On Mumford's construction of degenerating abelian varieties

We prove that a 1-dimnl family of abelian varieties with an ample sheaf defining principal polarization can be canonically compactified (after a finite base change) to a projective family with an ample sheaf. We show that the central fiber (P,L), which we call an SQAV, has a canonical Cartier theta divisor. We give a combinatorial definition for SQAVs and describe their geometrical properties, in particular compute cohomologies of L^n, n\ge0.Comment: Final version, to appear in Tohoku Math.

Knotoids

L'obiettivo di questa tesi è di studiare la teoria dei nodoidi. In particolare, troveremo alcuni interessanti risultati che collegano questa teoria con quella dei nodi e useremo questi risultati ottenuti per sviluppare alcune tecniche volte allo studio delle proteine

Prospects for Declarative Mathematical Modeling of Complex Biological Systems

Declarative modeling uses symbolic expressions to represent models. With such expressions one can formalize high-level mathematical computations on models that would be difficult or impossible to perform directly on a lower-level simulation program, in a general-purpose programming language. Examples of such computations on models include model analysis, relatively general-purpose model-reduction maps, and the initial phases of model implementation, all of which should preserve or approximate the mathematical semantics of a complex biological model. The potential advantages are particularly relevant in the case of developmental modeling, wherein complex spatial structures exhibit dynamics at molecular, cellular, and organogenic levels to relate genotype to multicellular phenotype. Multiscale modeling can benefit from both the expressive power of declarative modeling languages and the application of model reduction methods to link models across scale. Based on previous work, here we define declarative modeling of complex biological systems by defining the operator algebra semantics of an increasingly powerful series of declarative modeling languages including reaction-like dynamics of parameterized and extended objects; we define semantics-preserving implementation and semantics-approximating model reduction transformations; and we outline a "meta-hierarchy" for organizing declarative models and the mathematical methods that can fruitfully manipulate them

Doctor of Philosophy

dissertationWith the tremendous growth of data produced in the recent years, it is impossible to identify patterns or test hypotheses without reducing data size. Data mining is an area of science that extracts useful information from the data by discovering patterns and structures present in the data. In this dissertation, we will largely focus on clustering which is often the first step in any exploratory data mining task, where items that are similar to each other are grouped together, making downstream data analysis robust. Different clustering techniques have different strengths, and the resulting groupings provide different perspectives on the data. Due to the unsupervised nature i.e., the lack of domain experts who can label the data, validation of results is very difficult. While there are measures that compute "goodness" scores for clustering solutions as a whole, there are few methods that validate the assignment of individual data items to their clusters. To address these challenges we focus on developing a framework that can generate, compare, combine, and evaluate different solutions to make more robust and significant statements about the data. In the first part of this dissertation, we present fast and efficient techniques to generate and combine different clustering solutions. We build on some recent ideas on efficient representations of clusters of partitions to develop a well founded metric that is spatially aware to compare clusterings. With the ability to compare clusterings, we describe a heuristic to combine different solutions to produce a single high quality clustering. We also introduce a Markov chain Monte Carlo approach to sample different clusterings from the entire landscape to provide the users with a variety of choices. In the second part of this dissertation, we build certificates for individual data items and study their influence on effective data reduction. We present a geometric approach by defining regions of influence for data items and clusters and use this to develop adaptive sampling techniques to speedup machine learning algorithms. This dissertation is therefore a systematic approach to study the landscape of clusterings in an attempt to provide a better understanding of the data

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications

International audienceHybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows