We will talk about relations between ergodic properties of group actions and the structure of group characters. The latter is equivalent to the classification of all finite-type factor representations. The outstanding conjecture (often attributed to Vershik) is that for a large class of groups their group characters must have the form \mu(Fix(g)), where $\mu$ is a G-invariant measure for some special group action on a measure space, Fix(g) is the set of all fixed points of group element $g$. We will then establish Vershik's conjecture for the family of Higman-Thompson groups. Since these groups have no non-trivial ergodic measures, we get that they have no non-trivial factor representations. Examples of other classes of groups known to satisfy Vershik's conjecture will be also discussed. The talk will be based on two recent preprints by Dudko and Medynets, "Finite factor representations of Higman-Thompson Groups" ArXiv 1212.1230 and "On characters of inductive limits of symmetric groups" Arxiv 1105.6325

The characterization of all K_{2,3}-minor-free graphs is well-known: 2-connected K_{2,3}-minor-free graphs are either K_4 or outerplanar. In this talk, we provide a characterization of all K_{2,4}-minor-free graphs. For the 3-connected graphs, we have an infinite family which yields 2n-8 graphs on n vertices along with some small special examples on at most eight vertices. The 2-connected graphs are then formed by joining the 3-connected ones with outerplanar graphs subject to some restrictions. This is joint work with Mark Ellingham, Kenta Ozeki and Shoichi Tsuchiya.

Secondary invariants are important in geometry and topology. While primary invariants only depend on the topology of the underlining manifolds, secondary invariants also depend on certain auxiliary geometric data (e.g. metrics or connections etc. ) of the underlining manifolds. Some of the well-known secondary invariants are Chern-Simon invariants, eta invariant and rho invariant, where the latter two were introduced by Atiyah, Patodi and Singer. In this talk, I will discuss some of my recent work and joint work with others on these secondary invariants (and their higher versions). In particular, I shall talk about the higher eta invariant and the higher rho invariant, and their connections to the Baum-Connes conjecture and positive scalar curvature problems.

For a computable algebraic field F, the splitting set S_F of F is the set of polynomials with coefficients in F which factor over F, and the root set R_F of F is the set of polynomials with coefficients in F which have a root in F. In the first part of this talk, on October 2, 2012, we showed that under the bounded Turing (bT) reducibility, determining whether a polynomial has a root in a field F is more difficult than determining whether it factors over F, i.e. S_F is always bT-reducible to R_F, but there are fields F where R_F is not bT-reducible to S_F. In the second part, we will define a Rabin embedding g of a field into its algebraic closure, and for a computable algebraic field F, we compare the relative complexities of R_F, S_F, and g(F) under m-reducibility and under bT-reducibility.

The Four Color Theorem is easy to state:  If you have a (nice enough) map of countries, and you want to color them so that no two adjacent countries are the same color, then you never need more than four colors to do it!  This theorem was proved in the 70's, but its proof is controversial among mathematicians due to its heavy use of a computer.  We will discuss the history of this theorem, and then prove the Five Color Theorem, which states you never need more than five colors to color our map, which is much easier than four!

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Tea at 3:30 pm in 1425 Stevenson Center

We will show that the reverse mathematical strength of the statement "every commutative ring with identity has a prime ideal" is equivalent to WKL (Weak K\"onig's Lemma) over RCA (Recursive Comprehension Axiom).

Ever wonder how you can safely send your credit card number over the internet? The answer is RSA, the first widely used public-key cryptographic communications system. Using only elementary techniques from
number theory, RSA allows you to send secure communications over public channels without a pre-arranged code. In this talk, we discuss the difference between public-key and private-key cryptography, and cover some
basic ideas from number theory. Then we will show how to use RSA to encode and decode messages, and explain why this process works and why it is so difficult to crack.

Frames can be seen as generalized bases, that is, over-complete collections, which are used for stable representations of signals as linear combinations of basic building atoms. It is very useful when we can use locally adapted atoms, which in addition behave as elements of local bases. We explore the possibility of using localized parts of frames and bases when building a customized frame. After a review on Gabor frames and Wilson bases, we consider the question of combining parts of these collections into a multi-frame set and look at its properties.

Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes. Unfortunately, this representation is not local, and it does not reflect the way that music is actually generated by instruments playing individual notes at different times. We will discuss time-frequency representations, which are a type of local Fourier representation of signals. While such representations have limitations when it comes to music, they are powerful mathematical tools that appear widely throughout mathematics (e.g., partial differential equations and pseudodifferential operators), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering). We ask one very basic question: are the notes in this representation linearly independent? This seemingly trivial question leads to surprising mathematical difficulties. This talk is intended to be introductory and accessible to beginning graduate students. Tea at 3:30 pm in SC 1425.

In this talk I will discuss how strong forms of non-amenability are reflected in the asymptotic behavior of a group's Bernoulli action. Central to the discussion will be the notion of shift-minimality: A countable group G is called shift-minimal if every non-trivial measure preserving action weakly contained in the Bernoulli shift of G is free. I will discuss the connection between shift-minimality and certain properties of the reduced C*-algebra of G, and indicate the proof that if G admits a free pmp action of cost >1 then there is a finite normal subgroup N of G such that G/N is shift-minimal.

A series of informal talks (January-February, 2013), following the book by Peter J. Olver.

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.

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.

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.

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.

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.

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

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.

A quadrangulation is a 2-cell embedded graph where every face is a quadrangle. An even triangulation is a 2-cell embedded graph where every face is a triangle and every vertex degree is even. A triangulation on the sphere is 3-chromatic if and only if it is an even triangulation. In this talk, we show that any quadrangulation can be extended to an even triangulation by adding diagonal edges to all quadrangle faces. We also determine the number of distinct even triangulations. This is joint work with Atsuhiro Nakamoto and Kenta Ozeki.

We show that the explicit ALE Ricci-flat Kahler metrics constructed by Eguchi-Hanson, Gibbons-Hawking, Hitchin and Kronheimer, and their free quotients are Tian-Yau metrics. The proof relies on a construction of appropriate compactifications of Q-Gorenstein smoothings of quotient singularities as log del Pezzo surfaces. Time permitting, a geometric description of the compactifications will be provided. This is a joint work with I. Suvaina.

The main aim of talk is twofold. Firstly, to present an elementary method based on Farkas' lemma for rationals on how to embed any finite partial subalgebra of a linearly ordered MV-algebra into $\mathbb Q\cap [0,1]$ and then to establish a new elementary proof of the completeness of the Lukasiewicz axioms. Secondly, to present a direct proof of Di Nola's Representation Theorem for MV-algebras and to extend his results to the restriction of the standard MV-algebra on rational numbers.

nos-o-co-mi-al  (adjective)  - originating or occurring in a hospital
“get down R0, know your place, do not torment the human race”
primum non nocere – first, do no harm
Nosocomial epidemics are an increasing threat to society. The microbes that cause these epidemics are increasingly resistant to antibiotics. In the US they cause more than 100,000 deaths each year and that number is increasing. R0 (pronounced R-naught) is a quantity derived from mathematical models that predict the course of an epidemic. It is obtained from various parameters that determine the transmission dynamics of the infection. If R0 < 1, then the epidemic will abate. If R0 > 1, then the epidemic will worsen. The Hippocratic oath is the promise of physicians to not make the condition of a patient worse. It is the fundamental precept of medicine. I will tell you how all these are connected. I will also tell you how to avoid being infected by a nosocomial infection. Nosocomial infections occur in specific locations in the US, and only in these locations. You will never suffer a nosocomial infection if you do not go to one of these locations. I will tell you where these locations are. 

A character on a group is a positive definite function which takes the identity to 1 and is constant on conjugacy classes. Characters on a finite group gives an essential tool for understanding the representation theory of the group and motivated by this Thoma in 1964 initiated the study of characters on infinite groups. In 1966 Kirillov classified all characters on GL_n(k), and SL_n(k) for k a field and n \geq 2, excluding the case of SL_2(k). A number of other classification results have since been obtained for other groups by Ovcinnikov, Vershik, Kerov, and more recently by Bekka, Dudko, and Medynets, however the classification for SL_2(k) has not been completed. In my talk I will present the classification for SL_2(k) and SL_2 of some other rings and give some applications of these results. This is based on joint work with Andreas Thom.

A metric space is multiended if it admits a bounded subset whose complement has at least two unbounded connected components. For instance, the line is multiended but higher-dimensional Euclidean spaces are not. In the late sixties, Stallings has given a remarkable characterization of those finitely generated groups whose Cayley graph is multiended; the only such torsion-free groups are free products of two nontrivial groups! A key part of the proof is the construction of a action on a tree; in the seventies, the study of general group actions on trees was achieved by Bass and Serre. In the meantime, the study of multiended Schreier graphs was started by Abels and Houghton, and a remarkable connection with nonpositively curved cube complexes was discovered by Sageev twenty years later. While outstanding applications of cube complexes have been made since then, I will try to focus on the question of understanding which finitely generated groups admit a multiended Schreier graph. Tea at 3:30 pm in SC 1425.