### Graphical Designs and Graph Sparsifiers

** [1] **
Graphical designs and Gale duality
(with Catherine Babecki), * Mathematical Programming, * 200:2 (2023), 703-737.

** [2] ** Random Walks, Equidistribution and Graphical Designs
(with Stefan Steinerberger), June 2022.

** [3] **
Spectrahedral Geometry of Graph Sparsifiers (with Catherine Babecki and Stefan Steinerberger),
June 2023.

** [4] **
Conformally rigid graphs (with Stefan Steinerberger),
February 2024. new

### Algebraic Methods in Computer Vision

** [1] **
A QCQP Approach to Triangulation (with Chris Aholt and Sameer Agarwal),
* Computer Vision - ECCV 2012 *,
Lecture Notes in Computer Science, 7572 (2012), 654-667.

** [2] **
A Hilbert scheme in computer vision (with Chris Aholt and Bernd Sturmfels),
* Canadian Journal of Mathematics* 65:5 (2013), 961-988.

** [3] **
The Euclidean distance degree of an algebraic variety (with
Jan Draisma, Emil Horobet, Giorgio Ottaviani and Bernd Sturmfels),
* Foundations of Computational Mathematics* 16:1 (2016), 99-149.

** [4] **
Counting real critical points of the distance to spectral matrix sets (with
Dimitriy Drusvyatskiy and Hon-Leung Lee), * SIAM J. Matrix Analysis and Applications *, 36:3 (2015),1360-1380.

** [5] **
The Euclidean distance degree of orthogonally invariant matrix varieties (with
Dimitriy Drusvyatskiy, Hon-Leung Lee and Giorgio Ottaviani),
* Israel Journal of Mathematics,* 221:1 (2017), 291-316.

** [6] **
Certifying the existence of epipolar matrices (with
Sameer Agarwal, Hon-Leung Lee and Bernd Sturmfels),
July 2014.

** [7] **
On the existence of epipolar matrices (with
Sameer Agarwal, Hon-Leung Lee and Bernd Sturmfels),
* Int. J. Comput. Vis. * 121:3 (2017), 403-415.

** [8] **
Ideals of the Multiview Variety,
(with Sameer Agarwal and Andrew Pryhuber) *
IEEE Transactions on Pattern Analysis and Machine Intelligence, *
43:4 (2021), 1279-1292.

** [9] **
On the local stability of semidefinite relaxations
(with Diego Cifuentes, Sameer Agarwal and Pablo A. Parrilo),
* Mathematical Programming * 193:2 (2022), 629-663.

** [10] **
Existence of two view chiral reconstructions
(with Andrew Pryhuber and Rainer Sinn), * SIAM J. Applied Algebra and Geometry, * 6:1 (2022),
41-76.

** [11] **
The chiral domain of a camera arrangement
(with Sameer Agarwal, Andrew Pryhuber, and Rainer Sinn), * J. Mathematical Imaging and Vision, *
64 (2022), 948-967.

** [12] **
An Atlas for the Pinhole Camera
(with Sameer Agarwal, Timothy Duff and Max Lieblich), * Foundations of Computational Mathematics, * (2022), published online.

** [13]
**
The geometry of rank drop in a class of face-splitting matrix products: Part I
(with Erin Connelly, Sameer Agarwal and Alperen Ergur), * Advances in Geometry,* 24:3 (2024), 369-394.

** [14] **
Two Views of P3 ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation, Tromso, Norway,
(2023) 7-12.

** [15]
**
The geometry of rank drop in a class of face-splitting matrix products: Part II
(with Erin Connelly and Cynthia Vinzant),* Advances in Geometry, * 24:3, (2024), 395-420.

### Extremal Graph Theory

** [1] **
Symmetric sums of squares over k-subset hypercubes
(with Annie Raymond, James Saunderson and Mohit Singh),
* Mathematical Programming * Series A, 167:2 (2018), 315-354.

** [2] **
Symmetry in Turán sums of squares polynomials from flag algebras
(with Annie Raymond and Mohit Singh), * Algebraic Combinatorics * 1:2 (2018), 249-274.

** [3] **
Simple graph density inequalities with
no sum of squares proofs (with Greg Blekherman, Annie Raymond and Mohit Singh),
* Combinatorica, * 40:4 (2020), 455-471.

** [4] **
Tropicalization of graph profiles
(with Greg Blekherman, Annie Raymond and Mohit Singh), * Transactions of the AMS, * 375 (2022), 6281-6310.
link to article

### Lifts of Convex Sets

** [1] **
Theta bodies for polynomial ideals (with João Gouveia and Pablo A. Parrilo),
* SIAM Journal of Optimization *
20:4 (2010), 2097-2118.

** [2] ** Convex Hulls of Algebraic Sets , (with João Gouveia), chapter in the "Handbook of Semidefinite, Cone and Polynomial Optimization", International Series in Operations Research & Management Science, Vol. 166, Miguel Anjos and Jean-Bernard Lasserre (eds), Springer, 2012.

** [3] **
A new hierarchy of semidefinite programming relaxations for cycles in binary
matroids and cuts in graphs (with João Gouveia, Monique Laurent
and Pablo A. Parrilo), * Mathematical Programming, * Series A, 133:1-2 (2012), 203-225.

** [4] **
Lifts of convex sets and cone factorizations (with João Gouveia and Pablo A. Parrilo), * Mathematics of Operations Research * 38 (2013), 248-264.

** [5] **
Polytopes of minimum positive semidefinite rank
(with João Gouveia and Richard Robinson),
*Discrete & Computational Geometry * 50:3 (2013), 679-699.

** [6] **
Which nonnegative matrices are slack matrices? (with João Gouveia,
Roland Grappe, Volker Kaibel, Kanstanstin Pashkovich and Richard Robinson),
* Linear Algebra and its Applications * 439 (2013), 2921-2933.

** [7] **
Approximate cone factorizations and lifts of polytopes (with
João Gouveia and Pablo A. Parrilo),
Isaac Newton Institute preprint NI13069-POP,
* Mathematical Programming, * 151:2 (2015),
613-637.

** [8] **
Worst-case results for positive semidefinite rank (with
João Gouveia and Richard Robinson),
* Mathematical Programming * Series B, 153:1 (2015) 201-212.

** [9] **
Positive semidefinite rank (with
Hamza Fawzi, João Gouveia, Pablo A. Parrilo and Richard Z. Robinson),
* Math Programming Series B *, 153:1 (2015), 133-177.

** [10] **
Four dimensional polytopes of minimum positive semidefinite rank, (with
João Gouveia, Kanstanstin Pashkovich and Richard Z. Robinson),
* J. Combin. Theory Ser. A * 145 (2017), 184--226.

** [11] **
Spectrahedral lifts of convex sets
* Proceedings of the International Congress of Mathematicians -- 2018 Rio de Janeiro, * 3 (2018),
3807--3830.

** [12] **
The slack realization space of a polytope
(with João Gouveia, Antonio Macchia and Amy Wiebe),
* SIAM J. Discrete Math., * 33:3, (2019) 1637-1653.

** [13] **
Projectively unique polytopes and
toric slack ideals (with João Gouveia, Antonio Macchia and Amy Wiebe), * J. Pure and Applied Algebra, * 224:5 (2020), 106-229.

** [14] **
Lifting for simplicity: concise descriptions of convex sets
(with Hamza Fawzi, Joao Gouveia, Pablo A. Parrilo, and James Saunderson), * SIAM Review, *
64:4 (2022) published online

### Toric Ideals and Integer Programming

** [1] ** An algebraic geometry
algorithm for scheduling in presence
of setups and correlated demands (with Sridhar R. Tayur and
N.R.Natraj), *Mathematical Programming,* 69 (1995), 369-401.

**[2]** Groebner bases and
triangulations of the second hypersimplex
(with Jesus de Loera and Bernd Sturmfels), *Combinatorica*
15 (1995), 409-424.

**[3]** A geometric Buchberger
algorithm for integer programming,
*Mathematics of Operations Research,* 20 (1995),
864-884.

**[4]** Groebner bases in integer programming
(with Serkan Hosten),
*OPTIMA Mathematical Programming Society Newsletter,*
48 (1995).

**[5]** Test sets and inequalities for integer
programs (with Robert Weismantel)
* Proceedings of the 5th International IPCO conference, Vancouver *,
LNCS 1084 (1996) 16-30.

**[6]** Variation of cost
functions in integer programming
(with Bernd Sturmfels), *Mathematical Programming*
77 (1997), 357-387.

**[7]** Truncated Groebner bases
for integer programming (with Robert Weismantel), *Applicable
Algebra in Engineering, Communication and Computing* 8 (1997), 241-257.

**[8]** Applications to integer programming,
in * Applications of Computational Algebraic Geometry, *
D.A. Cox and B. Sturmfels eds.,
AMS Proceedings of Symposia in Applied Mathematics, 53
(1997), 119-142.

**[9]** Groebner bases in
integer programming,
* Handbook of Combinatorial Optimization Vol. 1*,
D.-Z. Du and P.M.Pardalos eds., Kluwer Acad. Publ.,
Boston, MA (1998) 533-572.

**[10]** Groebner bases and integer programming
(with Serkan Hosten), * Groebner Bases and
Applications*, B.Buchberger and F.Winkler eds.,
Cambridge University Press, London Math. Soc. Lecture Notes Series
251 (1998), 144-158.

**[11]** The associated primes
of initial ideals of lattice ideals (with Serkan Hosten), *
Mathematical Research Letters, * 6 (1999), 83-97.

**[12]** Standard pairs and group
relaxations in integer programming (with Serkan Hosten), *
Journal of Pure and Applied Algebra, * 139
(1999), 133-157.

**[13]** Algebraic methods in
integer programming, * Encyclopedia of
Optimization * (eds: C. Floudas and P. Pardalos), Kluwer Academic
Publishers, Dordrecht, 2001

**[14]** Gomory Integer Programs (with
Serkan Hosten), * Mathematical Programming*, Series B,
96 (2003), 271 - 292.

**[15] **
Toric initial ideals of $\Delta$-normal configurations:
Cohen-Macaulayness and degree bounds (with Edwin O'Shea),
* Journal of Algebraic Combinatorics, * 21
(2005), 247 - 268.

**[16]** The structure of group
relaxations *
Discrete Optimization (eds: K. Aardal, G. Nemhauser, R. Weismantel),
Handbooks in Operations Research and Management Science,*
12, Elsevier, 2005.

**[17]**
The circuit ideal of a vector configuration (with Tristram
Bogart and Anders Jensen), * Journal of Algebra, * 309:2 (2007), 518-542.

** [18] **
Small Chvatal rank (with Tristram Bogart and Annie Raymond),
* Mathematical Programming * Series B, 124:1-2 (2010), 45-68.

### Toric Hilbert Schemes

**[1]** Combinatorics of the toric Hilbert
scheme (with Diane Maclagan), * Discrete and Computational
Geometry * 27 (2002), 249-264.

**[2]** Algorithms for the toric Hilbert
scheme (with Mike Stillman and Bernd Sturmfels), a chapter in
* Computations in Algebraic Geometry with Macaulay 2 * (eds:
D. Eisenbud, D. Grayson, M. Stillman and B. Sturmfels), Algorithms and
Computation in Mathematics Vol 8, Springer 2002, 179-213

**[3] **
The toric Hilbert scheme of a rank two lattice
is smooth and irreducible (with Diane Maclagan),
* Journal of Combinatorial Theory, Series A, * 104:1 (2003), 29-48.

### Groebner bases, Groebner fans and Tropical Varieties

**[1]** Computing Groebner
fans of toric ideals (with Birk Huber),
* Experimental Mathematics * 9 (2000)
321-331.

**[2]**
The Hilbert zonotope and a polynomial
time algorithm for universal Groebner bases (with
Eric Babson and Shmuel Onn), * Advances in Applied Math *
30 (2003) 529-544.

**[3]** Computing tropical varieties (with Tristram Bogart, Anders Jensen,
David Speyer and Bernd Sturmfels), * Journal of Symbolic
Computation, * 42:1-2 (2007), 54-73.

**[4]**
The generic Groebner walk (with Komei Fukuda, Anders Jensen and
Niels Lauritzen), * Journal of Symbolic
Computation, * 43:3 (2007), 298-312.

**[5]** Computing Groebner fans (with Komei Fukuda and Anders
N. Jensen), * Mathematics of Computation, *
76 (2007), 2189-2212.

**[6]** Nice initial
complexes of some classical ideals (with Aldo Conca and Serkan
Hosten), * Contemporary Math., * 423
(2007), 11-42.

### Algebraic Combinatorics and Representation Theory

**[1]**
Symmetric iterated Betti numbers (with Eric Babson and Isabella
Novik),* Journal of Combinatorial Theory,
Series A, * 105 (2004), 233 - 254.

**[2] **
Reverse lexicographic and lexicographic shifting (with Eric
Babson and Isabella Novik), * Journal of Algebraic Combinatorics,
* 23:2 (2006), 107 - 123.

**[3]** Moduli spaces of graded representations of finite
dimensional algebras (with Eric Babson and Birge Huisgen-Zimmermann),
* Contemporary Math., * 419 (2006), 7-27.

**[4]**
Moduli of McKay quiver representations I: the coherent component
(with Alastair Craw and Diane Maclagan), * Proceedings
of the London Math. Society (3), * 95:1 (2007),
179-198.

**[5]**
Moduli of McKay quiver representations II: Groebner basis techniques
(with Alastair Craw and Diane Maclagan), *
Journal of Algebra, * 316:2 (2007), 514-535.

### Other Topics

**[1] ** Creating weekly timetables for
maximizing employee preferences (with Caleb Z. White, Youngbae
Lee, Yoonsoo Kim and Patrick Perkins), * The UMAP Journal, *
25:1 (2004), 5 - 24.