In past work, we define a notion of l^{p}-Dimension for uniformly bounded Banach space representations of a sofic group. This dimension is equal to the von Neuamn dimensnion, when H is a unitary representation of G contained in a multiple of the left-regular representation. We also computed this dimension for central natural representations of a sofic group, including direct sums of the translation action on l^{p}(G), and the multiplication action on L^{p}(L(G)). In this work, we shall explain how to define this notion of l^{p}-Dimension for representations of a sofic equivalence relation. When this equivalence relation satisfies a certain "finite presentation" assumption, we define an analogue of l^{2}-Betti numbers (or really l^{2}-Betti number +1) in the l^{p}-case. We can then connect some natural questions about this dimension with the cost versus l^{2}-Betti number conjecture.

Abstract is available at: http://sitemason.vanderbilt.edu/files/eVpAGc/TangVanderbilt.pdf

We discuss a directionally sensitive time-frequency decomposition and representation of functions. The coefficients of this representation allow one to measure the "amount'' of frequency the function (signal, image) contains in a certain time interval, and also in a certain direction. This has been previously achieved using a version of wavelets called ridge lets, but in this work we discuss an approach based on time-frequency or Gabor elements. Applications to image processing are discussed. Tea at 3:30 pm in SC 1425.

The human species has a unique age structure with extended juvenile and senescent phases.  What determines the age structure of humans and what could extend the human life span?  Some researchers believe we should focus on curing disease and replacing damaged body parts via stem cell therapies.  Others believe we must slow the aging process at the cellular and molecular levels.  All proposed longevity strategies, however, remain unproven.  The age structure of human population is shaped under common biological challenges, including environmental conditions, exposure to infectious diseases, distribution of resources to maintain the viability of the reproductively active populations, and replacement of reproductive populations by their offspring.  In this talk, we will explore the age structure of human populations over evolutionary time.

A porim is called divisible if it is naturally ordered, in the sense that a b i a = x b = b y for some x; y. Divisible porims are also known as pseudo-hoops, and divisible integral residuated lattices as integral GBL-algebras. We focus on divisibility in the setting of pseudo-BCK-algebras (or biresiduation algebras) that are the residuation subreducts of porims. We attempt to generalize some structural results proved by Blok and Ferreirim for hoops, and by Jipsen and Montagna for integral GBL-algebras. Since a porim is divisible i it satises the identities (xny)n(xnz) (ynx)n(ynz) and (z=x)=(y=x) (z=y)=(x=y), it seems natural to call pseudo-BCK-algebras satisfying these two identities divisible. Further, by a normal pseudo-BCK-algebra we mean a pseudo-BCK-algebra A = hA; n; =; 1i in which the 1-classes of the relative congruences can be characterized as the subsets K A satisfying: (i) 1 2 K, (ii) for all a; b 2 A, if a; anb 2 K, then b 2 K, and (iii) for all a; b 2 A, anb 2 K i b=a 2 K. The main result for normal divisible pseudo-BCK-algebras is the following: If A is a non-trivial subdirectly irreducible normal divisible pseudo-BCK-algebra, then A is the ordinal sum BC, where C is a non-trivial subdirectly irreducible linearly ordered cone algebra in the sense of Bosbach. We also generalize the concept of n-potent porims and prove that every n-potent divisible pseudo-BCK-algebra is a BCK-algebra.

There is a strong relation between the existence of non-singular solutions for the normalized Ricci flow and the underlying smooth structure of a 4-manifold. We are going to discuss an obstruction to the existence of non-singular solutions and its applications. The main examples are connected sums of complex projective planes and complex projective planes with reversed orientation. The key ingredients in our methods are the Seiberg-Witten Theory and symplectic topology. This is joint work with M. Ishida and R. Rasdeaconu.

An even embedding of a graph on a closed surface is a fixed 2-cell embedding such that each face is bounded by a closed walk of even length. It is known that a complete graph on n (> 6) vertices has an even embedding on a closed surface F^2 with Euler characteristic chi <=n(5-n)/4.  Such embeddings are called minimum genus even embeddings.  It is known that there is an invariant of even embeddings of graphs, which is called "cycle parity''.  It divides non-bipartite even embeddings of graphs into three classes on a fixed surface.  In this talk, we introduce cycle parity and consider relationships between cycle parity and minimum genus even embeddings of the complete graphs.