Algebraic, Tropical, and Nonarchimedean Analytic Geometry of Moduli Spaces (16w5153)

Arriving in Oaxaca, Mexico Sunday, May 1 and departing Friday May 6, 2016

Organizers

(Georgia Institute of Technology)

(Brown University)

(University of Kentucky)

(Yale University)

Objectives

This workshop will bring together leading researchers from three fields that rarely have overlapping conferences: tropical geometry, classical algebraic geometry of moduli spaces, and nonarchimedean analytic geometry. Recent breakthroughs have come from teams of mathematicians with a range of backgrounds spanning these disciplines. We believe that this trend will continue, and we will make a concerted effort to attract experts from all of these fields who are interested in collaborating across disciplines. Also, tropical geometry and nonarchimedean geometry (especially the theory of Berkovich spaces) are relatively young fields, so it will be easy and natural to include a significant number of advanced graduate students and recent PhDs in these areas. Our intended list of participants also includes a few carefully chosen geometric group theorists and topologists who have studied the topology of Outer Space and Teichm\"uller space (closely related to moduli spaces of tropical and algebraic curves [cmv}), and have also indicated an interest in learning about the algebraic and tropical perspectives. Topologist S{\o]ren Galatius, for instance, has already begun a collaboration with organizers Chan and Payne using tropical techniques to compute the homotopy type of the boundary complex and top weight cohomology for moduli spaces of curves with marked points.



Objective 1. Bring together experts on moduli spaces from algebraic, tropical, and nonarchimedean geometry to learn the state of the art in each area and develop a shared understanding of what has already been accomplished and the most promising directions for future development.



Objective 2. Explore possibilities for employing techniques from tropical and nonarchimedean analytic geometry to make progress on classical problems in algebraic geometry, such as the characteristic numbers of the projective plane, and relations among abelian cycles on the moduli space of curves.



Objective 3. Discuss opportunities and technical obstacles to be overcome in extending results connecting algebraic and tropical moduli spaces of curves to moduli spaces of surfaces and higher dimensional objects.

Our intended list of participants combines leading researchers from the algebraic geometry and topology of moduli spaces, nonarchimedean analytic geometry, and tropical geometry with advanced graduate students and recent PhDs working in these areas. The organizers will make special efforts to remind all speakers of the breadth of their audience, and will also ensure a relatively open schedule with ample break time when people can talk about more specialized topics in smaller groups, discuss particular problems of mutual interest, and even start collaborations. We will also supplement the research talks during the day with one or two informal open problem sessions in the evenings.

Bibliography



  1. [ab] O.~Amini and M.~Baker, Linear series on metrized complexes of algebraic curves, preprint, arXiv:1204.3508. To appear in Math. Ann.

  2. [acp] D.~Abramovich, L.~Caporaso, S.~Payne, The tropicalization of the moduli space of curves, preprint, arXiv:1212.0373, 2012.

  3. [baker08] M.~Baker, Specialization of linear systems from curves to graphs. Algebra Number Theory 2 (2008), no. 6, 613--653.

  4. [berk] V.~Berkovich, Smooth $p$-adic analytic spaces are locally contractible. Invent. Math. 137 (1999), no. 1, 1--84.

  5. [bbm] M.~Bestvina, K.-U. Bux, D.~Margalit, Dimension of the Torelli group for $\mathrm{Out}(F_n)$. Invent. Math. 170 (2007), no. 1, 1--32.

  6. [bmv] S. Brannetti, M. Melo, and F. Viviani. On the tropical Torelli map, Adv. Math. 226 (2011), no.~3, 2546--2586.

  7. [caporaso-connectivity] L.~Caporaso, Geometry of tropical moduli spaces and linkage of graphs, J. Combin. Theory Ser. A 119 (2012), no. 3, 579--598.

  8. [cap12] L.~Caporaso, Algebraic and tropical curves: comparing their moduli spaces, Handbook of Moduli, Vol. I, Adv.~Lect.~Math., 24, Int.~Press, Somerville, MA, 2013, 119--160.

  9. [cap13] L.~Caporaso, Rank of divisors on graphs: an algebro-geometric analysis. A celebration of algebraic geometry, 45--64, Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, 2013.

  10. [cjp] D.~Cartwright, D.~Jensen, S.~Payne, Lifting divisors on a generic chain of loops, preprint, arXiv:1404.4001. To appear in Canad. Math. Bull.

  11. [chmr] R.~Cavalieri, S.~Hampe, H.~Markwig, D.~Ranganathan, Moduli spaces of rational weighted stable curves and tropical geometry, preprint, arXiv:1404.7426, 2014.

  12. [cjm] R.~Cavalieri, P.~Johnson, H.~Markwig, Tropical Hurwitz numbers, J. Algebraic Combin. 32 (2010), no. 2, 241--265.

  13. [cmr] R.~Cavalieri, H.~Markwig, D.~Ranganathan, Tropicalizing the space of admissible covers, preprint, arXiv:1401.4626, 2014.

  14. [chan] M.~Chan, Combinatorics of the tropical Torelli map, Algebra Number Theory 6 (2012), no. 6, 1133--1169.

  15. [chan2] M.~Chan, Topology of the tropical moduli spaces $M_{2,n}$. In preparation.

  16. [cgp] M.~Chan, S.~Galatius, S.~Payne, The boundary and top weight cohomology of $M_{1,n}$. In preparation.

  17. [cmv] M.~Chan, M.~Melo, F.~Viviani, Tropical Teichm\"{u}ller and Siegel spaces, Algebraic and combinatorial aspects of tropical geometry, Contemporary Mathematics 589 (2013), 45--85.

  18. [cdpr] F.~Cools, J.~Draisma, S.~Payne, E.~Robeva, A tropical proof of the Brill-Noether theorem. Adv. Math. 230 (2012), no. 2, 759--776.

  19. [cv] M.~Culler, K.~Vogtmann, Moduli of graphs and automorphisms of free groups. Invent. Math. 84 (1986), no. 1, 91--119.

  20. [fgp] T.~Foster, P.~Gross, S.~Payne, Limits of tropicalizations, preprint, arXiv:1211.2718. To appear in Israel J. Math.

  21. [galatius] S.~Galatius, Stable homology of automorphism groups of free groups. Ann. of Math. (2) 173 (2011), no. 2, 705--768.

  22. [gross] A.~Gross, Correspondence theorems via tropicalizations of moduli spaces, preprint, arXiv:1406.1999.

  23. [jp] D.~Jensen, S.~Payne, Tropical multiplication maps and the Gieseker-Petri Theorem, preprint, arXiv:1401.2584.

  24. [liu-osserman] F.~Liu, B.~Osserman, Severi degrees on toric surfaces, preprint, arXiv:1401.7023, 2014.

  25. [mikhalkin-enum} G.~Mikhalkin, Enumerative tropical algebraic geometry in $R^2$. J.~Amer.~Math.~Soc.~18 (2005), no. 2, 313–-377. \bibitem{op] B.~Osserman and S.~Payne, Lifting tropical intersections. Doc. Math. 18 (2013), 121--175.

  26. [or] B.~Osserman and J.~Rabinoff, Lifting nonproper tropical intersections. Tropical and non-Archimedean geometry, 15--44, Contemp. Math., 605, Amer. Math. Soc., Providence, RI, 2013.

  27. [payne09] S.~Payne, Analytification is the limit of all tropicalizations. Math. Res. Lett. 16 (2009), no. 3, 543--556.

  28. [ulirsch14] M.~Ulirsch, Tropical geometry of moduli spaces of weighted stable curves, preprint, arXiv:1405.6940, 2014.

  29. [thuillier] A.~Thuillier, G\'eom\'etrie toro\"idale et g\'eom\'etrie analytique non archim\'edienne. Application au type d'homotopie de certains sch\'emas formels, Manuscripta Math. 123 (2007), no. 4, 381--451.

  30. [vogtmann-icm] K.~Vogtmann, The cohomology of automorphism groups of free groups. International Congress of Mathematicians. Vol. II, 1101--1117, Eur. Math. Soc., Z\"urich, 2006.

  31. [yang] Jihyeon Jessie Yang, Tropical Severi Varieties, to appear in Portugal.~Math.

  32. [yu14] Tony Yue Yu, Tropicalization of the moduli space of stable maps, preprint, arXiv:1407.8444, 2014.