# Fall 2019

**Triangle Lectures in Combinatorics (TLC)**

**Eighteenth meeting:** Saturday, November 16, 2019**Location:** North Carolina State University**Lecture Hall:** SAS 1102

**Speakers: **Georgia Benkart (University of Wisconsin at Madison),
Javier Peña (Carnegie Mellon University),
Margaret Readdy (University of Kentucky), Alexander Yong (University of Illinois at Urbana-Champaign)

**Organizing Committee:** Patricia Hersh (NCSU), Gabor Hetyei (UNC-Charlotte), Kailash Misra (NCSU), Gabor Pataki (UNC Chapel Hill)

**Registration and funding applications:**

To register and/or apply for funding, please fill out this form. Alternatively, you may register by emailing Patricia Hersh at plhersh@ncsu.edu.

We are asking that participants preregister if possible, as preregistration is very helpful for planning our coffee breaks and obtaining funding to support these meetings. We have some funding available for some participants, especially for early-career participants. Most of this is restricted to US citizens and permanent residents, and what is available to others still requires that the participants be employed at a U.S. university. You will have the opportunity to apply for funding when registering for the conference.

**Conference schedule:**

9:15-10am, bagels and coffee

10-11am, Georgia Benkart, McKay Quivers in Cooperation with Schur-Weyl Duality

11-11:30am, coffee break

11:30am-12:30pm, Javier Peña, Hoffman constants for systems of linear inequalities (handout, slides)

12:30-2:30pm, lunch break

2:30-3:30pm, Margaret Readdy, Geometric proofs of some combinatorial identities of Morel (slides)

3:30-4pm, coffee break

4-5pm, Alexander Yong, Complexity, combinatorial positivity, and Newton polytopes (slides)

6pm, informal dinner at the Flying Biscuit a 15-minute walk from SAS Hall

**Parking:** You may park right outside SAS Hall for free. Here is a map of the campus. On Saturdays, you can park anywhere on campus that is not specifically marked as being restricted (e.g. handicap spots are still off limits). We are hopeful that you won’t need any lot except the one by SAS Hall. SAS Hall is at the upper right of the map, and the parking lot is near the intersection of Stinson Drive and Boney Dr. A good back-up option for parking is the Coliseum Parking Deck.

**The room:** SAS Room 1102 is next to the lobby on the same floor when you enter from the parking lot. If you enter from the courtyard side, it is down one flight of stairs.

**Lunch/Food Recommendations:** Some of the good options for lunch, coffee, etc. within walking distance include (1) Guasaca (2512 Hillsborough Street, Peruvian), (2) Jasmin and Olivz (2430 Hillsborough Street, Mediterranean/Middle Eastern), (3) Kebab and Curry (2412 Hillsborough Street, Indian buffet), (4) Jubala Coffee (2100 Hillsborough Street, excellent biscuits, sandwiches, pastries and coffee), (5) Gonza Tacos y Tequila (2100 Hillsborough Street, Mexican), (6) Global Village Coffee (2428 Hillsborough Street), (7) Howling Cow Ice Cream (inside Hill library), and a bit farther away are more places in Cameron Village such as Cantina 18 (Mexican), Flying Biscuit Cafe (breakfast all day) and Tazza (brick oven pizza, small plates, etc.). Still farther away (requiring a car) is a really excellent Middle Eastern casual restaurant called Neomande (3817 Beryl Road).

**Hotel recommendations:** We have reserved a block of hotel rooms at Aloft that will be held until November 1.

Within short walk of the math department are several hotels, including the Doubletree by Hilton Raleigh – Brownstone (919-828-0811) and Aloft Raleigh (919-828-9900). About 1.5 miles away in downtown Raleigh (also walkable, but somewhat long walk) is the Clarion Raleigh Hotel (919-832-0501). Those with cars might also consider hotels farther away such as Holiday Inn Express (919-854-0001) 3741 Thistledown Drive (near Centennial Campus) as well as various hotel choices on Wake Town Drive, which is near numerous good restaurants; some such hotels (all right next to each other on Wake Towne Drive) are Marriott Courtyard (919-821-3400), Hampton Inn (919-828-1813), or Extended Stay America (919-829-7271).

**Airport:** Raleigh-Durham International Airport is 20-30 minutes drive from NCSU. Taxi fare is about $30.

**Participants: **

Jordan Almeter (NCSU)

Jai Aslam (NCSU)

Adrian Avalos (Coastal Carolina University)

Duff Baker-Jarvis (Wake Forest University)

Grant Barkley (NCSU)

Prakash Belkale (UNC Chapel Hill)

Georgia Benkart (University of Wisconsin at Madison)

Mark Bly (Coastal Carolina University)

Jane Coons (NCSU)

Joseph Cummings (University of Kentucky)

Spencer Daugherty (NCSU)

Robert Ferguson (University of Florida)

Darij Grinberg (Drexel University)

William Gustafson (University of Kentucky)

Derek Hanely (University of Kentucky)

Patricia Hersh (NCSU)

Gabor Hetyei (UNC Charlotte)

Ben Hollering (NCSU)

Sam Jeralds (UNC Chapel Hill)

Naihuan Jing (NCSU)

Joe Johnson (NCSU)

Bryson Kagy (North Carolina State)

Stephen Lacina (NCSU)

Mingming Lang (UNC Chapel Hill)

Ricky Liu (NCSU)

Baqiao Liu (UNC Chapel Hill)

Molly Lynch (Hollins University)

Aida Maraj (University of Kentucky)

Thomas McConville (UNC Greensboro)

Emily McGovern (NCSU)

Everett Meike (NCSU)

Kailash Misra (NCSU)

Evangelos Nastas (Syracuse University)

Gabor Pataki (UNC Chapel Hill)

Javier Peña (Carnegie Mellon University)

Robert Proctor (UNC Chapel Hill)

Margaret Readdy (University of Kentucky)

Nathan Reading (NCSU)

Richard Rimanyi (UNC Chapel Hill)

Heather Russell (University of Richmond)

Carla Savage (North Carolina State University)

Radmila Sazdanovic (NCSU)

Georgy Scholten (NCSU)

Michael Singer (NCSU)

Christian Smith (NCSU)

Cliff Smyth (UNC Greensboro)

Grace Stadnyk (Furman University)

Michael Strayer (Hampden-Sydney College)

Ashley Tharp (NCSU)

Vladimir Tonchev (Michigan Technological University)

Cynthia Vinzant (NCSU)

Matias von Bell (University of Kentucky)

Michael Weselcouch (Wake Forest University)

Alexander Yong (University of Illinois at Urbana-Champaign)

**Talk titles and abstracts:**

**Georgia Benkart** (University of Wisconsin-Madison)

Title: McKay Quivers in Cooperation with Schur-Weyl Duality

The well-known McKay Correspondence is a bijection between the finite subgroups G of SU_{2} and the simply-laced affine Dynkin diagrams. McKay’s insight was that the quivers determined by tensoring the simple G-modules with the G-module V = ℂ^{2} exactly correspond to the affine Dynkin diagrams of types A, D, E. McKay quivers are related to Auslander-Reiten quivers in the representation theory of finite-dimensional algebras and to a host of other topics such as singularity theory and orbifolds. This talk will focus on connecting McKay quivers to Schur-Weyl duality. For any finite group G (or finite-dimensional Hopf algebra) and any finite- dimensional G-module V, this combined theory provides results on the tensor product module V^{⊗k} and its G-invariants, and on the centralizer algebra End_{G}(V^{⊗k}), which often has a nice diagrammatic realization and a rich combinatorics. There are applications to chip-firing dynamics and Markov chains.

**Javier Peña** (Carnegie Mellon University)

Title: Hoffman constants for systems of linear inequalities

Abstract: A classical result of Hoffman (1952) shows that the distance from a point *u *to a non-empty polyhedron defined by the system of inequalities Ax ≤ b can be bounded above in terms of the “error” or “residual” (Au-b)_{+} = max(0,Au-b). More precisely, the distance from u to the nonempty polyhedron {x: Ax ≤ b} is bounded above by H(A)*|(Au-b)_{+}| for some Hoffman constant H(A) that depends only on the matrix A. This type of “error bound” plays a fundamental role in mathematical programming.

This talk will give a new and constructive proof of existence and characterization of the Hoffman constant H(A). We will discuss our developments in the following more general “relative” context. Suppose R is a “reference” polyhedron representing some easy-to-satisfy constraints such as box constraints. Then the distance from a point u ∈ R to the nonempty polyhedron {x ∈ R: Ax ≤ b} is bounded above by H(A|R)*|(Au-b)_{+}| for some “relative Hoffman constant” H(A|R).

Our results readily yield a novel combinatorial algorithm to compute Hoffman constants. The latter is a timely but notoriously difficult and largely unexplored computational problem.

**Margaret Readdy** (University of Kentucky)

Title: Geometric proofs of some combinatorial identities of Morel

Using the algebraic and geometric combinatorics of the permutahedron, we give proofs of combinatorial identities which arise in the technical heart of Morel’s computation of the intersection cohomology of Shimura varieties. No prior background will be assumed.

This is joint work with Richard Ehrenborg and Sophie Morel.

**Alexander Yong** (U. Illinois at Urbana-Champaign)

Title: Complexity, combinatorial positivity, and Newton polytopes

The Nonvanishing Problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, Nonvanishing is in the complexity class of problems with “good characterizations”. This suggests a new algebraic combinatorics viewpoint on complexity theory.

This talk discusses the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a tableau criterion for Nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid, together with a theorem of A. Fink, K. Meszaros and A. St. Dizier, which proved a conjecture of C. Monical, N. Tokcan and the speaker.

This is joint with Anshul Adve (U. California, Los Angeles, USA) and Colleen Robichaux (U. Illinois at Urbana-Champaign, USA).