The generalized Delta conjecture at t=0

We prove the cases q=0 and t=0 of the generalized Delta conjecture of Haglund, Remmel and Wilson involving the symmetric function $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$. Our theorem generalizes recent results by Garsia, Haglund, Remmel and Yoo. This proves also the case q=0 of our recent generalized Delta square conjecture.Comment: 21 pages, 3 figure

Area-width scaling in generalised Motzkin paths

13 pages, 3 figures19 pages, 3 figure

The Delta square conjecture

We conjecture a formula for the symmetric function $\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington (Loehr, Warrington 2007), recently proved by Sergel (Sergel 2017) after the breakthrough of Carlsson and Mellit (Carlsson, Mellit 2018). Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson (Haglund, Remmel, Wilson 2015), answering one of their questions. We support our conjecture by proving the specialization $m=q=0$, reducing it to the same case of the Delta conjecture, and the Schr\"{o}der case, i.e. the case $\langle \cdot ,e_{n-d}h_d\rangle$. The latter provides a broad generalization of the $q,t$-square theorem of Can and Loehr (Can, Loehr 2006). We give also a combinatorial involution, which allows to establish a linear relation among our conjectures (as well as the generalized Delta conjectures) with fixed $m$ and $n$. Finally, in the appendix, we give a new proof of the Delta conjecture at $q=0$.Comment: 27 pages, 6 figures. arXiv admin note: text overlap with arXiv:1807.0541

The Schr\"oder case of the generalized Delta conjecture

We prove the Schr\"oder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schr\"oder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schr\"oder of Haglund (Haglund 2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in (D'Adderio, Iraci 2017), which in turn extended results in (Aval et al. 2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations.Comment: 22 pages, 12 figure

3D mesh metamorphosis from spherical parameterization for conceptual design

Engineering product design is an information intensive decision-making process that consists of several phases including design specification definition, design concepts generation, detailed design and analysis, and manufacturing. Usually, generating geometry models for visualization is a big challenge for early stage conceptual design. Complexity of existing computer aided design packages constrains participation of people with various backgrounds in the design process. In addition, many design processes do not take advantage of the rich amount of legacy information available for new concepts creation. The research presented here explores the use of advanced graphical techniques to quickly and efficiently merge legacy information with new design concepts to rapidly create new conceptual product designs. 3D mesh metamorphosis framework 3DMeshMorpher was created to construct new models by navigating in a shape-space of registered design models. The framework is composed of: i) a fast spherical parameterization method to map a geometric model (genus-0) onto a unit sphere; ii) a geometric feature identification and picking technique based on 3D skeleton extraction; and iii) a LOD controllable 3D remeshing scheme with spherical mesh subdivision based on the developedspherical parameterization. This efficient software framework enables designers to create numerous geometric concepts in real time with a simple graphical user interface. The spherical parameterization method is focused on closed genus-zero meshes. It is based upon barycentric coordinates with convex boundary. Unlike most existing similar approaches which deal with each vertex in the mesh equally, the method developed in this research focuses primarily on resolving overlapping areas, which helps speed the parameterization process. The algorithm starts by normalizing the source mesh onto a unit sphere and followed by some initial relaxation via Gauss-Seidel iterations. Due to its emphasis on solving only challenging overlapping regions, this parameterization process is much faster than existing spherical mapping methods. To ensure the correspondence of features from different models, we introduce a skeleton based feature identification and picking method for features alignment. Unlike traditional methods that align single point for each feature, this method can provide alignments for complete feature areas. This could help users to create more reasonable intermediate morphing results with preserved topological features. This skeleton featuring framework could potentially be extended to automatic features alignment for geometries with similar topologies. The skeleton extracted could also be applied for other applications such as skeleton-based animations. The 3D remeshing algorithm with spherical mesh subdivision is developed to generate a common connectivity for different mesh models. This method is derived from the concept of spherical mesh subdivision. The local recursive subdivision can be set to match the desired LOD (level of details) for source spherical mesh. Such LOD is controllable and this allows various outputs with different resolutions. Such recursive subdivision then follows by a triangular correction process which ensures valid triangulations for the remeshing. And the final mesh merging and reconstruction process produces the remeshing model with desired LOD specified from user. Usually the final merged model contains all the geometric details from each model with reasonable amount of vertices, unlike other existing methods that result in big amount of vertices in the merged model. Such multi-resolution outputs with controllable LOD could also be applied in various other computer graphics applications such as computer games

Asymptotics and scaling analysis of 2-dimensional lattice models of vesicles and polymers

PhDThe subject of this thesis is the asymptotic behaviour of generating functions of different combinatorial models of two-dimensional lattice walks and polygons, enumerated with respect to different parameters, such as perimeter, number of steps and area. These models occur in various applications in physics, computer science and biology. In particular, they can be seen as simple models of biological vesicles or polymers. Of particular interest is the singular behaviour of the generating functions around special, so-called multicritical points in their parameter space, which correspond physically to phase transitions. The singular behaviour around the multicritical point is described by a scaling function, alongside a small set of critical exponents. Apart from some non-rigorous heuristics, our asymptotic analysis mainly consists in applying the method of steepest descents to a suitable integral expression for the exact solution for the generating function of a given model. The similar mathematical structure of the exact solutions of the different models allows for a unified treatment. In the saddle point analysis, the multicritical points correspond to points in the parameter space at which several saddle points of the integral kernels coalesce. Generically, two saddle points coalesce, in which case the scaling function is expressible in terms of the Airy function. As we will see, this is the case for Dyck and Schröder paths, directed column-convex polygons and partially directed self-avoiding walks. The result for Dyck paths also allows for the scaling analysis of Bernoulli meanders (also known as ballot paths). We then construct the model of deformed Dyck paths, where three saddle points coalesce in the corresponding integral kernel, thereby leading to an asymptotic expression in terms of a bivariate, generalised Airy integral.Universität Erlangen-Nürnberg Queen Mary Postgraduate Research Fun