I am an assistant professor in the Department of Mathematics and Statistics at the University of Massachusetts. My office is LGRT 1116, and I can be reached by email at raymond(at)math(dot)umass(dot)edu.

Here is my **CV**.

**Research interests:** I am interested in anything related to combinatorial optimization, extremal graph theory, applied real algebraic geometry, operations research, proof complexity and polyhedral combinatorics.

** Outreach interests:** I am interested in thinking of different ways of making math more diverse. I run _forall on Instagram and on www.mathisforall.com which features women and people of color in mathematics (currently on hiatus since the beginning of the pandemic). I am also invested in university-level education for incarcerate people.

**Past** - I spent the fall semester of 2017 at the Mathematical Sciences Research Institute in Berkeley as a Gamelin Endowed Postdoctoral Fellow in the Geometric and Topological Combinatorics program. From 2014 to 2017, I was an Acting Assistant Professor (i.e., a postdoc) in the Department of Mathematics at the University of Washington working with Rekha Thomas. I completed a Ph.D. in mathematics under the supervision of Martin Grötschel at the Technische Universität Berlin. My thesis was on ''Polyhedral Methods Applied to Extremal Combinatorics Problems''. During my time in Berlin, I was also a member of the Berlin Mathematical School and of the Zuse Institute Berlin, an interdisciplinary research institute for applied mathematics and data-intensive high-performance computing. Prior to that, I was a student at the Massachusetts Institute of Technology where I completed a bachelor of science in mathematics as well as a bachelor of science (!) in music. During that time, I participated in the Undergraduate Research Opportunities Program at MIT and in two summer research programs, one funded by the NSERC and the other by the LaCIM, both in my hometown at the Université du Québec à Montréal.

**Ubiquity of power sums in graph profiles** with Greg Blekherman

arXiv:2308.07422

**Tropicalizing the graph profile of some almost-stars** with Maria Dascălu, SIAM Journal on Discrete Mathematics, Volume 38, Issue 2 (2024) arXiv:2308.06242

**Generalized Pitman-Stanley polytope: vertices and faces** with William Dugan, Maura Hegarty* and Alejandro Morales

arXiv:2307.09925, related extended abstract at FPSAC 2023
(* denotes undergrad coauthors)

**Undecidability of polynomial inequalities in weighted graph homomorphism densities** with Greg Blekherman and Fan Wei, Forum of Mathematics, Sigma, Volume 12 (2024), e40 arXiv:2207.12378

**Non-Trivial Squares and Sidorenko's Conjecture** with Pranav Garg* and Amanda Redlich

arXiv:2206.10058 (* denotes undergrad coauthors)

**A Path Forward: Tropicalization in Extremal Combinatorics** with Greg Blekherman

Advances in Mathematics, Volume 407 (2022), 108561, pp. 1-68 arXiv:2108.06377

**Proof of the Erdős-Simonovits conjecture on walks** with Greg Blekherman

Graphs and Combinatorics, Volume 39, 53 (2023)
arXiv:2009.10845

**Tropicalization of Graph Profiles** with Greg Blekherman, Mohit Singh and Rekha Thomas

Transactions of the American Mathematical Society, Volume 375, Number 9 (2022), pp. 6281-6310 arXiv:2004.05207

**Linear Relaxation of an Integer Program for the Union-Closed Conjecture** with Brianna Amaral*, Lucien Dalton*, Drew Polakowski* and Bertram Thomas*.

arXiv:2004.05210 (* denotes undergrad coauthors)

**Simple Graph Density Inequalities with no Sum of Squares Proofs** with Greg Blekherman, Mohit Singh and Rekha Thomas

Combinatorica, Volume 40 (2020), pp. 455-471 arXiv:1812.08820

**The Bullet Problem with Discrete Speeds** with Brittany Dygert*, Matthew Junge, Christoph Kinzel*, Erik Slivken and Jennifer Zhu*

Electronic Communications in Probability, Volume 24, Issue 27 (2019), pp. 1-11 (* denotes undergrad coauthors)

**The Turán Polytope**

The Electronic Journal of Combinatorics, Volume 25, Issue 3 (2018), P3.43

**Symmetric Sums of Squares over k-Subset Hypercubes** with James Saunderson, Mohit Singh and Rekha Thomas

Mathematical Programming Series A, Volume 167, Issue 2 (2018), pp. 315-354

**Symmetry in Turan Sums of Squares Polynomials from Flag Algebras** with Mohit Singh and Rekha Thomas

Algebraic Combinatorics, Volume 1, Number 2 (2018), pp. 249-274.

**New Conjectures for Union-Closed Families** with Jonad Pulaj and Dirk Theis

The Electronic Journal of Combinatorics, Volume 23, Issue 3 (2016), P23.3

**Standings in Sports Competitions Using Integer Programming** with Christian Raack, Thomas Schlechte and Axel Werner

Journal of Quantitative Analysis in Sports, Volume 10, Issue 2 (2014), pp. 131-138

**Multiband Robust Optimization and its Adoption in Harvest Scheduling** with Fabio D'Andreagiovanni

FORMATH, Forest Resource Management and Mathematical Modeling International Symposium, Volume 13 (2013)

**0-1 Multiband Robust Optimization** with Christina Büsing and Fabio D'Andreagiovanni

Operations Research Proceedings 2013, pp. 89-96

**Robust Optimization under Multiband Uncertainty** with Christina Büsing and Fabio D'Andreagiovanni

Proceedings of the Workshop on Mixed Integer Programming 2013

**The Centers of Gravity of the Associahedron and of the Permutahedron Are the Same** with Christophe Hohlweg and Jonathan Lortie

The Electronic Journal of Combinatorics, Volume 17, Issue 1 (2010), R72

**Small Chvátal Rank** with Tristram Bogart and Rekha Thomas

Mathematical Programming, Volume 124, Issue 1-2 (2010), pp 45-68

**Spring Semester 2023: Math 697CM Combinatorial Optimization**: 14 students. In this grad course, we considered maximization and minimization problems in graphs and networks. We covered a broad range of topics such as matchings in bipartite graphs and in general graphs, assignment problem, polyhedral combinatorics,total unimodularity, matroids, matroid intersection, min arborescence, max flow/min cut, max cut, traveling salesman problem, stable sets and perfect graphs. One of our main tools was integer programming, and we also sometimes relied on semidefinite programming. Many of these problems come from real-world applications, so we also sometimes discussed the algorithms necessary to solve them. This was a rigorous mathematical introduction to combinatorial optimization with proofs.

**Fall Semester 2022: Math 456 Mathematical Modeling**: 29 students. This course is an introduction to mathematical modeling. The main goal of the class is to learn how to translate real-world problems into quantitative terms for interpretation, suggestions of improvement and future predictions. Since this is too broad of a topic for one semester, this class will focus on linear and integer programming to study real world problems that affect real people. The course will culminate in a final modeling project that will involve optimizing the logistics of the Food Bank of Western Massachusetts as well as optimizing course assignments within our math department.

**Spring Semester 2022: Math 557 Linear optimization and polytopes**: 18 students.
This proof-based course covers the fundamentals of linear optimization and polytopes and the relationship between them. The course will give a rigorous treatment of the algorithms used in linear optimization. The topics covered in linear optimization are graphical methods to find optimal solutions in two and three dimensions, the simplex algorithm, duality and Farkas' lemma, variation of cost functions, an introduction to integer programming and Chvatal-Gomory cuts. The topics covered simultaneously in polytopes are two- and three-dimensional polytopes, f-vectors, equivalence of the vertex and hyperplane descriptions of polytopes, the Hirsch conjecture, the secondary polytope, and an introduction to counting lattice points of polytopes.

**Fall Semester 2021: Math 456 Mathematical Modeling**: 30 students. This course is an introduction to mathematical modeling. The main goal of the class is to learn how to translate real-world problems into quantitative terms for interpretation, suggestions of improvement and future predictions. Since this is too broad of a topic for one semester, this class will focus on linear and integer programming to study real world problems that affect real people. The course will culminate in a final modeling project that will involve optimizing the logistics of the Food Bank of Western Massachusetts.

**Fall Semester 2020: Math 456 Mathematical Modeling**: 26 students. This course is an introduction to mathematical modeling. The main goal of the class is to learn how to translate real-world problems into quantitative terms for interpretation, suggestions of improvement and future predictions. Since this is too broad of a topic for one semester, this class will focus on linear and integer programming to study real world problems that affect real people. The course will culminate in a final modeling project that will involve optimizing the logistics of Pedal People.

**Spring Semester 2020: Math 697SS Sums of Squares: Theory and Applications**: 8 students + 4 auditors. The theory of sums of squares (SOS) blends exciting ideas from optimization, real algebraic geometry and convex geometry. Indeed, Hilbert's famous characterization of nonnegative polynomials that are SOS in 1888, and Artin's affirmative answer to Hilbert's 17th problem on whether all nonnegative polynomials are SOS of rational functions are at the origins of this topic. Over the last two decades, interest in the theory and application of SOS polynomials has exploded because of the work of Shor, Nesterov, Lasserre and Parrilo that connects SOS polynomials to modern optimization via semidefinite programming. Since then, there has been many thrilling applications in combinatorics, theoretical computer science, and engineering. This course will cover both the theory and some applications.

**Fall Semester 2019: Math 456 Mathematical Modeling**: 30 students. This course is an introduction to mathematical modeling. The main goal of the class is to learn how to translate real-world problems into quantitative terms for interpretation, suggestions of improvement and future predictions. Since this is too broad of a topic for one semester, this class will focus on linear and integer programming to study real world problems that affect real people. The course will culminate in a final modeling project that will involve optimizing the logistics of the Valley Bikeshare service.

**Spring Semester 2019: Math 697C Combinatorial Optimization**: 18 students. In this grad course, we considered maximization and minimization problems in graphs and networks. We covered a broad range of topics such as matchings in bipartite graphs and in general graphs, assignment problem, polyhedral combinatorics, total unimodularity, matroids, matroid intersection, min arborescence, max flow/min cut, max cut, traveling salesman problem, stable sets and perfect graphs. One of our main tools was integer programming, and we also sometimes relied on semidefinite programming. Many of these problems come from real-world applications, so we also sometimes discussed the algorithms necessary to solve them. This was a rigorous mathematical introduction to combinatorial optimization with proofs.

**Fall Semester 2018: Math 455 Discrete Mathematics**: 25 students. As below.

**Spring Semester 2018: Math 455 Discrete Mathematics**: 27 students. This was a rigorous introduction to some topics in mathematics that underlie areas in computer science and computer engineering, including graphs and trees, spanning trees, and matchings; the pigeonhole principle, induction and recursion, generating functions, and proofs involving bijections. The course integrated learning mathematical theories with applications to concrete problems from other disciplines using discrete modeling techniques. Student groups were formed to investigate a concept or an application related to discrete mathematics, and each group reported its findings to the class in a final presentation. The course was mostly flipped.

**Winter Semester 2022-2023: Math 100 Math for the Real World**: 10 students. UMass grad students Porter Morgan and Asher Supernaw were TAs. The students earned credits through UMass. This course covered some introductory topics in combinatorics, probability, statistics, linear algebra and optimization. Weekly homework and quizzes, one midterm, one final.

**Spring Semester 2020: Math 197C Intro to Coding and Mathematical Modeling**: 5 students. The course unfortunately stopped midway through because of the pandemic.

**Fall Semester 2019: Lecture Series** involving faculty members and grad students from different departments at UMass.

**Spring Semester 2019: Math 100 Math for the Real World**: 5 students, co-taught with Nathaniel Whitaker. UMass grad students Daniel Gallagher and Angelica Simonetti were TA's. The students earned credits through UMass. This course covered some introductory topics in combinatorics, probability, statistics, linear algebra and optimization. Weekly homework and quizzes, one midterm, one final.

**Fall Semester 2018: Biweekly Math Circle**: 15 students.

**Fall Semester 2017: Elementary Algebra** 28 students, co-taught with Dan Walls and Tomas Leon. This class was taught through the Prison University Project, an organization whose mission is to provide excellent higher education to people at San Quentin State Prison; to support increased access to higher education for incarcerated people; and to stimulate public awareness about higher education access and criminal justice. This course was meant to teach students the foundational tools needed for higher mathematics and a language for understanding the principles of science, engineering, economics, etc.

**Spring Semester 2017: MATH 106 College Algebra** 11 students + 3 auditors. This class is taught through University Beyond Bars, an organization providing higher education to people in prisons. The students earn credits for this course through Adams State University. This course is meant to be an introduction to the basic techniques of algebra. Topics include functions, systems of equations, matrix algebra, inequalities, and complex numbers. Weekly homework, four midterms, one final, two hands-on projects where the students applied what they learned to their daily lives.

**Fall Semester 2016: Lecture Series on Proofs from the Book** This lecture series was organized through University Beyond Bars. It provided an introduction to the world of mathematical proofs.

**Spring Semester 2016: MATH 104 Finite Mathematics** 16 students + 4 auditors. This class was taught through University Beyond Bars, and students earned credits through Adams State University. This course covered some introductory topics in combinatorics, probability, statistics, linear algebra and optimization. Weekly homework and quizzes, one midterm, one final, two hands-on projects where the students applied what they learned to their daily lives.

**Spring Quarter 2017: MATH 300 Introduction to Mathematical Reasoning** 32 students. An introduction to mathematical arguments and the writing of proofs through topics in elementary set theory, graph theory and number theory. Weekly homework and quizzes every other week, one midterm, one final.

**Spring Quarter 2017: MATH 324 Advanced Multivariable Calculus** 50 students. As below.

**Winter Quarter 2016: MATH 498A Special Topics in Mathematics ** 1 student. Reading course with an undergrad student on approximation algorithms.

**Fall Quarter 2016: MATH 324 Advanced Multivariable Calculus** 100 students over two sections. As below.

** Spring Quarter 2016: MATH 409 Combinatorial Optimization** 27 students. This course covered various topics connected to matchings and matroids from a polyhedral perspective. Weekly homework and quizzes, one midterm, one final.

**Spring Quarter 2016: MATH 498 Special Topics in Mathematics ** 1 student. Research with an undergrad student on the union-closed sets conjecture.

**Winter Quarter 2016: MATH 324 Advanced Multivariable Calculus** 92 students over two sections. As below with the addition of ASK or Annie's Survival Kit, three extra problems with carefully written out solutions sent weekly to better prepare the students for the quizzes. An extra ASK office hour right before the quiz was also added where I went over the problems and showed my thought process.

**Fall Quarter 2015: MATH 324 Advanced Multivariable Calculus ** 88 students over two sections. As below with the removal of one midterm and the addition of two extra-credit projects. The first one asked students to find out about an application of multivariable calculus relevant for their future career or their interests. The second one asked them to use Lagrange multipliers to optimize something in their lives.

**Spring Quarter 2015: MATH 308 Matrix Algebra With Applications** 49 students. As below with the addition of an extra-credit project where the students had to think about an advanced application of linear algebra (i.e., not just solving) relevant for their future career or their interests.

**Winter Quarter 2015: MATH 308 Matrix Algebra With Applications** 46 students. This is an introduction to linear algebra which covered solving systems of linear equations, matrix algebra, vector spaces, orthogonality, least squares, eigenvalues, diagonalization. Weekly homework through WebAssign as well as weekly quizzes, one midterm and one final.

**Fall Quarter 2014: MATH 324 Advanced Multivariable Calculus** 94 students over two sections. This course covered double and triple integrals in different coordinate systems, as well as line and surface integrals to culminate with Stokes' theorem and the Divergence Theorem. Weekly homework through WebAssign as well as weekly quizzes, two midterms and one final.

**Winter Semester 2011-2012: Proof Techniques in Polyhedral Combinatorics ** 10 students. Seminar for advanced undergrad students and master students co-taught with Martin Grötschel. Lectured the first few weeks and then helped the students to choose a topic of their own to study and then advised them in their research.