**Contemporary Mathematics**

Volume: 620;
2014;
328 pp;
Softcover

MSC: Primary 14; 18; 19; 55; 57; 60;

**Print ISBN: 978-0-8218-9474-3
Product Code: CONM/620**

List Price: $117.00

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MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-1855-7
Product Code: CONM/620.E**

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# Algebraic Topology: Applications and New Directions

Share this page *Edited by *
*Ulrike Tillmann; Søren Galatius; Dev Sinha*

This volume contains the proceedings of the
Stanford Symposium on Algebraic Topology: Applications and New
Directions, held from July 23–27, 2012, at Stanford University,
Stanford, California. The symposium was held in honor of Gunnar
Carlsson, Ralph Cohen and Ib Madsen, who celebrated their 60th and
70th birthdays that year. It showcased current research in Algebraic
Topology reflecting the celebrants' broad interests and profound
influence on the subject.

The topics varied broadly from stable equivariant homotopy theory
to persistent homology and application in data analysis, covering
topological aspects of quantum physics such as string topology and
geometric quantization, examining homology stability in algebraic and
geometric contexts, including algebraic \(K\)-theory and the theory of
operads.

#### Readership

Graduate students and research mathematicians interested in algebraic topology and its applications.

# Table of Contents

## Algebraic Topology: Applications and New Directions

- Preface ix10 free
- Scientific Programme xi12 free
- List of Participants and Conference Photo xv16 free
- Representation stability in cohomology and asymptotics for families of varieties over finite fields 120 free
- 1. Introduction 120
- 2. The twisted Grothendieck–Lefschetz Formula 1130
- 3. Hyperplane arrangements, their cohomology, and combinatorics of squarefree polynomials 1736
- 4. Statistics of squarefree polynomials and the cohomology of the pure braid group 2746
- 5. Maximal tori in \GL_{𝑛}(\F_{𝑞}) 4261
- References 5271

- A stability conjecture for the unstable cohomology of 𝑆𝐿_{𝑛}ℤ, mapping class groups, and 𝔸𝕦𝕥(𝔽_{𝕟}) 5574
- The Boardman-Vogt tensor product of operadic bimodules 7190
- Detecting and realising characteristic classes of manifold bundles 99118
- Controlled Algebraic 𝐺-theory, II 111130
- More examples of discrete co-compact group actions 133152
- On the 𝐾-theory of planar cuspical curves and a new family of polytopes 145164
- Equivariant multiplicative closure 183202
- Topology of random simplicial complexes: a survey 201220
- The definition of a non-commutative toric variety 223242
- The Stable Symplectic Category and Quantization 251270
- 1. Introduction 251270
- 2. Stabilized lagrangian immersions 253272
- 3. The Stable Symplectic homotopy category 257276
- 4. Internal structure of the stable symplectic homotopy category 261280
- 5. The algebra of observables and the Sympletomorphism Group 263282
- 6. The Stable Symplectic category as an 𝐴_{∞}-category: Background 264283
- 7. The Stable Symplectic category as an 𝐴_{∞}-category: Morphisms 268287
- 8. The Stable Metaplectic Category 273292
- 9. Appendix: Some computations and Remarks 275294
- References 278297

- A geometric perspective on quantum field theory 281300
- Sketches of a platypus: a survey of persistent homology and its algebraic foundations 295314
- 1. Introduction 296315
- 2. Persistence barcodes and diagrams 298317
- 3. Functions on a manifold 299318
- 3.1. A functional view of persistent homology 300319
- 3.2. Filtered complexes 301320
- 3.3. The stability meta-theorem 302321
- 3.4. Vector spaces with ordered bases 302321
- 3.5. Diagrams over (\R,≤) 302321
- 3.6. Directly to diagrams 305324
- 3.7. Measures on the real line 305324
- 3.8. Categorification 308327

- 4. Filtered topological spaces 308327
- 5. Shapes of theories, future directions 315334
- 6. Conclusion 316335
- 7. Acknowledgments 316335
- References 316335

- Cartier’s first theorem for Witt vectors on ℤ_{≥0}ⁿ-0 321340