Triangle Lectures in Combinatorics (TLC)
These are a series of combinatorial workshops, held once per
semester, each on a Saturday. The series began with its first
meeting in spring 2010. They rotate among the universities
in the Research Triangle. Participants come from numerous
colleges and universities within a few hours drive, and some from
even farther away. These workshops are funded by the National Science
Foundation, in particular enabling us to bring in four exciting
speakers to give one hour talks each time.
TLC steering committee:
Patricia Hersh (NCSU),
Ezra Miller (Duke),
Scott Provan (UNC Chapel Hill), and
Nathan Reading (NCSU)
Upcoming Spring 2012 meeting: February 11, 2012
at Duke University
Speakers:
Alex Fink (NCSU), Matroid Valuations;
Sergey Fomin (Michigan), Cluster structures in rings of SL_3 invariants;
Nets Katz (Indiana University), Erdos Distinct Distances problem in the plane; and
Isabella Novik (University of Washington), Face numbers of centrally symmetric polyopes.
Location: Talks will be in Physics Building Room 128, with coffee breaks in Physics Building Room 101, both at Duke University, Durham, NC
Parking: available in the Bryan Center Parking Garage (PG4) for $5 for the day. This is across the street from the Physics Building.
Schedule:
Friday 4-5pm, NCSU Algebra and Combinatorics Seminar: Leo Mihalcea, Curve Neighborhoods of Schubert Varieties, SAS Hall 4201, NCSU (Raleigh, NC)
Friday 6pm, Seminar Dinner, Details to be determined at the seminar
Saturday, 9:15-10am: coffee
10-11am, Nets Katz, Erdos Distinct Distance Problem in the Plane
11-11:30am, coffee break
11:30am-12:30pm, Alex Fink, Matroid Valuations
12:30-2:30pm, lunch break
2:30-3:30pm, Isabella Novik, Face Numbers of Centrally Symmetric Polytopes
3:30-4pm, coffee break
4-5pm, Sergey Fomin, Cluster Structures in Rings of SL_3 Invariants
6pm-??, Conference dinner at
Revolution, located at 107 West Main Street (with street parking and nearby parking garage).
To preregister: send email to Patricia Hersh, plhersh@ncsu.edu (there is
no registration fee).
Participant expense reimbursement: There is some limited funding available to reimburse participant hotel and mileage expenses -- if this would help you to be able to attend, please send email to Patricia Hersh, plhersh@ncsu.edu, to apply for funding. Include in this email (1) expected hotel expense for 1-2 nights, (2) total roundtrip mileage (or cost of airfare), (3) a sentence or two of justification, (4) list of other sources of funding available to you. This is primarily intended for participants driving a substantial distance or with compelling reasons it would help their research to be able to stay overnight in the Triangle area.
Hotel suggestions:
Hilton Durham, 919-383-8033, 3800 Hillsborough Rd, Durham, NC (rooms for $130 per night)
Millenium Hotel, 919-383-8495, 2800 Campus Walk Ave, Durham, NC (rooms for $130 per night)
Brookwood Inn, 877-616-2306, 2306 Elba St, Durham, NC (rooms for $100 per night)
Talk titles and abstracts:
Speaker:Alex Fink
Title:Matroid valuations
Abstract:
Many important invariants for matroids, such as the Tutte polynomial,
the Billera-Jia-Reiner quasi-symmetric function, Derksen's universal
invariant, and Speyer's invariant describing the combinatorics of the
Bergman fan, are _valuative_; that is, they are well-behaved under
decompositions of the polytopes.
I will construct the module of matroid valuations, and describe an
explicit combinatorial basis, starting from Brion's theorem; these
results can be extended to polymatroids, and there is a natural Hopf
algebra structure as well.
From a similar starting point, lattice point combinatorics can be used
to associate to every matroid a class in the $K$-theory of the
Grassmannian. This, in particular, provide an algebro-geometric
interpretation of the Tutte polynomial parallel to Speyer's invariant.
Speaker: Sergey Fomin
Title: Cluster structures in rings of SL_3 invariants
Abstract:
The rings of polynomial SL(V)-invariants of configurations of vectors and
linear forms in a k-dimensional complex vector space V have been
explicitly described by Hermann Weyl in the 1930s. Each such ring
conjecturally carries a natural cluster algebra structure (typically, many
of them) whose cluster variables include Weyl's generators. In joint work
with Pavlo Pylyavskyy, we describe and explore these cluster structures in
the case k=3.
Speaker: Nets Katz
Title: Erdos Distinct Distances problem in the plane
Abstract:
In joint work with L. Guth, we show that there is a universal constant
C>0 so that any set of N points in the plane determines at least {N
\over C log N} distinct distances. This settles a longstanding problem
of Erd\"os regarding the best exponent of N that one can obtain in
that estimate.
Speaker: Isabella Novik
Title: Face numbers of centrally symmetric polytopes
Abstract:
How neighborly can a centrally symmetric d-dimensional polytope with n
vertices be? What is the largest number of i-dimensional faces that such a
polytope can have? Which of these polytopes have the largest number of faces? In
this talk I'll report on some recent progress (joint with Alexander Barvinok,
Seung Jin Lee, and Nati Linial) on these questions.
Preregistered participants (so far):
Geir Agnarsson, George Mason University (Fairfax, VA)
Ed Allen, Wake Forest University (Winston-Salem, NC)
Taylor Allison, NCSU (NC)
Cammie Smith Barnes, Sweet Briar College (Sweet Briar, VA)
Christine Berkesch, Duke University (NC)
Sarah Birdsong, UNC Charlotte (Charlotte, NC)
Yue Cai, University of Kentucky (Lexington, KY)
Shaoshi Chen, NCSU (NC)
Ruth Davidson, NCSU (NC)
Alex Fink, NCSU (NC)
Sergey Fomin, Michigan (Ann Arbor, MI)
Jennifer Gamble, NCSU (NC)
Patricia Hersh, NCSU (NC)
Gabor Hetyei, UNC Charlotte (Charlotte, NC)
JT Hird, NCSU (NC)
Johnny Humphries (NC)
John Hutchens, NCSU (NC)
Garrett Johnson, NCSU (NC)
Austin Jones, NCSU (NC)
JiYoon Jung, University of Kentucky (Lexington, KY)
Nets Katz, Indiana University (Bloomington, IN)
Sangwook Kim, George Mason University (Fairfax, VA)
Yonggu Kim, visiting NCSU from Korea (NC)
Patricia Klein, NCSU (NC)
David Lax, UNC Chapel Hill (NC)
Matthew Macauley, Clemson University (Clemson, SC)
Sonja Mapes, Duke University (NC)
Sarah Mason, Wake Forest University (Winston-Salem, NC)
Leo Mihalcea, Virginia Tech (Blacksburg, VA)
Jed Mihalisin, Meredith College (Raleigh, NC)
Ezra Miller, Duke University (NC)
Kailash Misra, NCSU (NC)
Michael Mossinghoff, Davidson College (Davidson, NC)
Liz Munch, Duke University (NC)
Elizabeth Niese, Marshall University (Huntington, WV)
Asamoah Nkwanta, Morgan State University (Baltimore, MD)
Isaballa Novik, Univ. of Washington (Seattle, WA)
Matthew O'Meara, UNC Chapel Hill (NC)
Lindsay Piechnik, High Point University (High Point, NC)
Nathan Reading, NCSU (NC)
Carla Savage, NCSU (NC)
Seth Sullivant, NCSU (NC)
Ryan Vinroot, College of William and Mary (Williamsburg, VA)
Mirko Visontai, U Penn (Philadelphia, PA)
Weikun Wang, NCSU (NC)
Adam Wilkerson, NCSU (NC)
Recent meeting: November 5, 2011
at UNC Chapel Hill
Talk slides (for the computer talk):
Jesus De Loera (UC Davis), Algebraic-Geometric ideas in Discrete Optimization
Speakers:
Thomas Lam (Michigan),
Jesus De Loera (UC Davis),
Ezra Miller (Duke) and
Doron Zeilberger (Rutgers)
Lecture hall for Saturday workshop: Hanes 120 (main floor). Refreshments not permitted
in the classrooms. Hanes is just east of Phillips Hall and just south of
Cameron Ave (not to be confused with Hanes Art Center!)
UNC Chapel Hill Campus Map: as pdf file.
Friday Afternoon Seminar Schedule:
Friday, 5:15-6:15pm, UNC-CH math seminar talk (Phillips Hall 367): Thomas Lam, Projected Richardson Varieties
Jesus De Loera -- Friday seminar talk cancelled
Friday evening: dinner (location TBA)
Saturday Workshop Schedule:
9:15am-10am, coffee, tea and bagels (third floor of Hanes)
10-11am, Jesus de Loera, Algebraic-Geometric Algorithms in Discrete Optimization
11-11:30am, cofee break
11:30am-12:30pm, Doron Zeilberger, Statistical Combinatorics
12:30-2:30pm, lunch
2:30-3:30pm, Thomas Lam, Electrical Networks and Lie Theory
3:30-4pm, cofee break
4-5pm, Ezra Miller, Binomial Irreducible Decomposition
Saturday evening: dinner
(Talullas, 456 W. Franklin Street) -- please email plhersh@ncsu.edu
to RSVP for dinner if you plan to attend the dinner.
Talk titles and abstracts:
Speaker: Jesus A. De Loera, Univ. of California, Davis
Title: Algebraic-Geometric Algorithms in Discrete Optimization
Abstract: It is common knowledge that the understanding of the combinatorial geometry of convex bodies has helped speed up algorithms in discrete optimization. For example, cutting planes and facet-description of polyhedra have been crucial in the success of branch-and-bound algorithms for mixed integer linear programming. Another example, is how the ellipsoid method can be used to prove polynomiality results in combinatorial optimization. For the future, the importance of algebraic-combinatorial geometry in optimization appears even greater.
In the past 5 years two beautiful geometric algorithms on polyhedral have been used to prove unexpected new results on the computation of integer programs (both linearly *and* non-linearly constrained). The first is Barvinok's algorithm for polytopes, the second is Graver's bases method on polyhedral cones. I will describe these two algorithms and explain why we can now prove theorems that were beyond our reach before, mostly about integer optimization with non- linear objectives. I will also describe attempts to turn these two algorithms into practical computation, not just in theoretical results.
This a nice story collecting results contained in several papers joint work with various subsets of the following people: R. Hemmecke, M. Koeppe, S. Onn, U. Rothblum, and R. Weismantel. Our monograph with the same title is scheduled to be published by SIAM-MOS series next year.
Speaker: Thomas Lam, University of Michigan
Title: Electrical Networks and Lie Theory
Abstract: Electrical networks consisting only of resistors are modeled in combinatorics by undirected weighted graphs, where the weight of an edge is the resistance of a resistor. Some basic questions one asks are: (1) to compute the electrical properties of the network: for example what current flows through when certain voltages are applied at particular vertices, (2) when two electrical networks have identical electrical properties: for example two resistors in series or in parallel can be replaced by a single resistor, (3) to what extent an electrical network can be reconstructed if its electrical properties are known.
In this talk we will discuss these problems in a combinatorial and algebraic context. In particular, I'll explain how certain simple combinatorial operations on electrical networks give rise to a Lie group action on the space of electrical networks, allowing one to apply ideas from Lie theory and representation theory.
This talk is based on joint work with Pavlo Pylyavskyy.
Speaker: Ezra Miller, Duke University
Title: Binomial Irreducible Decomposition
Abstract: Monomial irreducible decomposition combinatorially expresses the (exponents on the) monomials outside of a monomial ideal as a union of box-shaped sets of lattice points. Binomial irreducible decomposition aims for a similar outcome when the input is a binomial ideal, but its existence has until now remained open. This talk is about equivalence relations and partial orders on commutative monoids, explicitly described in terms of lattice points, as in the monomial case. The resulting combinatorics, along with a modicum of abelian group character theory, yields binomial irreducible decomposition. This is joint work with Thomas Kahle.
Speaker: Doron Zeilberger, Rutgers University
Title: Statistical Combinatorics
Abstract: A large part of traditional "continuous"
probability and statistics can be redone (and improved!)
using purely combinatorial arguments plus a computer algebra system.
Friday seminar talk:
Geometric Methods in Representation Theory Seminar Talk: Thomas Lam, Projected Richardson Varieties
Abstract: Richardson varieties are intersections of a Schubert variety with an opposite Schubert variety. I will discuss some of the remarkable geometric and combinatorial properties of the projections of Richardson varieties from the full flag variety to partial flag varieties. These projected Richardson varieties play a role in a surprising number of subjects: total positivity, Poisson geometry, Frobenius splitting, quantum cohomology, and so on.
This talk is based on joint work with Knutson and Speyer, and with He.
Directions and parking information: 2 page pdf file
Suggested hotels:
Chapel Hill University Inn Hotel - Chapel Hill, North Carolina (NC)
1301 N. Fordham Blvd, Chapel Hill
Reservations: 1-888-452-5765; Contact: 919-929-2171
Rate: $89.50 for a room with one king or 2 double beds
Days Inn Chapel Hill,
1312 North Fordham Blvd,
Chapel Hill, NC 27514
Phone: 919-929-3090
Rate: $71.40 for room with king or 2 double beds
Preregistered participants:
Geir Agnarsson, George Mason University (VA)
Ed Allen, Wake Forest (NC)
Taylor Allison, NCSU (NC)
Alyssa Armstrong, NCSU (NC)
Camillia Smith Barnes, Sweet Briar College (VA)
Jonathan Beagley, George Mason University (VA)
Melissa Bechard, Wake Forest (NC)
Prakash Belkale, UNC Chapel Hill (NC)
Christine Berkesch, Duke (NC)
Hoda Bidkhori, NCSU (NC)
Sarah Birdsong, UNC Charlotte (NC)
Shaoshi Chen, NCSU (NC)
Ruth Davidson, NCSU (NC)
Sajal Dash, UNC Chapel Hill (NC)
Jesus De Loera, UC Davis (CA)
Graham Enos, UNC Charlotte (NC)
Alex Fink, NCSU (NC)
Jennifer Gamble, NCSU (NC)
Anant Godbole, East Tennessee State University (TN)
Ian Haywood, NCSU (NC)
Allison Hedges, NCSU (NC)
Patricia Hersh, NCSU (NC)
Gabor Hetyei, UNC Charlotte (NC)
JT Hird, NCSU (NC)
John Hutchens, NCSU (NC)
Garrett Johnson, NCSU (NC)
Austin Jones, NCSU (NC)
Borworn Khuhirun, NCSU (NC)
Patricia Klein, NCSU (NC)
Irina Kogan, NCSU (NC)
Thomas Lam, Michigan (MI)
Shirley Law, NCSU (NC)
David Lax, UNC Chapel Hill (NC)
Colby Long, NCSU (NC)
Jian Luo, NCSU (NC)
Sonja Mapes, Duke (NC)
Sarah Mason, Wake Forest (NC)
Chris McDaniel, Endicott College (MA)
Ezra Miller, Duke (NC)
Kailash Misra, NCSU (NC)
Fatemeh Mohammadi, MSRI (CA)
Walter Morris, George Mason University (VA)
Michael Mossinghoff, Davidson College (NC)
Swarnava Mukhopadhyay, UNC Chapel Hill (NC)
Asamoah Nkwanta, Morgan State University (MD)
Elizabeth Niese, Marshall University (WV)
Chris O'Neill, Duke (NC)
Daniel Orr, UNC Chapel Hill (NC)
Gabor Pataki, UNC Chapel Hill (NC)
Lindsay Piechnik, High Point University (NC)
Imre Polik, SAS Corporation (NC)
Bob Proctor, UNC Chapel Hill (NC)
Scott Provan, UNC Chapel Hill (NC)
Margaret Rahmoeller, NCSU (NC)
Margaret Readdy, University of Kentucky (KY)
Nathan Reading, NCSU (NC)
Richard Rimanyi, UNC Chapel Hill (NC)
Carla Savage, NCSU (NC)
Mike Schuster, UNC Chapel Hill (NC)
Joe Seaborn, UNC Chapel Hill (NC)
Sean Skwerer, UNC Chapel Hill (NC)
Jack Snoeyink, UNC Chapel Hill (NC)
Seth Sullivant, NCSU (NC)
Chris Thunes, NCSU (NC)
John Trowbridge, NCSU (NC)
Mirko Visontal, U Penn (PA)
Weikun Wang, NCSU (NC)
William Wheeless, NCSU (NC)
Matt Willis, UNC Chapel Hill (NC)
Fateme Yousefi
Doron Zeilberger, Rutgers (NJ)
Mingfu Zhu, Duke (NC)
Organizing committee: Prakash Belkale, Gabor Pataki, Bob Proctor, Scott Provan, Richard Rimanyi
Past meetings:
Third meeting: April 9, 2011 at NC State
Speakers:
Prakash Belkale (UNC Chapel Hill),
Vic Reiner (University of Minnesota),
John Stembridge (University of Michigan),
Stephanie van Willigenberg (UBC).
Organizing committee:
Hoda Bidkhori (NCSU), Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).
More information
Second meeting: September 25, 2010 at Duke
Speakers:
Alexander Barvinok (University of Michigan), Anne Shiu (Duke),
Sami Assaf (MIT),
Persi Diaconis (Stanford).
Organizing committee:
Patricia Hersh (NCSU),
Sonja Mapes (Duke),
Ezra Miller (Duke).
More information
First meeting: February 6, 2010 at NC State
Speakers:
Carla Savage (NCSU),
Bernd Sturmfels (UC Berkeley),
Ed Swartz (Cornell),
Laszlo Szekely (University of South Carolina).
Organizing committee:
Patricia Hersh (NCSU),
Ezra Miller (Duke),
Scott Provan (UNC)
Nathan Reading (NCSU).
More information (including talk slides)