### An Introduction to Families, Deformations and Moduli

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It is known that in every dimension the number of deformation families of smooth Fano varieties is finite, but the classification of Fano varieties of dimension at least 4 is completely open. Ideas coming from Mirror Symmetry constitute a new way to tackle this problem.

Altmann has deeply studied homogeneous deformations of affine toric varieties, by noting that Minkowski decompositions of polyhedra induce deformations. In this poster, which is based an ongoing project with Alessio Corti and Paul Hacking, I will present an approach to construct non-homogeneous deformations of Fano toric varieties of dimension 3, with Gorenstein singularities. This approach lies in the context of the Gross-Siebert programme.

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Compactifications of moduli spaces of points and lines. When considering degenerations of such objects, it is interesting to compare the resulting compactifications. This is joint work in progress with Jenia Tevelev. Birational Geometry and Moduli Spaces. Indam Workshop. Schedule and Slides. The Workshop will start on Monday 11th June after lunch about 2.

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- Next talks.
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- What Is Young's Modulus?.

More Information about talks, posters, abstracts and timetable will come as soon as possible. This is a joint work with Misha Verbitsky. Let X be an irreducible holomorphic symplectic manifold and z a Beauville-Bogomolov negative 1,1 -class. Using the ergodicity of the monodromy action, we prove some deformation-invariance statements for the locus covered by rational curves of class z in X.

Carrying out the Minimal Model Program for moduli spaces is a classical and extremely challenging problem. In this talk, we will deal with a particular moduli space, namely the Hilbert scheme of points on a surface with irregularity zero.

## algebraic geometry - Good books/expository papers in moduli theory - Mathematics Stack Exchange

After explaining the connection between the birational models of a variety and the combinatorics of its Nef cone, we will show how Bridgeland stability conditions are a powerful machinery to produce extremal rays in the Nef cone of the Hilbert scheme. Time permitting, we will give a complete description of the Nef cone in some examples of low Picard rank. This is joint work with J. Huizenga, Y. Lin, E. Riedl, B. Schmidt, M.

## Topological methods in moduli theory

Woolf and X. We will talk about a work in progress with C. Knutsen and T. In particular we will focus on the existence problem and the regularity problem. I will discuss how to generalise known nonvanishing and semiampleness conjectures from various contexts. This is joint work with Thomas Peternell. The theory of linear series on tropical curves, since its introduction by Baker and Norine about 10 years ago, has seen spectacular developments in recent years. In fact, the combinatorial systematic treatment of degenerations of classical linear series that the theory has led to the proof of many important results on algebraic curves.

On the other hand, the introduction and study of a number of tropical moduli spaces of curves along with its realization as skeletons of their classical compactified counterparts allows for a deeper understanding of combinatorial aspects of moduli spaces and in particular of their compactifications. In this case, we have some stability results for the perfect cone compactification and the matroidal partial compactification.

The features of stable cohomology classes are very different from those in the classical examples, although they have some combinatorial aspects in common with them. This is joint work with Sam Grushevsky and Klaus Hulek. In this poster, I will present the intersection Betti numbers of the moduli space of non-hyperelliptic Petri-general genus four curves.

### An introduction to families, deformations and moduli

This space has a canonical compactification as GIT quotient, which was proven to be the final step in the Hassett-Keel log MMP for stable genus four curves. The strategy of the cohomological computation relies on a general method developed by F.

Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on stratifications, a partial desingularisation and the decomposition theorem. However their techniques fail when the curve is singular. We use degeneration techniques from Hodge theory to prove an analogous result when the underlying curve is irreducible nodal with a single node. This is joint work with A. Dan and S. The dynamics of an automorphism preserving a fibration.

Recent advances have made possible to apply the machinery of wall-crossing and stability conditions on derived categories introduced by Bridgeland in in order to understand the birational geometry of moduli space of shaves M of a K3 surface X. In will describe how to refine this analysis and describe the geometry of the exceptional locus of the contraction in terms of wall-crossing. More deformation occurs in a flexible material compared to that of a stiff material.

The basic concept behind Young's modulus was described by Swiss scientist and engineer Leonhard Euler in In , Italian scientist Giordano Riccati performed experiments leading to modern calculations of the modulus. It should probably be called Riccati's modulus, in light of the modern understanding of its history, but that would lead to confusion. The Young's modulus often depends on the orientation of a material. Isotropic materials display mechanical properties that are the same in all directions.

Examples include pure metals and ceramics. However, working a material or adding impurities to it can produce grain structures that make mechanical properties directional. These anisotopic materials may have very different Young's modulus values, depending on whether force is loaded along the grain or perpendicular to it.

Good examples of anisotropic materials include wood, reinforced concrete, and carbon fiber. This table contains representative values for samples of various materials. Keep in mind, the precise value for a sample may be somewhat different, since the test method and sample composition affect the data. In general, most synthetic fibers have low Young's modulus values.