**Email:**riliu (at) ncsu.edu**Office:**SAS Hall 3264**Phone:**919-515-0681

I am currently an assistant professor at **North Carolina State University**. Previously I was a T. H. Hildebrandt Research Assistant Professor at the **University of Michigan, Ann Arbor** and an NSF postdoctoral fellow at the **University of Minnesota, Twin Cities**. I completed my Ph.D. at the **Massachusetts Institute of Technology** under the guidance of Alexander Postnikov in 2010. (**CV**)

My research interests lie in **algebraic combinatorics**, especially its relationship to **algebraic geometry**, **combinatorial geometry**, and **representation theory**.

- Flow polytopes and the space of diagonal harmonics (with K. Mészáros and A. Morales), submitted.
We consider a generalization of Haglund's formula for the Hilbert series of the space of diagonal harmonics as a sum over Tesler matrices by summing over flow polytopes of threshold graphs.

- Kronecker coefficients and noncommutative super Schur functions (with J. Blasiak), submitted.
We show how the theory of noncommutative super Schur functions (based on work of Fomin and Greene) gives a combinatorial formula for the Kronecker coefficient $g_{\lambda\mu\nu}$ when $\mu$ is a hook.

- A simplified Kronecker rule for one hook shape,
*Proceedings of the American Mathematical Society*, to appear.We give a simple combinatorial rule for the Kronecker coefficient $g_{\lambda\mu\nu}$ when $\mu$ is a hook.

- Subalgebras of the Fomin-Kirillov algebra (with J. Blasiak and K. Mészáros),
*Journal of Algebraic Combinatorics*, Volume 44, Issue 3, November 2016, pp. 785–829.We define subalgebras of the Fomin-Kirillov algebra for any finite graph and show that they have a surprising number of nice properties analogous to those of Coxeter groups and nil-Coxeter algebras. We explicitly compute the Hilbert series for the subalgebras corresponding to the Dynkin diagrams $A_n$, $D_n$, $E_n$ ($n=6,7,8$), and $\tilde A_{n-1}$.

- On the commutative quotient of Fomin-Kirillov algebras,
*European Journal of Combinatorics*, Volume 54, May 2016, pp. 65–75.We prove that the commutative quotient of the Fomin-Kirillov algebra of a graph $G$ on $n$ vertices is isomorphic to the Orlik-Terao algebra of $G$. In particular, its Hilbert series is $(-t)^n \cdot \chi_G(-t^{-1})$, where $\chi_G(t)$ is the chromatic polynomial of $G$.

- Complete branching rules for Specht modules,
*Journal of Algebra*, Volume 446, January 2016, pp. 77–102.We give a combinatorial criterion for when the Specht module of an arbitrary diagram admits a (complete) branching rule. We also show that the only relations in such Specht modules are given by generalized Garnir relations.

- Positive expressions for skew divided difference operators,
*Journal of Algebraic Combinatorics*, Volume 42, Issue 3, November 2015, pp. 861–874.We give a positive formula for the skew divided difference operators (defined by Macdonald) in terms of divided difference operators $\partial_{ij}$ with $i < j$, settling a conjecture of Kirillov.

- Coefficients of a relative of cyclotomic polynomials,
*Acta Arithmetica*, Volume 165, Number 4, 2014, pp. 301–325.We give an explicit description of the coefficients of $(1-x)\Phi_{pqr}(x)$. We also define an analogous polynomial for any number of primes and describe their coefficients and growth rate as the number of primes increases.

- Laurent polynomials, Eulerian numbers, and Bernstein's theorem,
*Journal of Combinatorial Theory Series A*, Volume 124, May 2014, pp. 244–250.We give a simple proof of a result of Erman, Smith, and Várilly-Alvarado relating Laurent polynomials and Eulerian numbers using Bernstein's theorem. We also show that a refinement of the Eulerian numbers gives a combinatorial interpretation for volumes of certain (rational) hyperplane sections of the hypercube.

- Nonconvexity of the set of hypergraph degree sequences,
*Electronic Journal of Combinatorics*, Volume 20, Issue 1, January 2013.We show that the set of hypergraph degree sequences is not the intersection of a lattice and a convex polytope. We also prove an analogous result for multipartite hypergraphs.

- Matching polytopes and Specht modules,
*Transactions of the American Mathematical Society*, Volume 364, Number 2, February 2012, pp. 1089–1107.We prove that the normalized volume of the matching polytope of a forest equals the dimension of the corresponding Specht module. We also give $S_n$- and $GL_n$-branching rules, thereby defining a notion of Schur functions and standard/semistandard tableaux for forests.

- Matrices with restricted patterns and $q$-analogues of permutations (with J. Lewis, A. Morales, G. Panova, S. Sam, and Y. Zhang),
*Journal of Combinatorics*, Volume 2, Number 3, 2011, pp. 355–395.We study the number of matrices of a given rank with specified zero entries over a finite field. We show that these numbers give a $q$-analogue of certain restricted permutations and obtain explicit $q$-analogues for derangement numbers. We also consider the analogous questions for symmetric and skew-symmetric cases and discuss related questions.

- Specht modules and Schubert varieties for general diagrams.
**Thesis.**May 2010.We present a conjecture relating the cohomology class of certain subvarieties of the Grassmannian to the structure of certain representations of the symmetric group and give evidence towards this conjecture.

- An algorithmic Littlewood-Richardson rule,
*Journal of Algebraic Combinatorics*, Volume 31, Issue 2, February 2010, pp. 253–266.We give a Littlewood-Richardson rule based on iteratively deforming a skew Young diagram into a straight shape. This rule is based on a geometric rule by Izzet Coskun.

- Counting subrings of $\mathbf Z^n$ of index $k$,
*Journal of Combinatorial Theory Series A*, Volume 114, Issue 2, February 2007, pp. 278–299We consider the problem of counting subrings of $\mathbf Z^n$ of a given index $k$. We show that a decomposition theorem holds and give a precise result when $n$ is at most 4 or $k$ is not divisible by the sixth power of any prime.

At **North Carolina State University**, I will be teaching the following course in Spring 2018:

Previously, I taught the following courses:

- Math 493: Combinatorial Game Theory, Fall 2017
- Math 416: Introduction to Combinatorics, Spring 2017
- Math 591: Algebraic Combinatorics, Fall 2016
- Math 724: Combinatorics II, Spring 2016
- Math 524: Combinatorics I, Fall 2015
- Math 416: Introduction to Combinatorics, Spring 2015
- Math 437: Applications of Algebra, Spring 2015

At the **University of Michigan**, I taught the following courses:

- Math 465: Introduction to Combinatorics, Winter 2014
- Math 215: Calculus III, Fall 2013
- Math 412: Introduction to Modern Algebra, Winter 2013
- Math 465: Introduction to Combinatorics, Fall 2012
- Math 217: Linear Algebra, Winter 2012
- Math 115: Calculus I, Fall 2011

In Fall 2009, I was a teaching assistant for 18.02: Multivariable Calculus at MIT.

I have been an instructor at the **Mathematical Olympiad Summer Program** since 2007. I was also a research adviser at Joe Gallian's **Research Experience for Undergraduates** at the **University of Minnesota, Duluth** for the summers 2006–2008.