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Home » Positivity Questions in Geometric Combinatorics

Summer Graduate School

Positivity Questions in Geometric Combinatorics July 10, 2017 - July 21, 2017
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Location: MSRI: Simons Auditorium, Atrium
Organizers Eran Nevo (The Hebrew University of Jerusalem), Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
Lecturer(s)

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  • Eran Nevo (The Hebrew University of Jerusalem)
  • Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
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Description
McMullen’s g-Conjecture from 1970 is a shining example of mathematical foresight that combined all results available at that time to conjure a complete characterization of face numbers of convex simple/simplicial polytopes. The key statement in its verification is that certain combinatorial numbers associated to geometric (or topological) objects are non-negative. The aim of this workshop is to introduce graduate students to selected contemporary topics in geometric combinatorics with an emphasis on positivity questions. It is fascinating that the dual notions of simple and simplicial polytopes lead to different but equally powerful algebraic frameworks to treat such questions. A key feature of the lectures will be the simultaneous development of these algebraic frameworks from complementary perspectives: combinatorial-topological and convex-geometric.  General concepts (such as Lefschetz elements, Hodge–Riemann–Minkowski inequalities) will be developed side-by-side, and analogies will be drawn to concepts in algebraic geometry, Fourier analysis, rigidity theory and measure theory. This allows for entry points for students with varying backgrounds.  The courses will be supplemented with guest lectures highlighting further connections to other fields. Suggested prerequisites For eligibility and how to apply, see the Summer Graduate Schools homepage     Group Picture
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Schedule, Notes/Handouts & Videos
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Jul 10, 2017
Monday
08:45 AM - 09:00 AM
  Introduction to MSRI
09:00 AM - 10:00 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 11, 2017
Tuesday
09:00 AM - 10:00 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 12, 2017
Wednesday
09:00 AM - 10:00 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Centrally symmetric polytopes and spheres: towards the upper bound conjectures
Steven Klee (Seattle University), Isabella Novik (University of Washington)
11:30 AM - 02:00 PM
  BBQ at Tilden Park
02:00 PM - 03:00 PM
  Lower Bound Theorems for centrally symmetric complexes
Steven Klee (Seattle University), Isabella Novik (University of Washington)
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 13, 2017
Thursday
09:00 AM - 10:00 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Presentations
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 14, 2017
Friday
09:00 AM - 10:00 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Presentations
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 17, 2017
Monday
09:00 AM - 10:00 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 18, 2017
Tuesday
09:00 AM - 10:00 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 19, 2017
Wednesday
09:00 AM - 10:00 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Classification of combinatorial polynomials (in particular, Ehrhart polynomials of zonotopes)
Matthias Beck (San Francisco State University)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  The algebraic and combinatorial structure of generalized permutahedra
Federico Ardila (San Francisco State University)
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 20, 2017
Thursday
09:00 AM - 10:00 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Presentations
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises
Jul 21, 2017
Friday
09:00 AM - 10:00 AM
  Lecture
Eran Nevo (The Hebrew University of Jerusalem)
10:00 AM - 10:30 AM
  Coffee Break
10:30 AM - 11:30 AM
  Lecture
Raman Sanyal (Johann Wolfgang Goethe-Universität Frankfurt)
11:30 AM - 12:30 PM
  Exercises
12:30 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Presentations
03:00 PM - 03:30 PM
  Tea
03:30 PM - 05:00 PM
  Exercises