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Post-edited  The non-archimedean SYZ fibration and Igusa zeta functions - Part 1
Nicaise, Johannes (Auteur de la Conférence) | CIRM (Editeur )

The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint work with Mircea Mustata and Chenyang Xu. The SYZ fibration is a conjectural geometric explanation for the phenomenon of mirror symmetry for maximal degenerations of complex Calabi-Yau varieties. I will explain Kontsevich and Soibelman's construction of the SYZ fibration in the world of non-archimedean geometry, and its relations with the Minimal Model Program and Igusa's p-adic zeta functions. No prior knowledge of non-archimedean geometry is assumed. These lectures are based on joint ...

14B05 ; 14D06 ; 14E30 ; 14E18 ; 14G10 ; 14G22

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Le troisième groupe de cohomologie non ramifiée d'une variété lisse, à coefficients dans les racines de l'unité tordues deux fois, intervient dans plusieurs articles récents, en particulier en relation avec le groupe de Chow de codimension 2. On fera un tour d'horizon : espaces homogènes de groupes algébriques linéaires; variétés rationnellement connexes sur les complexes; images d'applications cycle sur les complexes, sur un corps fini, sur un corps de nombres. Le troisième groupe de cohomologie non ramifiée d'une variété lisse, à coefficients dans les racines de l'unité tordues deux fois, intervient dans plusieurs articles récents, en particulier en relation avec le groupe de Chow de codimension 2. On fera un tour d'horizon : espaces homogènes de groupes algébriques linéaires; variétés rationnellement connexes sur les complexes; images d'applications cycle sur les complexes, sur un corps fini, sur un ...

19E15 ; 14C35 ; 14C25 ; 14E08

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Post-edited  Whitney problems and real algebraic geometry
Fefferman, Charles (Auteur de la Conférence) | CIRM (Editeur )

This talk sketches connections between Whitney problems and e.g. the problem of deciding whether a given rational function on $\mathbb{R}^n$ belongs to $C^m$.

26Bxx ; 46E10 ; 58A20 ; 14Qxx

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Post-edited  On the remodeling conjecture for toric Calabi-Yau 3-orbifolds
Liu, Chiu-Chu Melissa (Auteur de la Conférence) | CIRM (Editeur )

The remodeling conjecture proposed by Bouchard-Klemm-Marino-Pasquetti relates Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-orbifold to Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. It can be viewed as a version of all genus open-closed mirror symmetry. In this talk, I will describe results on this conjecture based on joint work with Bohan Fang and Zhengyu Zong.

14J33 ; 14N35

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Post-edited  Stability and applications to birational and hyperkaehler geometry - Lecture 1
Bayer, Arend (Auteur de la Conférence) | CIRM (Editeur )

This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland’s notion of stability conditions on derived categories [2, 5, 6]. I will then proceed to explain the concept of wall-crossing, both in theory, and in examples [1, 2, 4, 6].

- Wall-crossing and birational geometry. Every moduli space of Bridgeland-stable objects comes equipped with a canonically defined nef line bundle. This systematically explains the connection between wall-crossing and birational geometry of moduli spaces. I will explain and illustrate the underlying construction [7].

- Applications : Moduli spaces of sheaves on $K3$ surfaces. I will explain how one can use the theory explained in the previous talk in order to systematically study the birational geometry of moduli spaces of sheaves, focussing on $K3$ surfaces [1, 8].
This lecture series will be an introduction to stability conditions on derived categories, wall-crossing, and its applications to birational geometry of moduli spaces of sheaves. I will assume a passing familiarity with derived categories.

- Introduction to stability conditions. I will start with a gentle review of aspects of derived categories. Then an informal introduction to Bridgeland’s notion of stability conditions on derived categories ...

14D20 ; 14E30 ; 14J28 ; 18E30

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Post-edited  Rank 3 rigid representations of projective fundamental groups
Simpson, Carlos (Auteur de la Conférence) | CIRM (Editeur )

This is joint with Adrian Langer. Let $X$ be a smooth complex projective variety. We show that every rigid integral irreducible representation $ \pi_1(X,x) \to SL(3,\mathbb{C})$ is of geometric origin, i.e. it comes from a family of smooth projective varieties. The underlying theorem is a classification of VHS of type $(1,1,1)$ using some ideas from birational geometry.

14F35 ; 14D07 ; 58E20 ; 22E40

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Post-edited  Invariants of determinantal varieties
Ruas, Maria Aparecida Soares (Auteur de la Conférence) | CIRM (Editeur )

We review basic results on determinantal varieties and show how to apply methods of singularity theory of matrices to study their invariants and geometry. The Nash transformation and the Euler obstruction of Essentially Isolated Determinantal Singularities (EIDS) are discussed. To illustrate the results we compute the Euler obstruction of corank one EIDS with non isolated singularities.

14B05 ; 32S05