**Triangle Lectures in Combinatorics (TLC)**

** Tenth meeting: ** October 4, 2014

**Location:** High Point University in High Point, North Carolina (one hour west of the Research Triangle by car).

**Lecture Hall: **206 Cogdon Hall MAP (The map also contains parking information.)

**Speakers: **
Richard Ehrenborg (Kentucky), Robert Ghrist (Penn), Luis Serrano (U Quebec a Montreal),
and Rekha Thomas (U. Washington).

**Organizing Committee:** Lindsay Piechnik (chair, High Point), Ed Allen (Wake Forest), Sarah Mason (Wake Forest), Seth Sullivant (NCSU),
and Laurie Zack (High Point)

**Preregistration: **please send email to Patricia Hersh (plhersh@ncsu.edu) to preregister. This is very helpful in our planning how much coffee, etc. to have at coffee breaks and for our obtaining funding to support these meetings.

**Participant travel funding:** We have funds available for travel and hotel for participants, especially for early career participants, thanks to grants from the NSF and the NSA. Much of this is restricted to U.S. citizens and permanent residents, and what is available to others still requires that the participants be employed at a U.S. university. To apply for funding
fill out the form here: Travel Funding Request Form If you have questions, please contact Nathan Reading (nathan_reading@ncsu.edu)

**Saturday Triangle Lectures in Combinatorics Schedule:**

9:15-10am, coffee and bagels

10-11am, Rob Ghrist, *Poincare Duality in Network Flow Optimization*

11-11:30am, coffee break

11:30am-12:30pm, Luis Serrano, *The immaculate basis of the non-commutative symmetric functions*

12:30-2:30pm, lunch break

2:30-3:30pm, Rekha Thomas, *The Euclidean distance degree of an algebraic variety*

3:30-4pm, coffee break

4-5pm, Richard Ehrenborg, *Euler enumeration*

6pm, somewhat informal conference dinner at Sumela

**Logistical information:** The nearest airport is Greensboro/High Point (GSO), 20 minutes drive away. This is also known as the Piedmont Triad Airport. Within 90 minutes drive are both Raleigh-Durham (RDU) Airport and Charlotte (CLT) Airport. A cab from GSO costs about $30 each way. Hotels are about 1.5 to 3 miles from campus. Thus, a car rental may make sense for anyone flying to the meeting.

A block of hotel rooms has been reserved at the Courtyard Marriott High Point (336-882-3600), which is 1.7 miles from campus. The rooms are blocked off under "The TLC" until September 19th. However, if rooms are still available, they will honor our group rate of $119 per night (+tax) after this date. (The room includes two queen beds, wireless internet, and either a hot breakfast sandwich or two continental breakfast items, plus beverage, for 2 adults).
There are many other local options including:
-High Point Plaza , 1 mile from campus (more downtown). 336-889-8888
-Crestwood Suites, about 2.7 miles from campus (more economical). 336-886-5665.

**Preregistered Participants (so far):**

Geir Agnarsson, George Mason University

Ed Allen, Wake Forest

Taylor Allison, UNC Chapel Hill

Justin Allman, Virgina Tech

Jennifer Anderson, Marshall University

Emily Barnard, NCSU

Yue Cai, U Kentucky

Dustin Cartwright, U Tennessee

Shihwei Chao, Clemson University

Paul Cubre, Clemson University

Brian Davis, U Kentucky

Robert Davis, U Kentucky

Rafael S. Gonzalez D'Leon, U Kentucky

Anahita Davoudi, U Central Florida

Vivek Dhand, Charlottesville, VA

Richard Ehrenborg, U Kentucky

Norman Fox, U Kentucky

Amy Grady, Clemson University

Rob Ghrist, U Penn

Alex Happ, U Kentucky

Isaiah Harney, U Kentucky

Qijun He, Clemson

Patricia Hersh, NCSU

Qijun He, Clemson University

Cyrus Hettle, U Kentucky

Gabor Hetyei, UNC Charlotte

Elliot Hollifield, Wake Forest U

Chetak Hossain, NCSU

Wesley Hough, U Kentucky

Andy Jenkins, Clemson University

Yvonne Kemper, NIST

Florian Kohl, U Kentucky

Rebecca Kotsonism, Wake Forest U

David Lax, UNC Chapel Hill

David Mason, UNC Chapel Hill

Sarah Mason, Wake Forest

Marie Meyer, U Kentucky

Frank Moore, Wake Forest U

John Mosley, U Kentucky

Sayan Mukherjee, Duke

Vidit Nanda, U Penn

Yusra Naqvi, Rutgers

Sarah Nelson, U Kentucky

Elizabeth Niese, Marshall University

Jason Parsley, Wake Forest U

Gabor Pataki, UNC Chapel Hill

Svetlana Poznanovikj, Clemson U

Sarah Orchard, U Kentucky

Lindsay Piechnik, High Point

Richard Rimanyi, UNC Chapel Hill

Radmila Sazdanovic, NCSU

Luis Serrano, UQAM

Cliff Smyth, UNC Greensboro

Liam Solus, U Kentucky

Jeremiah Southwich, Wake Forest U

Kara Stasikelis, Clemson University

Seth Sullivant, NCSU

Breeanne Swart, The Citadel

Clifford Taylor, U Kentucky

Rekha Thomas, U Washington

Tzvetalin Vassilev, Nippising U.

Devin Wilmott, U Kentucky

Martha Yip, U Kentucky

Laurie Zack, High Point

**Talk titles and abstracts:**

Richard Ehrenborg (U Kentucky)

Title: Euler enumeration

Abstract: The flag vector contains all the face incidence data of a polytope,
and in the poset setting, the chain enumerative data. It is a
classical result due to Bayer and Klapper that for face lattices of
polytopes, and more generally, Eulerian graded posets, the flag vector
can be written as a cd-index, a non-commutative polynomial which
removes all the linear redundancies among the flag vector
entries. This result holds for regular CW complexes. We relax the
regularity conditions to show the cd-index exists for manifolds whose
boundary has a Whitney stratification. The setting of Whitney
stratifications allows us to give shorter proofs of identities
involving the cd-index and opens inequality questions for manifolds.
This is joint work with Mark Goresky and Margaret Readdy.

Rob Ghrist (U Pennsylvania)

Title: Poincare Duality in Network Flow Optimization

Abstract: One of the classical cornerstones of optimization theory is LP (linear
programming) duality,
and one of its simplest applications is to the classical
max-flow-min-cut theorem, which
expresses a duality between optimal network flow values and optimal
cut capacities.
This talk argues that flow-cut duality is really topological in nature
-- an expression of
Poincare duality. A recent proof by S. Krishnan shows that
Poincare duality on sheaves
of semimodules implies a sheaf-theoretic extension of the classical
max-flow-min-cut
theorem. This greatly expands the types of problems to which flow-cut
dualities can be
applied, as well as explains the source of duality gaps in more advanced
settings. This talk will give details.

Luis Serrano (U Quebec a Montreal)

Title: The immaculate basis of the non-commutative symmetric functions

Abstract: We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula. These bases have many interesting properties similar to those of the Schur basis, and we will outline a few of them.

Rekha Thomas (U Washington)

Title: The Euclidean distance degree of an algebraic variety

Abstract: It is a common problem in optimization to minimize the Euclidean distance from a given data point u to some set X. In this talk I will consider the situation in which X is an algebraic variety, a common situation in practice. The number of critical points of the objective function on X is called the Euclidean distance degree of X, and is an intrinsic measure of the complexity of this polynomial optimization problem. Using algebraic geometry we obtain formulas and methods to calculate this degree in many situations. I will explain the results and illustrate the formulas that can be obtained in several situations ranging from matrix analysis to control theory to computer vision. Joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani and Bernd Sturmfels.