classification (2000) 32W50 · 15A66 1 Introduction The development of function theories over Clifford algebras has proved a useful set- ting for generalizing many aspects of one variable complex function theory to high- er dimensions. The study of these function theories is referred to as Clifford analysis [Brackx et al(1982),Huang et al(2006),Gong et al(2009),Ryan(2000)]. This analysis is closely related to a number of studies made in mathematical physics, and many appli- cations have been found in this area in recent years. In [Ryan(1995)], Ryan considered solutions of the polynomial Dirac operator, which affords an integral representation.