3. Postdoctoral Program - MSRI

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2 TRISTRAM BOGART and its facets are known [Zie95], but otherwise it is little-understood. We verify several categorical properties, prove a structural result in the case where (P,Q) is a pair of generic polygons, and compute several families of examples. (3) Properties of Smooth Lattice Polytopes An important reason for studying (lattice) polytopes is their connection to toric geometry. Among the properties of a projective toric variety X that are determined by its lattice polytope P are whether (a) X is smooth: if and only if the primitive edge directions at each vertex of P form a lattice basis, (b) X is normal (integrally closed): if and only if the lattice points in (P,1) ⊂ R n+1 generate the monoid of lattice points in the cone over P, and (c) the ideal of X has a square-free quadratic initial ideal: if and only if P possesses a regular unimodular triangulation. It is known that (1) =⇒ (2) =⇒ (3), but it is unknown whether (2) or (3) implies (1). This question was the subject of a workshop I attended at the American Institute for Mathematics in 2009 and I was among a group of participants that wrote a paper [BHH + ] verifying the reverse implications for lattice three-polytopes with up to 12 lattice points. My mentor Federico Ardila, along with Florian Block, introduced a new construction of polytopes in [AB]: given any d-polytope P and any partition λ with d parts, there is a ×· · ·×∆λd . polytope P λ that surjects onto P with each fiber being a product of simplices ∆λ1 If P is smooth then so is P λ for any λ. Our current project is to show that the other properties carry over from P to P λ . In particular, we can show that normality carries over, and we are currently working regular unimodular triangulations. 3. Travel With the help of MSRI travel funding, I was able to present the tropical lifting result at a conference in Osaka, Japan and at seminars in Saarbrücken, Germany and in Davis, California. I also gave a colloqium talk at Sonoma State University and research talks on other topics at Bay Area Discrete Mathematics Day (BADMath Day) 22 at MSRI and at a Society for Industrial and Applied Mathematics conference on optimization in Darmstadt, Germany. I attended the Canadian Mathematical Society winter meeting in Vancouver and will attend and speak at the CMS summer meeting in Edmonton. 4. Teaching and Mentoring In fall 2010, I taught a section of calculus I at San Francisco State with 35 students and one teaching assistant. Teaching this material was not new for me, but San Francisco State is a little different in its unusually diverse student population. There are many non-traditional students who haven’t taken a math class in many years. Helping these students keep up was a challenge, but a rewarding one. I also assisted Prof. Ardila in his graduate course on polytopes and discrete geometry by giving two lectures, answering questions, and helping a pair of students on their final project, related to my work on mapping polytopes. Along with Prof. Matthias Beck, I coadvised a masters’ thesis on Goulomb rulers. Goulomb rulers arise in number theory (of which I know little) but the project was to study them using discrete geometry. Specifically, the set of such rulers with fixed parameters is enumerated by the set of lattice points that lie inside a polytope but not on any of a collection of hyperplanes. The student created a new type of labelled graphs that index the regions of this hyperplane arrangement.
MSRI FELLOWSHIP FINAL REPORT 3 5. Service I served as the local organizer for BadMath Day 23, which took place at SFSU on April 30, 2011. BadMath Day is a one-day free conference that aims to introduce a variety of research topics in discrete mathematics to students around the region and to facilitate communication among mathematicians and students at all levels. The recent conference featured four half-hour talks and two one-hour talks and was attended by 63 people, about two-thirds of them students. I also coorganized and regularly attended the Algebra-Geometry-Combinatorics seminar at SFSU, and assisted Ardila and Beck in organizing the larger conference Formal Power Series and Algebraic Combinatorics (FPSAC) ’10, held on August 2-6, 2010 at SFSU. References [AB] Federico Ardila and Florian Block. Universal polynomials for severi degrees of toric surfaces. arXiv:1012.5305. [BHH + ] Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill, Andreas Paffenholz, Francisco Santos, and Hal Schenck. Few smooth d-polytopes with N lattice points. arXiv:1010.3887. [BK] Tristram Bogart and Eric Katz. Obstructions to lifting tropical curves in hypersurfaces. arXiv:1101.0297. [Zie95] G.M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
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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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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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Currently Available Videos � Sam
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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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Currently Available Videos � Sam
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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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Research Workshop: Symplectic and C
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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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REPORT ON THE MSRI WORKSHOP “HOMO
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of the theory via webs and foams mo
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Schedule Connections for Women: Hom
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Connections for Women: Homology The
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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, FINAL REPORT
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Introductory Workshop: Homology The
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Connections for Women: Homology The
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Research Workshop: Homology Theorie
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2 REPORT ON THE MSRI WORKSHOP “HO
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Schedule Research Workshop: Homolog
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Currently Available Videos Research
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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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Yes.. I would like to talk to 4 - 6
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No, but I have continued to make my
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5. Are you planning to work with co
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(1) change the order of my curricul
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I will show my 6th grade teacher co
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to teaching and planning. Perhaps b
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III - Develop a lesson on similarit
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4. What are you going to implement/
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matics? * How are people teaching t
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Evaluation #23 (I guess this is Cla
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Evaluation #24 1. What were the ben
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2010 Mathematical Sciences Research
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During the 2007-2009 summers, 46 st
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MSRI-UP peers promoted mathematical
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The outcome of all the students’
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school in mathematics.” They prov
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16. Evidence Already Pointing to Lo
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II. Research Projects 1. Your resea
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5. Is there a workshop, discussion
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V. Non-Academic Aspects of MSRI-UP
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MSRI-UP 2010 Calendar, Weeks 1 and
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MSRI-UP 2010 Calendar, Weeks 5 and
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In 1945 Stanislaw Ulam posed a ques
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2.2 Circle Theorem 2. Let r ∈ Q w
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parabola is (a) The parabola and tw
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Proof. For each 1 ≤ i < j ≤ 4 w
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2.5 Non-Concyclic Points on a Parab
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⎛ ⎜ 1 ⎜ 0 ⎜ 0 ⎜ 0 ⎜ 0
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following rational points: n23 = Y1
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Toric Varieties Cox, David Invited
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Toric varieties Held: June 15-26, 2
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the graduate participants to hear a
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This lecture series addressed topic
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Participant List MSRI Workshop: Sym
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Example of a Summer Graduate Worksh
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Random Matrix theory July 6, 2009 t
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Random Matrix Theory Held: July 6 -
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Participant List MSRI Workshop: Ran
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Below is the list of topics conside
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stability estimates for the inverse
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Participant List MSRI Workshop: Inv
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The Arithmetic of L-Functions IAS/P
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ackground material on group represe
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Computational Theory of Real Reduct
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3. Postdoctoral Program - MSRI

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