Generating series of a new class of orthogonal Shimura varieties

For a new class of Shimura varieties of orthogonal type over a totally real number field, we construct special cycles and show the the modularity of Kudla's generating series in the cohomology group.Comment: Improved exposition, same resul

Constructing families of moderate-rank elliptic curves over number fields

We generalize a construction of families of moderate rank elliptic curves over $\mathbb{Q}$ to number fields $K/\mathbb{Q}$. The construction, originally due to Steven J. Miller, \'Alvaro Lozano-Robledo and Scott Arms, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius, the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.Comment: Version 1.0, 4 pages, sequel to arXiv:math/040657

Special Cycles on GSpin Shimura Varieties

In this thesis we prove that a certain generating function of special cycles on GSpin Shimura varieties is modular. More specifically, we consider the Shimura variety corresponding to the reductive group \Res_{F/\Q} G, where G=\GSpin(V) the GSpin group for $V$, a quadratic space over a totally real number field $F$, [F:\QQ]=d with certain conditions at the infinite places. We construct a generating function in the sense of Kudla and Millson and show that its image in cohomology is an automorphic form

Special Cycles on GSpin Shimura Varieties

In this thesis we prove that a certain generating function of special cycles on GSpin Shimura varieties is modular. More specifically, we consider the Shimura variety corresponding to the reductive group \Res_{F/\Q} G, where G=\GSpin(V) the GSpin group for $V$, a quadratic space over a totally real number field $F$, [F:\QQ]=d with certain conditions at the infinite places. We construct a generating function in the sense of Kudla and Millson and show that its image in cohomology is an automorphic form

Analyzing self-similar and fractal properties of the C. elegans neural network.

The brain is one of the most studied and highly complex systems in the biological world. While much research has concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural connectivity of the brain should elucidate some of its functional properties. In this paper we analyze the connectome of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian Matrix of the 279-neuron "giant component" of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot visualizations of the neural network in eigenfunction coordinates. Small-world properties of the system are examined, including average path length and clustering coefficient. We test for localization of eigenfunctions, using graph energy and spacial variance on these functions. To better understand results, all calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been "rewired" to have small-world properties. We propose algorithms for generating Laplacian matrices of each of these graphs

Weyl Ratios

<p>(a) <i>C. elegans</i> neural network (b) Sierpinski Gasket, Level 5 (c) Random Tree (d) Hexacarpet Level 3 (e) Random Network (f) Sierpinski Gasket Rewiring .</p

Normalizing the Laplacian

<p><i>C. elegans</i> neural network: (a) Un-normalized Laplacian, (b) Normalized Laplacian .</p

Weyl Ratios

<p>(a) <i>C. elegans</i> neural network (b) Sierpinski Gasket, Level 5 (c) Random Tree (d) Hexacarpet Level 3 (e) Random Network (f) Sierpinski Gasket Rewiring .</p