**Contemporary Mathematics**

Volume: 685;
2017;
277 pp;
Softcover

MSC: Primary 00; 13; 20; 46; 51; 52; 62; 65; 91;

Print ISBN: 978-1-4704-2321-6

Product Code: CONM/685

List Price: $111.00

AMS Member Price: $88.80

MAA member Price: $99.90

**Electronic ISBN: 978-1-4704-3743-5
Product Code: CONM/685.E**

List Price: $111.00

AMS Member Price: $88.80

MAA member Price: $99.90

# Algebraic and Geometric Methods in Discrete Mathematics

Share this page *Edited by *
*Heather A. Harrington; Mohamed Omar; Matthew Wright*

This volume contains the proceedings of the AMS
Special Session on Algebraic and Geometric Methods in Applied Discrete
Mathematics, held on January 11, 2015, in San Antonio, Texas.

The papers present connections between techniques from
“pure” mathematics and various applications amenable to
the analysis of discrete models, encompassing applications of
combinatorics, topology, algebra, geometry, optimization, and
representation theory. Papers not only present novel results, but also
survey the current state of knowledge of important topics in applied
discrete mathematics.

Particular highlights include: a new computational framework, based
on geometric combinatorics, for structure prediction from RNA
sequences; a new method for approximating the optimal solution of a
sum of squares problem; a survey of recent Helly-type geometric
theorems; applications of representation theory to voting theory and
game theory; a study of fixed points of tensors; and exponential
random graph models from the perspective of algebraic statistics with
applications to networks.

This volume was written for those trained in areas such as algebra,
topology, geometry, and combinatorics who are interested in tackling
problems in fields such as biology, the social sciences, data
analysis, and optimization. It may be useful not only for experts, but
also for students who wish to gain an applied or interdisciplinary
perspective.

#### Readership

Graduate students and research mathematicians interested in applications of algebra and geometry.

# Table of Contents

## Algebraic and Geometric Methods in Discrete Mathematics

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Eigenconfigurations of tensors 110
- Sum of squares basis pursuit with linear and second order cone programming 2736
- Helly’s theorem: New variations and applications 5564
- Representation theory of the symmetric group in voting theory and game theory 97106
- Modeling the distribution of distance data in Euclidean space 117126
- 1. Introduction to phylogenetics 117126
- 2. The space of additive distance matrices embedded in ℝ^{𝕟\choose2} 118127
- 3. Geometry of distance-based methods 120129
- 4. Modeling dissimilarity maps using statistical distributions and geometry 123132
- 5. Developing Model Family (1) 123132
- 6. Model Family (2) 124133
- 7. Data analysis 129138
- 8. Motivations: choice of distributions in Model Families (1) and (2) 130139
- 9. Non-ultrametric trees and application to larger data sets 131140
- 10. Applications and future work 132141
- 11. Supplementary material 133142
- Acknowledgements 133142
- References 133142

- Geometric combinatorics and computational molecular biology: Branching polytopes for RNA sequences 137146
- Polyhedral approaches to learning Bayesian networks 155164
- 1. Introduction 155164
- 2. Background 157166
- 3. BN Representations & BN Polytopes 161170
- 4. Learning BN Structure 164173
- 5. Facets & Valid Inequalities of BN Polytopes 166175
- 6. Example Family Variable Polytopes 175184
- 7. Example Characteristic Imset Polytopes 177186
- 8. Practical Approaches to Learning BN 180189
- References 185194

- Neural network coding of natural images with applications to pure mathematics 189198
- 1. Introduction 189198
- 2. Discrete memory models in neuroscience 192201
- 3. Natural image coding with a discrete neural network 198207
- 4. Natural image patch modeling 200209
- 5. Rate-distortion theory 203212
- 6. Near-optimal DRNN coding of ternarized natural image patches 207216
- 7. Optimal codings of continuous variables are usually discrete 208217
- 8. Conclusion 216225
- Acknowledgment 216225
- References 217226

- Proving Tucker’s Lemma with a volume argument 223232
- Factorization invariants in numerical monoids 231240
- A survey of discrete methods in (algebraic) statistics for networks 251260
- Back Cover Back Cover1290