# Mathematical Research

## Interests

My research interests are in algebraic, geometric, and topological combinatorics, including:

- Integer points in rational polytopes and cones
- Quasi-polynomials and generating functions
- Integer partitions and compositions
- Simplicial and cell complexes arising in combinatorics
- Graphs and associated matrices
- Graded algebras and associated resolutions
- Groebner bases

I am also interested in aspects of mathematical computation and regularly use the following software:

## Publications

- Rationality of Poincare Series for a Family of Lattice Simplices, (joint with Brian Davis), submitted.
- Laplacian Simplices, (joint with Marie Meyer), submitted.
- Counting Arithmetical Structures on Paths and Cycles, (joint with Hugo Corrales, Scott Corry, Luis David Garcia Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, and Carlos E. Valencia), submitted.
- Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices, (joint with Robert Davis and Liam Solus), submitted.
- Euler-Mahonian Statistics and Descent Bases for Semigroup Algebras, (joint with McCabe Olsen), European J. Combin. 69 (2018), 237-254.
- Matching and Independence Complexes Related to Small Grids, (joint with Wesley K. Hough), Electron. J. Comb., 24(4) (2017), #P4.18
- Generating Functions and Triangulations for Lecture Hall Cones, (joint with Matthias Beck, Matthias Koeppe, Carla Savage, and Zafeirakis Zafeirakopoulos), SIAM J. Discrete Math., 30(3), 1470-1479.
- Unimodality Problems in Ehrhart Theory, in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27
- r-Stable Hypersimplices, (joint with Liam Solus), to appear in J. Combin. Theory Ser. A
- Ehrhart series, unimodality, and integrally closed reflexive polytopes, (joint with Robert Davis), Ann. Comb. 20 (2016), no. 4, 705-717.
- Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes, (joint with Sarah Crown Rundell), Electron. J. Comb., 21(2) (2014), #P2.35
- s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones, (joint with Matthias Beck, Matthias Koeppe, Carla Savage, and Zafeirakis Zafeirakopoulos), The Ramanujan Journal, February 2015, Volume 36, Issue 1-2, pp 123-147.
- Lattice Point Generating Functions for Symmetric Cones, (joint with Matthias Beck, Thomas Bliem, and Carla Savage), Journal of Algebraic Combinatorics 38 (2013), 543-566.
- Compositions constrained by graph Laplacian minors, (joint with Robert Davis, Jessica Doering, Ashley Harrison, Jenna Noll, and Clifford Taylor), INTEGERS 13 (2013), paper A41.
- Euler-Mahonian Statistics via Polyhedral Geometry, (joint with Matthias Beck), Advances in Mathematics 244 (2013), 925-954.
- Mahonian Partition Identities via Polyhedral Geometry, (joint with Matthias Beck and Nguyen Le), From Fourier Analysis and Number Theory to Radon Transforms and Geometry: In Memory of Leon Ehrenpreis, (H. Farkas, R. Gunning, M. Knopp, and B. A. Taylor, eds.), Developments in Mathematics 28 (2013), 41--54.
- Cellular Resolutions of Ideals Defined by Simplicial Homomorphisms, (joint with Jonathan Browder and Steven Klee), Israel J. Math. 196 (2013), no. 1, 321-344. DOI: 10.1007/s11856-012-0149-2.
- Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs, (joint with Matthew Zeckner), Proc. Amer. Math. Soc. 142 (2014), 413-427.
- Independence Complexes of Stable Kneser Graphs, Electronic Journal of Combinatorics, 18, no. 1 (2011), P118.
- Nowhere-Harmonic Colorings of Graphs, (joint with Matthias Beck), Proc. Amer. Math. Soc. 140 (2012), 47-63.
- Symmetries of the Stable Kneser Graphs, Adv. in Appl. Math., 45 (2010), no. 1, 12 - 14.
- The Complex of Non-Crossing Diagonals of a Polygon, (joint with Richard Ehrenborg), J. Combin. Theory Ser. A, 117 (2010), no. 6, 642 - 649.
- Ehrhart Polynomial Roots and Stanley's Non-negativity Theorem, (joint with Mike Develin), Integer Points in Polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, AMS Contemporary Mathematics 2008, Volume: 452, pp 67-78.
- Norm Bounds For Ehrhart Polynomial Roots, Discrete and Computational Geometry, 39 (2008), no. 1-3, 191-193.
- An Ehrhart Series Formula For Reflexive Polytopes, Electronic Journal of Combinatorics, 13, no. 1 (2006), N 15.

## PhD Thesis

Ehrhart Theory for Lattice Polytopes

My thesis is a combination of the papers "Norm Bounds...," "An Ehrhart Series Formula...," and "Ehrhart Polynomial Roots..." shown above.

Braun_Dissertation.pdf