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| Monday 3 |
3:10-4 pm, room 1432. Graph Theory and Combinatorics Seminar. Petar Markovic, University of Novi Sad, Serbia. Characterizations of bounded width and few generators
Constraint Satisfaction Problem. In this talk I will show exactly in which cases can each of the two
known algorithms for proving tractability of CSP be applied. The case of
few generators is an algebraically-motivated algorithm which was
characterized in 2006. The bounded width (i.e. bounded treewidth) is a
logic/graph-theoretic motivated algorithm which was characterized in a
groundbreaking new result by L. Barto and M. Kozik last week. I will
concentrate in this lecture on the second algorithm, and show some of the
techniques they used for characterization, including the result on graphs
with no sources and no sinks which I mentioned towards the end of the
previous lecture. It would be helpful if the audience was present at my prevous talk, or
at least at some CSP lecture, but not essential, as I will repeat (most
of) the definitions.
4:10-5 pm, room 1432. NCGOA Research Training Group Seminar. Qayum Khan, Vanderbilt University. The fundamental theorem of Bass--Heller--Swan. We shall study the proof of the fundamental theorem of H. Bass, A. Heller, and R. Swan (1964) in the classical algebraic K-theory of rings. The punchline is that the functor K_0 can be recovered from the functor K_1. Specifically, we prove a naturally split, short exact sequence involving a Laurent extension. This result led Bass to the definition of the lower K-functors and led Quillen to the justification of the higher K-functors. The other summands in the decomposition are Nil-groups, which vanish for regular rings. |
| Tuesday 4 |
3:20 pm, room 1425. Graduate Student Tea.
4:10-5 pm, room 1432. Noncommutative Geometry Seminar. Dan Ramras, Vanderbilt University. An introduction to Quillen's algebraic K-theory II. After reviewing the Q-construction, I'll explain the proof that it yields the correct notion of K_0. We'll then discuss the relationship between the Q-construction and Quillen's second approach to K-theory, the plus construction. Time permitting, I'll say a little bit about how negative K-groups can be brought into the picture. This relies on the theory of bounded modules, developed by Pederson and Weibel. 4:10-5:00 pm, room 1312. Computational Analysis Seminar. Brigitte Forster, Technische Universität München. Shift-invariant spaces from rotation-covariant functions. We consider shift-invariant multiresolution spaces generated by rotation-covariant functions $\rho$ in $\mathbb{R}^2$. To construct corresponding scaling and wavelet functions, $\rho$ has to be localized with an appropriate multiplier, such that the localized version is an element of $L^2(\mathbb{R}^2)$. We consider several classes of multipliers and show a new method to improve regularity and decay properties of the corresponding scaling functions and wavelets. The wavelets are complex-valued functions, which are approximately rotation-covariant and therefore behave as Wirtinger differential operators. Moreover, our class of multipliers gives a novel approach for the construction of polyharmonic B-splines with better polynomial reconstruction properties. The method works not only on classical lattices, such as the cartesian or the quincunx, but also on hexagonal lattices. 4:30-5:30 pm, room 1308. Universal Algebra and Logic Seminar. Ralph McKenzie, Vanderbilt University. The complexity of constraint satisfaction problems: classifying the complexity of CSP problems using finite algebras. Many natural combinatorial problems can be expressed as constraint sat- isfaction problems. This class of problems is known to be NP-complete in general (it includes graph three-coloring), but certain restrictions on the form of constraints can ensure tractability. Feder and Vardi (1998) conjectured that every “specific” instance of the CSP (such as graph three-coloring) is either tractable or NP-complete. Bulyatov and Jeavons (2002) showed that to every specific instance of CSP is correlated a finite algebra and that rela- tive complexity of two CSP problems is directly tied to structural relations between the correlated algebras. They proposed general algebraic criteria for an instance of CSP to be tractable, or NP-complete. Thus they created an algebraic version of the dichotomy conjecture of Feder and Vardi, which is currently the focus of a lot of research. Remarkable progress has been made on the dichotomy conjecture during the past two years. Using a combination of universal-algebraic and graph- theoretic ideas, a team of young central european mathematicians has shown that for a finite algebra A for which no non-trivial Abelian congruences occur in the variety it generates, all CSP problems involving admissible relations over A are tractable. My talks will discuss in detail an easier proof of a result the team proved two months ago, on the way to the mentioned result: if a relational structure R has J ´onsson terms among its polymorphisms, then the (2p, 3p)-consistency algorithm solves CSP(R) in polynomial time. 7-8 pm, room 1206. Undergraduate Seminar in Mathematics. Alex Wires, Vanderbilt University. When is the Whole Equal to the Sum of Its Parts? The number theoretic work of Euclid's Elements culminates with the perfect numbers. But the ancient Greek geometers could only describe perfect numbers which were even. Are there infinitely many even perfect numbers? Are there any odd perfect numbers? In asking these two questions Edmund Landau wrote, "Modern mathematics has solved many (apparently) difficult problems, even in number theory; but we stand powerless in the face of such (apparently) simple problems as these." Drop in to learn about the oldest unsolved math problem. There will also be the inaugural announcement of the Odd Perfect Cash Prize. |
| Wednesday 5 | 4:10 pm, room 1310. Topology & Group Theory Seminar. Claus Ernst, Western Kentucky University. On the rope-length of a knot. One can ask the question how long does a rope need to be in order to tie a given knot. In order to make this length independent of the thickness of the rope we express the length in units of the radius of the rope. In this talk we explain several bounds on the rope-length in terms of the crossing number of the knot. Research in this area of physical knots has been motivated by large knotted molecules such as DNA or polymer knots. Such examples will be given as well. |
| Thursday 6 | 4:10-5 pm, room 5211. Colloquium. Brigitte Forster-Heinlein, Centre for Mathematical Sciences, Technische Universität München. Complex B-splines, Dirichlet means, and divided differences for multi-dimensions. Complex B-splines are a natural extension of classical B-splines. We show their relations to fractional divided differences and fractional difference operators, and present a generalized Hermite-Genocchi formula. This formula then allows the definition of a larger class of complex B-splines. The notion of complex B-spline is extended to a multivariate setting by means of ridge functions employing the known geometric relationship between ordinary B-splines and multivariate B-splines. To derive properties of complex B-splines in R^n, 1 < n, the Dirichlet average has to be generalized to include infinite dimensional simplices. Based on this generalization several identities of multivariate complex B-splines are presented. This is joint work with Peter Massopust. |
| Friday 7 | 4:10 pm, room 1307. Partial Differential Equations Seminar. Michael O'Leary, Towson University. A diffusion model in population genetics with mutation and dynamic fitness. We analyze a degenerate diffusion equation with singular boundary data, modeling the evolution of a polygenic trait under selection, drift and mutation. The equation models the contributions of a large but finite number of loci (genes) to the trait and at the same time allows the population trait mean to vary in a way that affects the strength of selection at individual loci; in this respect it differs from other population-genetic models that have been rigorously analyzed. We present existence, uniqueness and stability results for solutions of the system provided the mutation rate is sufficiently small. We also analyze the long term limit of the genetic variance in the system. |