 Laura Caravenna: Fine properties and regularity in conservation laws
Conservation laws are first order partial differential equations which appear in the modeling of physical and engineering problems: they describe constraints on how certain quantities (for example mass, momentum, energy) vary in time and in the space variables. Even if the initial datum is smooth, due to nonlinearity a smooth solution of a conservation law will in general break down in finite time, owing to the formation of discontinuities, and this qualitative feature agrees with the phenomena that the equations model. From the mathematical viewpoint, the formation of discontinuities and of other types of irregular behaviors poses severe challenges in the analysis of this class of equations. Despite severe restrictions, some regularity results are available for conservation laws: these results are relevant from both theoretical and applicative purposes.
In my presentation I will highlight few major achievements in the general theory of 1d conservation laws. I will then report on some of the present research concerning fine properties and regularity issues, better focusing on the following topic. In 1973 Schaeffer established a regularity theorem that applies to scalar (i.e., realvalued) conservation laws with uniformly convex fluxes and which can be roughly speaking formulated as follows: for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the timespace plane. Here the term "generic" should be interpreted in a suitable topological sense, related to the Baire Category Theorem. In my talk I will introduce the positive and negative results which are available, illustrating a recent explicit mathematical counterexample that shows that Schaffer's Theorem does not extend to nonlinear hyperbolic systems of conservation laws.
 Elisa Davoli: Title
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 Giovanni Faonte: Title
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 Matteo Gallet: Planar linkages following a prescribed motion
Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. I will report about a work together with the colleagues of the Symbolic Computation group in Linz that provides a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. I will show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.
 Aleks Jevnikar: Liouvilletype problems on compact surfaces: a variational approach
A large number of important second order nonlinear elliptic equations involve exponential nonlinearities. We focus on the Liouville equation which is related to the prescribed Gaussian curvature problem and discuss its variational aspects. We next consider the Toda system which arises in mathematical physics in the ChernSimons theory and in geometry in the description of holomorphic curves and discuss how the variational arguments can be adapted to the system case.
 Andrea Mondino: Smooth and nonsmooth aspects of Ricci curvature lower bounds
After recalling the basic notions coming from differential geometry, the talk will be focused on spaces satisfying Ricci curvature lower bounds. The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80s and was pushed by Cheeger and Colding in the '90s who investigated the fine structure of possibly nonsmooth limit spaces. A completely new approach via optimal transportation was proposed by LottVillani and Sturm almost ten years ago. Via such an approach one can give a precise definition of what means for a nonsmooth space to have Ricci curvature bounded below. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seems to be new even for smooth Riemannian manifolds.
 Nicola Pagani: The Franchetta conjecture and some of its recent consequences
A general approach in mathematics to attack a complicated problem is by "approximation to the first order". One incarnation of this approach is to study the geometry of a smooth space X by studying its tangent bundle T_X, the space whose points are pairs (p,v) with p a point of X, and v a tangent vector of X at p. When X has dimension one, the tangent space at each point is a line, which makes the tangent bundle T_X a line bundle.
The Franchetta conjecture roughly says that, for smooth algebraic curves X, the only line bundles that can be chosen uniformly in X are the tangent bundle T_X and its tensor powers. The conjecture was stated by Franchetta as a theorem (with an incorrect proof) in the forties, and it was proven in the eighties by combined works of Harer, ArbarelloCornalba and Mestrano.
We will begin by formulating the precise statement of Franchetta's conjecture. This involves introducing the moduli space of (smooth, algebraic) curves M_g, the moduli space of line bundles on a curve and a combination of the two: the universal moduli space of line bundles on curves, called the universal Jacobian J_g. The Franchetta conjecture describes all algebraic (or holomorphic) sections of the natural map J_g > M_g that forgets the line bundle.
A similar and slightly more general version of the conjecture describes the sections of the natural forgetful map from the universal Jacobian over the moduli space of curves with marked points. We will conclude by discussing some of our recent results (joint with Jesse Kass), that describe how and when these sections uniquely extend to the natural compactifications of these moduli spaces.
 Marta Panizzut: Title
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 Fabio Tanturri: Degeneracy loci in algebraic geometry
In algebraic geometry, degeneracy loci of morphisms between vector bundles are ubiquitous: the simplest examples are complete intersection varieties, but other classical varieties such as Segre, Veronese varieties or rational normal scrolls can be interpreted as such. Starting from the easy example of determinantal varieties, I will introduce the notion of degeneracy loci and present some of the situations in which they appear, together with a few related questions and (partial) answers. Finally, I will introduce the notion of orbital degeneracy locus, which generalizes the classical definition and gives us a tool to construct several interesting varieties.
 Dragana Vuckovic: Title
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