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Tropical and Non-Archimedean Geometry: Tentative Schedule of Talks and Topics
- Saturday AM: Baker -- Introduction to tropical geometry [SS, MS, R]
- Newton polygons and Newton complexes
- Initial degenerations and the balancing condition
- The "fundamental theorem of tropical geometry"
- Higher-dimensional Newton polygons, multiplicity formula
- Saturday PM: Ducros -- Introduction to Berkovich spaces and rigid analysis [Con, D1, D2, Be1, Be2]
- Sunday AM: Thuillier -- Potential theory on Berkovich curves [Ba1, T]
- Sunday PM: Baker -- Tropical geometry and Berkovich spaces [Pay, BPR]
- Extended tropicalizations
- Payne's homeomorphism theorem
- Structure of Berkovich curves
- Statement of main results of [BPR]
- Monday PM: Maclagan -- Polyhedral structure on tropical varieties [MS]
- Monday PM: Rabinoff -- Rigid analysis and tropicalization [R]
- Tuesday AM: Baker -- Linear series on tropical curves [BN, GK, MZ, BF, Ba2, CDPR]
- Tropical Riemann-Roch
- Tropical Abel-Jacobi map
- The Specialization Lemma
- Tropical Brill-Noether theory
- Tuesday PM: Norin -- Tropical Riemann-Roch and reduced divisors [BN, HKN, L, A]
- Tuesday PM: Cools -- Tropical Brill-Noether theory, real linear series, gonality of tropical curves [CDPR, CoCo, CaCo]
- Wednesday AM: Baker --Drawing tropical curves [BPR]
- The Poincare-Lelong formula
- Slopes as orders of vanishing
- Tropical elliptic curves
- Tropical implicitization
- Wednesday PM: Papikian -- Mumford uniformization theory for curves and abelian varieties [P, FvdP, GvdP]
- Wednesday PM: Viviani -- Tropical Torelli theorem, moduli spaces of tropical curves and tropical Abelian varieties [CV, BMV, Ch, Cap]
- Thursday AM: Rabinoff -- Proofs of the main results from [BPR]
- Admissible R-algebras
- The projection formula
- Comparing relative and tropical multiplicities
- Existence of faithful representations
- Thursday PM: Chambert-Loir -- Tropicalization of Abelian varieties, canonical measures,
Gubler's work on the Bogomolov conjecture [C-L, G1, G2, G3]
Background Reading and References
(Click
here to return to the tentative schedule.)
- [A] O. Amini. Reduced divisors and embeddings of tropical curves.
Preprint. (arXiv:1007.5364v1)
- [Ba1] M. Baker. An introduction to Berkovich analytic spaces and non-Archimedean potential theory on curves.
p-adic geometry, 123-174, Univ. Lecture Ser., 45, Amer. Math. Soc., Providence, RI, 2008.
(Preprint from author's web page)
- [Ba2] M. Baker. Specialization of linear systems from curves to graphs.
With an appendix by Brian Conrad. Algebra Number Theory 2 (2008), no. 6, 613-653.
(arXiv:math/0701075)
- [BF] M. Baker and X. Faber. Metric properties of the tropical Abel-Jacobi map.
To appear in J. Algebraic Combin. (arXiv:0905.1679v4)
- [BN] M. Baker and S. Norine. Riemann-Roch and Abel-Jacobi theory on a finite graph.
Adv. Math. 215 (2007), no. 2, 766-788.
(arXiv:math/0608360)
- [BPR] M. Baker, S. Payne, and J. Rabinoff. Non-Archimedean geometry, tropicalization, and metrics on curves.
Preprint. (arXiv:1104.0320)
- [Be1] V. G. Berkovich. Spectral theory and analytic geometry over non-Archimedean fields.
Mathematical Surveys and Monographs, 33. American Mathematical Society, Providence, RI, 1990.
- [Be2] V. G. Berkovich. p-adic analytic spaces.
Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998).
- [Bo] S. Bosch. Lectures on formal and rigid geometry.
Preprint. (Manuscript available here)
- [BMV] S. Brannetti, M. Melo, and F. Viviani. On the tropical Torelli map.
Adv. Math. 226 (2011), no. 3, 2546-2586.
(arXiv:0907.3324v5)
- [Cap] L. Caporaso. Algebraic and tropical curves: comparing their moduli spaces.
Preprint. (arXiv:1101.4821v1)
- [CV] L. Caporaso and F. Viviani. Torelli theorem for graphs and tropical curves.
Duke Math. J. 153 (2010), no. 1, 129-171.
(arXiv:0901.1389v3)
- [CaCo] W. Castryck and F. Cools. Newton polygons and curve gonalities.
Preprint. (Preprint from author's web page)
- [C-L] A. Chambert-Loir. Mesures et équidistribution sur les espaces de Berkovich.
J. Reine Angew. Math. 595 (2006), 215-235.
(arXiv:math/0304023v3)
- [Ch] M. Chan. Combinatorics of the tropical Torelli map.
Preprint. (arXiv:1012.4539v2)
- [Con] B. Conrad. Several approaches to non-Archimedean geometry.
p-adic geometry, 9-63, Univ. Lecture Ser., 45, Amer. Math. Soc., Providence, RI, 2008.
(Preprint from author's web page)
- [CoCo] F. Cools and M. Coppens. Linear pencils on graphs and on real curves.
Preprint. (arXiv:1006.1770v1)
- [CDPR] F. Cools, J. Draisma, S. Payne, and E. Robeva. A tropical proof of the Brill-Noether Theorem.
Preprint. (arXiv:1001.2774v2)
- [D1] A. Ducros. Espaces analytiques p-adiques au sens de Berkovich.
Séminaire Bourbaki. Vol. 2005/2006. Astérisque No. 311 (2007), Exp. No. 958, viii, 137-176.
(Preprint from author's web page)
- [D2] A. Ducros. Géométries analytique p-adique: la théorie de Berkovich.
Gaz. Math. No. 111 (2007), 12-27.
(Preprint from author's web page)
- [FvdP] J. Fresnel and M. van der Put. Rigid analytic geometry and its applications.
Progress in Mathematics, 218. Birkhäuser Boston, Inc., Boston, MA, 2004.
- [GK] A. Gathmann and M. Kerber. A Riemann-Roch theorem in tropical geometry.
Math. Z. 259 (2008), no. 1, 217-230.
(arXiv:math/0612129v2)
- [GvdP] L. Gerritzen and M. van der Put. Schottky groups and Mumford curves.
Lecture Notes in Mathematics, 817. Springer, Berlin, 1980.
- [G1] W. Gubler. Tropical varieties for non-Archimedean analytic spaces.
Invent. Math. 169 (2007), no. 2, 321-376.
(arXiv:math/0609383)
- [G2] W. Gubler. The Bogomolov conjecture for totally degenerate abelian varieties.
Invent. Math. 169 (2007), no. 2, 377-400.
(arXiv:math/0609387)
- [G3] W. Gubler. Non-Archimedean canonical measures on abelian varieties.
Compos. Math. 146 (2010), no. 3, 683-730.
(arXiv:0801.4503)
- [HKN] J. Hladký, D. Král', and S. Norine. Rank of divisors on tropical curves.
Preprint. (arXiv:0709.4485)
- [L] Y. Luo. Rank-determining sets of metric graphs.
Preprint. (arXiv:0906.2807v2)
- [MS] D. Maclagan and B. Sturmfels. Introduction to Tropical Geometry.
Book in progress.
(Manuscript from author's web page)
- [MZ] G. Mikhalkin and I. Zharkov. Tropical curves, their Jacobians and theta functions.
Curves and abelian varieties, 203-230, Contemp. Math., 465, Amer. Math. Soc., Providence, RI, 2008.
(arXiv:math/0612267v2)
- [Pap] M. Papikian. Rigid-analytic geometry and the uniformization of abelian varieties.
Snowbird lectures in algebraic geometry, 145-160, Contemp. Math., 388, Amer. Math. Soc., Providence, RI, 2005. (Preprint from author's web page)
- [Pay] S. Payne. Analytification is the limit of all tropicalizations.
Math. Res. Lett. 16 (2009), no. 3, 543-556 .
(arXiv:0805.1916)
- [R] J. Rabinoff. Tropical analytic geometry, Newton polygons, and tropical intersections.
Preprint. (arXiv:1007.2665v1)
- [SS] D. Speyer and B. Sturmfels. Tropical mathematics.
Math. Mag. 82 (2009), no. 3, 163-173.
(arXiv:math/0408099v1)
- [T] A. Thuillier. Théorie du potentiel sur les courbes en géométrie analytique non
archimédienne:
Applications à la théorie d'Arakelov. Thèse de
l'Université de Rennes 1 (2005). (Manuscript from author's web page)