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## Tropical Aspects in Geometry, Topology and Physics

Oberwolfach :
2015.

Under the general editorship: T. Ekholm, H. Markwig

The workshop *Tropical Aspects in Geometry, Topology and Physics* was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject. The development of tropical geometry is based on deep links between problems in real and complex enumerative geometry, symplectic geometry, quantum fields theory, mirror symmetry, dynamical systems and other research areas. On the other hand, new interesting phenomena discovered in the framework of tropical geometry (like refined tropical enumerative invariants) pose the problem of a conceptual understanding of these phenomena in the “classical” geometry and mathematical physics.

Galkin S., , in : Tropical Aspects in Geometry, Topology and Physics. : Oberwolfach: [б.и.], 2015. P. 1269–1273.

Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations help with computing Gromov–Witten invariants, and the role of this story in “Fanosearch” programme. The challenge is to formulate enumerative symplectic geometry of complex 3-folds in a way suitable for extracting invariants ...

Added: December 2, 2015

A. Levin, Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2013.

We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ with regular singularities of connections at
marked points. The bundles are classified by their characteristic classes.
These classes are elements of the second cohomology group
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of
$G$. For any complex simple Lie group $G$ and arbitrary class we define ...

Added: December 27, 2013

Braverman A., Rybnikov L. G., Feigin B. L. et al., Communications in Mathematical Physics 2011 Vol. 308 No. 2 P. 457–478

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts ...

Added: May 12, 2012

Kharlamov V., Viktor Kulikov, / Cornell University. Series math "arxiv.org". 2013.

In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli ...

Added: December 27, 2013

Fedor Bogomolov, Böhning C., / Cornell University. Series math "arxiv.org". 2012.

In this article we determine the stable cohomology groups H^i_s (A_n, Z/p) of the alternating groups A_n for all integers n and i, and all primes p. ...

Added: December 4, 2013

Fedor Bogomolov, Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2013.

We discuss the problem of stable conjugacy of finite subgroups of Cremona
groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant
and compute this group in some cases. ...

Added: November 21, 2014

Galkin S., Shinder E., / Cornell University. Series math "arxiv.org". 2012. No. 1210.3339.

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections ...

Added: September 14, 2013

Akhmedov E., Physical Review D - Particles, Fields, Gravitation and Cosmology 2013 Vol. 87 No. 4 P. 044049

Following Krotov and Polyakov [ Nucl. Phys. B849 410 (2011)], we show that in global de Sitter space its isometry is broken by the loop IR divergences for any invariant vacuum state of the massive scalars. We derive a kinetic equation in global de Sitter space that follows from the Dyson-Schwinger equation of the Schwinger-Keldysh ...

Added: February 27, 2013

Lev Soukhanov, / Cornell University. Series math "arxiv.org". 2014.

We consider the systems of diffusion-orthogonal polynomials, defined in the
work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why
these systems with boundary of maximal possible degree should always come from
the group, generated by reflections. Our proof works for the dimensions $2$ (on
which this phenomena was discovered) and $3$, and fails in ...

Added: September 19, 2014

Rybakov S., / Cornell University. Series math "arxiv.org". 2014.

A k-isogeny class of abelian varieties over a finite field k is uniquely determined by the Weil polynomial f of any variety from this class. When we consider classification problems concerning abelian varieties inside an isogeny class, the classification can be given in terms of the corresponding Weil polynomial. In this paper we improve our ...

Added: January 21, 2014

Efimov A., / Cornell University. Series math "arxiv.org". 2013.

In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary ...

Added: October 31, 2013

Michael Finkelberg, Leonid Rybnikov, / Cornell University. Series math "arxiv.org". 2013.

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic ...

Added: December 27, 2013

Positselski L., Efimov A., / Cornell University. Series math "arxiv.org". 2013. No. arXiv:1102.0261.

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations ...

Added: December 22, 2013

Positselski L., / Cornell University. Series math "arxiv.org". 2014. No. 1209.2995.

Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version ...

Added: February 6, 2013

Lee K., Shabalin T., / Cornell University. Series math "arxiv.org". 2014.

We construct exceptional collections of maximal length on four families of
surfaces of general type with $p_g=q=0$ which are isogenous to a product of
curves. From these constructions we obtain new examples of quasiphantom
categories as their orthogonal complements. ...

Added: October 17, 2014

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005–1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014

F.A. Bogomolov, Vik.S. Kulikov, / Cornell University. Series math "arxiv.org". 2014.

In \cite{Ku0}, the ambiguity index $a_{(G,O)}$ was introduced for each
equipped finite group $(G,O)$. It is equal to the number of connected
components of a Hurwitz space parametrizing coverings of a projective line with
Galois group $G$ assuming that all local monodromies belong to conjugacy
classes $O$ in $G$ and the number of branch points is greater than some
constant. ...

Added: November 21, 2014

Campana F., Demailly J., Misha Verbitsky, / Cornell University. Series math "arxiv.org". 2013.

We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of ...

Added: May 13, 2013

Romaskevich O. L., L'Enseignement Mathématique 2014

We consider 3 -periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the complexification of the problem coupled with the complex law of reflection. ...

Added: December 25, 2014

Fedor Bogomolov, De Oliveira B., / Cornell University. Series math "arxiv.org". 2014.

In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of ...

Added: November 21, 2014

Ivan Cheltsov, Park J., Won J., / Cornell University. Series math "arxiv.org". 2013.

For each del Pezzo surface $S$ with du Val singularities, we determine
whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we
present an effective divisor $D$ that is $\mathbb{Q}$-linearly equivalent to
$-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder.
As a corollary, we classify all the del Pezzo surfaces with du ...

Added: December 27, 2013

Akhmedov E., Popov F., Astrakhantsev N., Journal of High Energy Physics 2014 Vol. 1409 P. 071

We calculate one--loop corrections to the vertexes and propagators of photons and charged particles in the strong electric field backgrounds. We use the Schwinger--Keldysh diagrammatic technique. We observe that photon's Keldysh propagator receives growing with time infrared contribution. As the result, loop corrections are not suppressed in comparison with tree--level contribution. This effect substantially changes ...

Added: September 25, 2014

Victor Kulikov, Shustin E., / Cornell University. Series math "arxiv.org". 2014.

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for ...

Added: February 2, 2015

Bezrukavnikov R., Finkelberg M. V., / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Gorchinskiy Sergey, Rosly Alexei, / Cornell University. Series math "arxiv.org". 2012.

We construct the so-called polar complex for an arbitrary locally free sheaf on a smooth variety over a field of characteristic zero. This complex is built from logarithmic forms on all irreducible subvarieties with values in a locally free sheaf. We prove that cohomology groups of the polar complex are canonically isomorphic to the cohomology ...

Added: October 31, 2013