3. Postdoctoral Program - MSRI

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Proof. For each 1 ≤ i < j ≤ 4 where xi + xj = g(mij), we have six equations: x1 + x2 = g(m12) x1 + x3 = g(m13) x1 + x4 = g(m14) x2 + x3 = g(m23) x2 + x4 = g(m24) x3 + x4 = g(m34) This is equivalent to the following row reduced augmented matrix: ⎛ g(m12)+g(m13)−g(m23) 2 ⎜ 1 ⎜ 0 ⎜ 0 ⎜ 0 ⎜ 0 ⎜ ⎝ 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ⎟ g(m12)−g(m13)+g(m23) ⎟ 2 ⎟ −g(m12)+g(m13)+g(m23) ⎟ 2 ⎟ −g(m12)−g(m13)+g(m23)+2g(m14) ⎟ 2 ⎟ g(m13) + g(m24) − g(m23) − g(m14) ⎟ ⎠ 0 0 0 0 g(m12) + g(m34) − g(m23) − g(m14) which gives us our desired equations. Definition 2. A set of points is concyclic if they lie on a common circle. Similarly, a set of points is nonconcyclic if one cannot construct a common circle through them. We will now examine the cases of concyclic and nonconcyclic points on a parabola. 8 ⎞
2.4 Concyclic Points on a Parabola Proposition 1. If Pi for 1 ≤ i ≤ 4 are four rational points on the parabola, then they are concyclic if and only if x1 + x2 + x3 + x4 = − 2b a . Proof. Let Cα,β,ρ be the circle (x − α) 2 + (y − β) 2 = ρ 2 , where α, β, ρ ∈ R. This circle intersects y = ax 2 + bx + c at the points whose x-coordinates are the roots (x − α) 2 + (ax 2 + bx + c − β) 2 − ρ 2 or a 2 x 4 + 2abx 3 + (2a(c − β) + b 2 + 1)x 2 + 2b(c − β)x + (c − β) 2 . Since the coefficient of x 3 is 2b a , −(x1 + x2 + x3 + x4) = 2b a . Now if we have xi, 1 ≤ i ≤ 4 such that −(x1 + x2 + x3 + x4) = 2b, then we must a have the following: a 2 (x−x1)(x−x2)(x−x3)(x−x4) = a 2 x 4 +2abx 3 +(2a(c−β)+b 2 +1)x 2 +2b(c−β)x+(c−β) 2 . So, we see that x1 + x2 + x3 + x4 = − 2b, for some α, β, and ρ. a Theorem 5. Suppose that P1, P2, P3, and P4 are rational and concyclic. Then, these points are at rational distance if and only if there are nonzero rational values m12, m13, and m23 such that the equations of Theorem 4 hold. If we have this condition, then x4 = − 1 2 (g(m12) + g(m13) + g(m23) + 4b a ). Proof. By proposition 1, x4 = −(x1 + x2 + x3 + 2b). Therefore, we can directly a solve for x4 using the values given for the x1, x2, x<strong>3.</strong> Inputting, our values in, we get x4 = − 1 2 (g(m12) + g(m13) + g(m23) + 4b a ). 9
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Final Report, 2005-2010 and Annual
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6. Appendix - Final Reports .......
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• Spring 2008: Representation The
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place the highest value on the oppo
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which establishes the relation betw
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Public Understanding of Mathematics
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1.3 Scientific Programs and their A
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The first week of May 2010 coincide
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The goal of this program was, throu
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"An illustration of the tropical Ab
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• lead to new breakthroughs in th
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Other mathematical breakthroughs th
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• Advice on preparing for intervi
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An off-site workshop partially fund
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1.7 Other Scientific Workshops Hot
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experienced circle leaders. Togethe
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2. Program and Workshop Data 2.1 Pr
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Jöricke Burglind Institut des Haut
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Smith Ivan Centre for Mathematical
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Home Institution Classified by Stat
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2.4 Workshop Participant List (See
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2.7 Program Publication List Last N
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Ghiggini Paolo Sutures and contact
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Lekili Yanki Geometric composition
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Draisma Jan The tropical projective
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Krasner Daniel Patterns in sl(2)-li
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Sarkar Sucharit Murasugi sums Matt
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Falk Michael Feichtner Eva Tropical
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Manolescu Ciprian Manolescu Ciprian
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Steingrimsson Einar Steingrimsson E
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Here are additional details on the
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Kutluhan, Cagatay Lekili, Yanki Cag
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Savelyev, Yakov Vertesi, Vera Yakov
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Katz, Eric Lopez de Medrano, Lucia
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Grigsby, Elisenda Kutluhan, Cagatay
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Sazdanovic, Radmila Wehrli, Stephan
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Angeltveit, Vigleik Crofts, Scott H
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3.2 Postdoctoral Fellow Placement L
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3.4 Postdoctoral Fellow Demographic
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Home Institution Classified by Coun
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SGS 1: IAS/PCMI Summer Workshop: Th
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Jafarov, Elchin University of Alask
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Xie, Yu Purdue University Graduate
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4.3 Program Associates Program Asso
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P r o g r a m A s s o c i a t e s D
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Home Institution Classified by Coun
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5. Undergraduate Program (MSRI-UP)
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This project seeks to use elliptic
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6. Appendix - Final Reports 97
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2 2. Participants The response to o
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4 • Gleb Koshevoy: Bases of tropi
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6 While at MSRI, Gathmann extended
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8 In conclusion, my postdoc at MSRI
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10 -with Christian Haase, who staye
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12 “I was working on the relation
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14 Josephine Yu PhD: UC Berkeley, 2
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16 6. Alex Fink was an invited spea
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Tropical Geometry Postdoctoral Fell
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Tropical Geometry Demographic Summa
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Ozsváth-Szabó Heegaard homology t
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field. Nevertheless, some of it is
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developments which occurred in this
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the audience would know a little ab
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on the group of Hamiltonian diffeom
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German Research Foundation Mentor:
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Mark spend his time at MSRI working
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PhD: University of Texas, 2007 Posi
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example Gadbled, Mandini, Ma’u, H
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Symplectic and Contact Geometry and
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Symplectic and Contact Geometry and
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2 REPORT ON THE MSRI WORKSHOP “HO
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Homology Theory of Knots and Links
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Complementary Program August 17, 20
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Complementary Program 2009-10 Postd
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Complementary Program Demographic S
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MID YEAR REPORT - DECEMBER 2009 VIG
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California. We talked about several
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EXTERNAL POSTDOCTORAL FELLOW REPORT
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on. While his research area is some
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POSTDOCTORAL FELLOW REPORT December
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[1] (with Garcia-Puente, Martin del
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REPORT FROM MENTOR June 2010 Year e
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I have also begun working on relate
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2 Professional Travel Sep. 6 - 26,
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REPORT FROM MENTOR June 2010 Karl h
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POSTDOCTORAL FELLOW REPORT June 201
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REPORT FROM MENTOR June 2010 ------
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Postdoctoral Program supported by t
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RO(S 1 )-graded homotopy groups of
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MSRI POSTDOCTORAL FELLOWSHIP REPORT
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2 TRISTRAM BOGART and its facets ar
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To Whom It May Concern: During the
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MSRI FELLOWSHIP FINAL REPORT SCOTT
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As his mentor at UCSC, I had freque
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- UC Davis Combinatorics Seminar -
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Christopher J. Hillar University of
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G. Isely, C. Hillar, and F. Sommer,
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Postdoc Report for: Dr. Christopher
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2 NSF REPORT FOR ERIC KATZ linear s
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Year-End Report 2010-11 NSF Math In
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Jan. 25 - 28, 2011 Recruitment Visi
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MSRI POSTDOCTORAL FELLOWSHIP REPORT
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Final report: mentoring Sikimeti Ma
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eginning of the Fall semester, and
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MSRI FINAL REPORT JARED SPECK 1. Ba
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External Postdoctoral Fellows 2009-
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External Postdoctoral Fellows 2009-
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2 connections to geometry, topology
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Connections for Women: Tropical Geo
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Participant List MSRI Workshop: Con
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Connections for Women: Tropical Geo
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2 Combinatorics, Several Complex Va
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Introductory Workshop: Tropical Geo
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Currently Available Videos � Sam
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Name Institution Hering, Milena Uni
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Introductory Workshop: Tropical Geo
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analysis. The clarity of Payne’s
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Tropical Geometry in Combinatorics
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Name Institution Kharlamov, Viatche
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Tropical Geometry in Combinatorics
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Above satisfactory 35 73% Somewhat
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Tropical structures in geometry and
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Research Workshop: Tropical Structu
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Research Workshop: Tropical Structu
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Algebraic Structures in the Theory
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2 WORKSHOP REPORT toric fibrations,
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Algebraic Structures in the Theory
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Wednesday November 18, 2009 09:00AM
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Participant List MSRI Workshop: Alg
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Name Institution Smith, Ivan Centre
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Research Workshop: Symplectic and C
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Detailed review: We will now discus
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Symplectic and Contact Topology and
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Final Report MSRI Connections for W
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Connections : Symplectic and Contac
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Currently Available Videos � Siki
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Milin, Isidora University of Illino
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Final Report Introductory Workshop
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at a variety of levels. Since a hig
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Introductory Workshop: Symplectic a
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Participant List MSRI Workshop: Int
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Park, Jongil Seoul National Univers
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Symplectic and Poisson Geometry in
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Symplectic and Poisson Geometry in
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Symplectic Geometry, Noncommutative
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09:00AM - 10:00AM Registration 10:0
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of the theory via webs and foams mo
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04/29/2010 Connections for Women: H
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REPORT ON THE MSRI WORKSHOP “INTR
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Introductory Workshop: Homology The
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Research Workshop: Homology Theorie
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04/29/2010 Research Workshop: Homol
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comprehensive lectures explaining t
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Hot Topics: Black Holes in Relativi
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Currently Available Videos � Miha
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Shao, Shuanglin Institute for Advan
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MATHEMATICAL SCIENCES RESEARCH INST
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MPDI Summer Institute 2009: Final R
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school teachers (in the case of sim
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in length)? The audience would be e
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Evaluation #1 1. What were the bene
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6. Do you have to report to an admi
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your class because your way of teac
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employee with a summary of what you
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what have you learned in this insti
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Evaluation #9 (signed Kelly L.) 1.
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therefore should not be given the j
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would you like them organized? Plea
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Evaluation #11 1. What were the ben
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Evaluation #12 (signed Jake Disston
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ideas are insignificant in light of
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manipulative and pictorial level. F
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3. Postdoctoral Program - MSRI

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