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Forbidden Subgraph Characterization of Quasi-Line Graphs Medha Dhurandhar [email protected]
Forbidden Subgraph Characterization of Quasi-line Graphs Medha Dhurandhar [email protected] Abstract: Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques. 1. Introduction: A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. In [2] Chudnovsky and Seymour gave a constructive characterization of quasi-line graphs. An alternative characterization of quasi-line graphs is given in [3] stating that a graph has a fuzzy reconstruction iff it is a quasi-line graph and also in [4] using the concept of sums of Hoffman graphs. Here we characterize quasi-line graphs in terms of the forbidden induced subgraphs like line graphs. We consider in this paper only finite, simple, connected, undirected graphs. The vertex set of G is denoted by V(G), the edge set by E(G), the maximum degree of vertices in G by Δ(G), the maximum clique size by (G) and the chromatic number by G). N(u) denotes the neighbourhood of u and N(u) = N(u) + u. For further notation please refer to Harary [3]. 2. Main Result: Before proving the main result we prove some lemmas, which will be used later. Lemma 1: If G is {3K1, C5}-free, then either 1) G ~ K|V(G)| or 2) If v, w V(G) are s.t. -
After Ramanujan Left Us– a Stock-Taking Exercise S
Ref: after-ramanujanls.tex Ver. Ref.: : 20200426a After Ramanujan left us– a stock-taking exercise S. Parthasarathy [email protected] 1 Remembering a giant This article is a sequel to my article on Ramanujan [14]. April 2020 will mark the death centenary of the legendary Indian mathe- matician – Srinivasa Ramanujan (22 December 1887 – 26 April 1920). There will be celebrations of course, but one way to honour Ramanujan would be to do some introspection and stock-taking. This is a short survey of notable achievements and contributions to mathematics by Indian institutions and by Indian mathematicians (born in India) and in the last hundred years since Ramanujan left us. It would be highly unfair to compare the achievements of an individual, Ramanujan, during his short life span (32 years), with the achievements of an entire nation over a century. We should also consider the context in which Ramanujan lived, and the most unfavourable and discouraging situation in which he grew up. We will still attempt a stock-taking, to record how far we have moved after Ramanujan left us. Note : The table below should not be used to compare the relative impor- tance or significance of the contributions listed there. It is impossible to list out the entire galaxy of mathematicians for a whole century. The table below may seem incomplete and may contain some inad- vertant errors. If you notice any major lacunae or omissions, or if you have any suggestions, please let me know at [email protected]. 1 April 1920 – April 2020 Year Name/instit. Topic Recognition 1 1949 Dattatreya Kaprekar constant, Ramchandra Kaprekar number Kaprekar [1] [2] 2 1968 P.C. -
Clay Mathematics Institute 2005 James A
Contents Clay Mathematics Institute 2005 James A. Carlson Letter from the President 2 The Prize Problems The Millennium Prize Problems 3 Recognizing Achievement 2005 Clay Research Awards 4 CMI Researchers Summary of 2005 6 Workshops & Conferences CMI Programs & Research Activities Student Programs Collected Works James Arthur Archive 9 Raoul Bott Library CMI Profile Interview with Research Fellow 10 Maria Chudnovsky Feature Article Can Biology Lead to New Theorems? 13 by Bernd Sturmfels CMI Summer Schools Summary 14 Ricci Flow, 3–Manifolds, and Geometry 15 at MSRI Program Overview CMI Senior Scholars Program 17 Institute News Euclid and His Heritage Meeting 18 Appointments & Honors 20 CMI Publications Selected Articles by Research Fellows 27 Books & Videos 28 About CMI Profile of Bow Street Staff 30 CMI Activities 2006 Institute Calendar 32 2005 Euclid: www.claymath.org/euclid James Arthur Collected Works: www.claymath.org/cw/arthur Hanoi Institute of Mathematics: www.math.ac.vn Ramanujan Society: www.ramanujanmathsociety.org $.* $MBZ.BUIFNBUJDT*OTUJUVUF ".4 "NFSJDBO.BUIFNBUJDBM4PDJFUZ In addition to major,0O"VHVTU BUUIFTFDPOE*OUFSOBUJPOBM$POHSFTTPG.BUIFNBUJDJBOT ongoing activities such as JO1BSJT %BWJE)JMCFSUEFMJWFSFEIJTGBNPVTMFDUVSFJOXIJDIIFEFTDSJCFE the summer schools,UXFOUZUISFFQSPCMFNTUIBUXFSFUPQMBZBOJOnVFOUJBMSPMFJONBUIFNBUJDBM the Institute undertakes a 5IF.JMMFOOJVN1SJ[F1SPCMFNT SFTFBSDI"DFOUVSZMBUFS PO.BZ BUBNFFUJOHBUUIF$PMMÒHFEF number of smaller'SBODF UIF$MBZ.BUIFNBUJDT*OTUJUVUF $.* BOOPVODFEUIFDSFBUJPOPGB special projects -
Prahladh Harsha
Prahladh Harsha Toyota Technological Institute, Chicago phone : +1-773-834-2549 University Press Building fax : +1-773-834-9881 1427 East 60th Street, Second Floor Chicago, IL 60637 email : [email protected] http://ttic.uchicago.edu/∼prahladh Research Interests Computational Complexity, Probabilistically Checkable Proofs (PCPs), Information Theory, Prop- erty Testing, Proof Complexity, Communication Complexity. Education • Doctor of Philosophy (PhD) Computer Science, Massachusetts Institute of Technology, 2004 Research Advisor : Professor Madhu Sudan PhD Thesis: Robust PCPs of Proximity and Shorter PCPs • Master of Science (SM) Computer Science, Massachusetts Institute of Technology, 2000 • Bachelor of Technology (BTech) Computer Science and Engineering, Indian Institute of Technology, Madras, 1998 Work Experience • Toyota Technological Institute, Chicago September 2004 – Present Research Assistant Professor • Technion, Israel Institute of Technology, Haifa February 2007 – May 2007 Visiting Scientist • Microsoft Research, Silicon Valley January 2005 – September 2005 Postdoctoral Researcher Honours and Awards • Summer Research Fellow 1997, Jawaharlal Nehru Center for Advanced Scientific Research, Ban- galore. • Rajiv Gandhi Science Talent Research Fellow 1997, Jawaharlal Nehru Center for Advanced Scien- tific Research, Bangalore. • Award Winner in the Indian National Mathematical Olympiad (INMO) 1993, National Board of Higher Mathematics (NBHM). • National Board of Higher Mathematics (NBHM) Nurture Program award 1995-1998. The Nurture program 1995-1998, coordinated by Prof. Alladi Sitaram, Indian Statistical Institute, Bangalore involves various topics in higher mathematics. 1 • Ranked 7th in the All India Joint Entrance Examination (JEE) for admission into the Indian Institutes of Technology (among the 100,000 candidates who appeared for the examination). • Papers invited to special issues – “Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding” (with Eli Ben- Sasson, Oded Goldreich, Madhu Sudan, and Salil Vadhan). -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Some Remarks on Even-Hole-Free Graphs
Some remarks on even-hole-free graphs Zi-Xia Song∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Abstract A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger’s conjecture and the Erd˝os-Lov´asz Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all k ≥ 7, every even-hole-free graph with no Kk minor is (2k − 5)-colorable; every even-hole-free graph G with ω(G) < χ(G) = s + t − 1 satisfies the Erd˝os-Lov´asz Tihany conjecture provided that t ≥ s>χ(G)/3. Furthermore, we prove that every 9-chromatic graph G with ω(G) ≤ 8 has a K4 ∪ K6 minor. Our proofs rely heavily on the structural result of Chudnovsky and Seymour on even-hole-free graphs. 1 Introduction All graphs in this paper are finite and simple. For a graph G, we use V (G) to denote the vertex set, E(G) the edge set, |G| the number of vertices, e(G) the number of edges, δ(G) the minimum degree, ∆(G) the maximum degree, α(G) the independence number, ω(G) the clique number and χ(G) the chromatic number. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. -
Mathematical Sciences 2016
Infosys Prize Mathematical Sciences 2016 Number theory is the branch Ancient civilizations developed intricate of mathematics that deals with methods of counting. Sumerians, Mayans properties of whole numbers or and Greeks all show evidence of elaborate positive integers. mathematical calculations. Akshay Venkatesh Professor, Department of Mathematics, Stanford University, USA • B.Sc. in Mathematics from The University of Western Australia, Perth, Australia • Ph.D. in Mathematics from Princeton University, USA Numbers are Prof. Akshay Venkatesh is a very broad mathematician everything who has worked at the highest level in number theory, arithmetic geometry, topology, automorphic forms and Number theorists are particularly interested in ergodic theory. He is almost unique in his ability to fuse hyperbolic ‘tilings’. These ‘tiles’ carry a great deal of information that are significant in algebraic and analytic ideas to solve concrete and hard number theory. For example they are very problems in number theory. In addition, Venkatesh’s work on the interested in the characteristic frequencies of cohomology of arithmetic groups the tiles. These are the frequencies the tiles studies the shape of these tiles and would vibrate at, if they were used as drums. “I think there’s a lot of math in the world that’s not connects it with other areas of math. at university. Pure math is only one part of math but math is used in a lot of other subjects and I think that’s just as interesting. So learn as much as you can, about all the subjects around math and then see what strikes you as the most interesting.” Prof. -
The Structure of Claw-Free Graphs
The structure of claw-free graphs Maria Chudnovsky and Paul Seymour Abstract A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. At first sight, there seem to be a great variety of types of claw-free graphs. For instance, there are line graphs, the graph of the icosahedron, complements of triangle-free graphs, and the Schl¨afli graph (an amazingly highly-symmetric graph with 27 vertices), and more; for instance, if we arrange vertices in a circle, choose some intervals from the circle, and make the vertices in each interval adjacent to each other, the graph we produce is claw-free. There are several other such examples, which we regard as \basic" claw-free graphs. Nevertheless, it is possible to prove a complete structure theorem for claw- free graphs. We have shown that every connected claw-free graph can be ob- tained from one of the basic claw-free graphs by simple expansion operations. In this paper we explain the precise statement of the theorem, sketch the proof, and give a few applications. 1 Introduction A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. (Graphs in this paper are finite and simple.) Line graphs are claw-free, and it has long been recognized that claw-free graphs are an interesting generalization of line graphs, sharing some of the same properties. For instance, Minty [16] showed in 1980 that there is a polynomial-time algorithm to find a stable set of maximum weight in a claw-free graph, generalizing the algorithm of Edmonds [9, 10] to find a maximum weight matching in a graph. -
2004 Research Fellows
I Institute News 2004 Research Fellows On February 23, 2004, the Clay Mathematics Institute announced the appointment of four Research Fellows: Ciprian Manolescu and Maryam Mirzakhani of Harvard University, and András Vasy and Akshay Venkatesh of MIT. These outstanding mathematicians were selected for their research achievements and their potential to make signifi cant future contributions. Ci ian Man lescu 1 a nati e R mania is c m letin his h at Ha a ni Ciprian Manolescu pr o (b. 978), v of o , o p g P .D. rv rd U - versity under the direction of Peter B. Kronheimer. In his undergraduate thesis he gave an elegant new construction of Seiberg-Witten Floer homology, and in his Ph.D. thesis he gave a remarkable gluing formula for the Bauer-Furuta invariants of four-manifolds. His research interests span the areas of gauge theory, low-dimensional topology, symplectic geometry and algebraic topology. Manolescu will begin his four-year appointment as a Research Fellow at Princeton University beginning July 1, 2004. Maryam Mirzakhani Maryam Mirzakhani (b. 1977), a native of Iran, is completing her Ph.D. at Harvard under the direction of Curtis T. McMullen. In her thesis she showed how to compute the Weil- Petersson volume of the moduli space of bordered Riemann surfaces. Her research interests include Teichmuller theory, hyperbolic geometry, ergodic theory and symplectic geometry. As a high school student, Mirzakhani entered and won the International Mathematical Olympiad on two occasions (in 1994 and 1995). Mirzakhani will conduct her research at Princeton University at the start of her four-year appointment as a Research Fellow beginning July 1, 2004. -
Mathematics People
Mathematics People Akshay Venkatesh was born in New Delhi in 1981 but Venkatesh Awarded 2008 was raised in Perth, Australia. He showed his brillance in SASTRA Ramanujan Prize mathematics very early and was awarded the Woods Me- morial Prize in 1997, when he finished his undergraduate Akshay Venkatesh of Stanford University has been studies at the University of Western Australia. He did his awarded the 2008 SASTRA Ramanujan Prize. This annual doctoral studies at Princeton under Peter Sarnak, complet- prize is given for outstanding contributions to areas of ing his Ph.D. in 2002. He was C.L.E. Moore Instructor at the mathematics influenced by the Indian genius Srinivasa Massachusetts Institute of Technology for two years and Ramanujan. The age limit for the prize has been set at was selected as a Clay Research Fellow in 2004. He served thirty-two, because Ramanujan achieved so much in his as associate professor at the Courant Institute of Math- brief life of thirty-two years. The prize carries a cash award ematical Sciences at New York University and received the of US$10,000. Salem Prize and a Packard Fellowship in 2007. He is now professor of mathematics at Stanford University. The 2008 SASTRA Prize Citation reads as follows: “Ak- The 2008 SASTRA Ramanujan Prize Committee con- shay Venkatesh is awarded the 2008 SASTRA Ramanujan sisted of Krishnaswami Alladi (chair), Manjul Bhargava, Prize for his phenomenal contributions to a wide variety Bruce Berndt, Jonathan Borwein, Stephen Milne, Kannan of areas in mathematics, including number theory, auto- Soundararajan, and Michel Waldschmidt. Previous winners morphic forms, representation theory, locally symmetric of the SASTRA Ramanujan Prize are Manjul Bhargava and spaces, and ergodic theory, by himself and in collabora- Kannan Soundararajan (2005), Terence Tao (2006), and tion with several mathematicians. -
Professor Shang-Hua Teng
Professor Shang-Hua Teng Department of Computer Science, University of Southern California 2054 N. New Hampshire Ave 941 Bloom Walk, SAL 300, Los Angeles, CA 90089 Los Angeles, CA 90027 (213) 440-4498, [email protected] (617) 440-4281 EMPLOYMENT: Seeley G. Mudd Professor and Chair (Computer Science) USC (Fall 2009 – ) Professor (Computer Science) Boston University (2002 – 2009) Research Affiliate Professor (Mathematics) MIT (1999 – Present) Visiting Research Scientist Microsoft Research Asia and New England (2004 – Present) Senior Research Scientist Akamai (1997 – Present) Visiting Professor (Computer Science) Tsinghua University (2004 – 2008) Professor (Computer Science)[Associate Professor before 1999] UIUC (2000 – 2002) Research Scientist IBM Almaden Research Center (1997 – 1999) Assistant Professor (Computer Science) Univ. of Minnesota (1994 – 1997) Research Scientist Intel Corporation (Summer 1994, 1995) Instructor (Mathematics) MIT (1992 – 1994) Computer Scientist NASA Ames Research Center (Summer 1993) Research Scientist Xerox Palo Alto Research Center (1991 – 1992) EDUCATION: Ph.D. (Computer Science) Carnegie Mellon University (1991) Thesis: “A Unified Geometric Approach to Graph Partitioning.” Advisor: Gary Lee Miller M.S. (Computer Science) University of Southern California (1988) Thesis: “Security, Verifiablity, and Universality in Distributed Computing”. Advised by Leonard Adleman and Ming-Deh Huang B.S. (Computer Science) & B.A. (Electrical Engineering) Shanghai Jiao-Tong University, China (1985) Thesis: “Efficient Methods for Implementing Logic Programs.” Advisor: Yixing Hou HONORS/AWARDS: 2011 ACM STOC Best Paper 2009 ACM Fellow, 2009 2009 Fulkerson Prize (American Mathematical Society/Mathematical Programming Society) 2008 G¨odel Prize (ACM-EATCS). 2003 The NSF’s 2003 Budget Request to Congress listed the Smoothed Analysis of Spielman-Teng as one of three most significant accomplishments funded by its computer science division. -
2019-2020 Award #1440415 August 10, 2020
Institute for Pure and Applied Mathematics, UCLA Annual Progress Report for 2019-2020 Award #1440415 August 10, 2020 TABLE OF CONTENTS EXECUTIVE SUMMARY .................................................................................................................... 2 A. PARTICIPANT LIST ....................................................................................................................... 4 B. FINANCE SUPPORT LIST ............................................................................................................. 4 C. INCOME AND EXPENDITURE REPORT .................................................................................... 4 D. POSTDOCTORAL PLACEMENT LIST ........................................................................................ 5 E. MATH INSTITUTE DIRECTORS’ MEETING REPORT ............................................................. 6 F. PARTICIPANT SUMMARY .......................................................................................................... 20 G. POSTDOCTORAL PROGRAM SUMMARY............................................................................... 23 H. GRADUATE STUDENT PROGRAM SUMMARY ...................................................................... 24 I. UNDERGRADUATE STUDENT PROGRAM SUMMARY .......................................................... 25 J. PROGRAM DESCRIPTION .......................................................................................................... 25 K. PROGRAM CONSULTANT LIST ...............................................................................................