Set-Codes with Small Intersections and Small Discrepancies

We are concerned with the problem of designing large families of subsets over a common labeled ground set that have small pairwise intersections and the property that the maximum discrepancy of the label values within each of the sets is less than or equal to one. Our results, based on transversal designs, factorizations of packings and Latin rectangles, show that by jointly constructing the sets and labeling scheme, one can achieve optimal family sizes for many parameter choices. Probabilistic arguments akin to those used for pseudorandom generators lead to significantly suboptimal results when compared to the proposed combinatorial methods. The design problem considered is motivated by applications in molecular data storage and theoretical computer science

Generalized packing designs

Generalized $t$-designs, which form a common generalization of objects such as $t$-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of $t$-designs, \emph{Discrete Math.}\ {\bf 309} (2009), 4835--4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with $t=2$ and block size $k=3$ or 4.Comment: 38 pages, 2 figures, 5 tables, 2 appendices. Presented at 23rd British Combinatorial Conference, July 201

Which groups are amenable to proving exponent two for matrix multiplication?

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $\omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $\omega$ and methods for ruling them out.Comment: 23 pages, 1 figur

Using Structural Bioinformatics to Model and Design Membrane Proteins

Cells require membrane proteins for a wide spectrum of critical functions. Transmembrane proteins enable cells to communicate with its environment, catalysis, ion transport and scaffolding. The functional roles of membrane proteins are specified by their sequence composition and precise three dimensional folding. The exact mechanisms driving folding of membrane proteins is still not fully understood. Further, the association between membrane proteins occurs with pinpoint specificity. For example, there exists common sequence features within families of transmembrane receptors, yet there is little cross talk between families. Therefore, we ask how membrane proteins dial in their specificity and what factors are responsible for adoption of native structure. Advancements in membrane protein structure determination methods has been followed by a sharp increase in three dimensional structures. Structural bioinfomatics has been utilized effectively to study water soluble proteins. The field is now entering an era where structural bioinformatics can be applied to modeling membrane proteins without structure and engineering novel membrane proteins. The transmembrane domains of membrane proteins were first categorized structurally. From this analysis, we are able to describe the ways in which membrane proteins fold and associate. We further derived sequence profiles for the commonly occurring structural motifs, enabling us to investigate the role of amino acids within the bilayer. Utilizing these tools, a transmembrane structural model was constructed of principle cell surface receptors (integrins). The structural model enabled understanding of possible mechanisms used to signal and to propose a novel membrane protein packing motif. In addition, novel scoring functions for membrane proteins were developed and applied to modeling membrane proteins. We derived the first all-atom membrane statistical potential and introduced the usage of exposed volume. These potentials allowed modeling of complex interactions in membrane proteins, such as salt bridges. To understand the geometric preferences of salt bridges, we surveyed a structural database. We learned about large biases in salt bridge orientations that will be useful in modeling and design. Lastly, we combine these structural bioinformatic efforts, enabling us to model membrane proteins in ways which were previously inaccessible