Dalhousie University Mathematics Colloquium, 2009/10Mathematics Colloquiums are on Mondays, 3:30pm in room 319 in the Chase Building. There is an alternate time on Thursdays, 2:30pm.
AbstractsToby Kenney (Univerzita Mateja Bela): "Graphical composition of equivalence relations, and its relevance to congruence lattices"
In 1970, H. Werner showed that the lattices of equivalence relations can be described as the lattices closed under arbitrary joins, and under a new operation that he called graphical composition of equivalence relations. In this talk, I will explain this graphical composition of equivalence relations, and present some of its properties as an additional operation on an abstract lattice. I will also present some sufficient conditions for an abstract operation to be representable as graphical composition of equivalence relations. I hope in future to obtain a complete classification of which operations on an abstract lattice can be represented as graphical compositions of equivalence relations. This should be useful for improving our understanding of congruence lattices, which are until now poorly understood.
I will discuss some graph theoretic aspects of calculating the residues of Feynman integrals in simple cases, and consider the interesting transcendental numbers which result.
Groupoidification is a program, proposed by Baez and Dolan, for recovering structures in linear algebra as "shadows" of structures in setting involving groupoids (which can be understood as sets with local symmetry operations). 2-linearization is a way of capturing even more structure from this category, using "2-linear algebra". In this talk, I will motivate both of these processes first by describing a simple construction for pushing and pulling functions on sets through set maps, and then show how "degroupoidification" and "2-linearization" generalize this construction from sets to groupoids.
Pattern formation problems arise in many physical and biological systems as orderly outcomes of self-organization principles. Examples include animal coats, skin pigmentation, and morphological phases in a block copolymer. Recent advances in singular perturbation theory and asymptotic analysis have made it possible to study these problems rigorously. In this talk I will discuss a central theme in the construction of various patterns as solutions to some well known PDE and geometric problems: how a single piece of structure built on the entire space can be used as an Ansatz to produce a near periodic pattern on a bounded domain. We start with the simple disc and show how the spot pattern in morphogenesis and the cylindrical phase in diblock copolymers can be mathematically explained. More complex are the ring structure and the oval structure which can also be used to construct solutions on bounded domains. Finally we will discuss the newly discovered smoke-ring structure and the toroidal tube structure in space. The results presented in this lecture come from my joint works with Kang, Kolokolnikov, and Wei.
Over the past several years, a significant shift in mathematical signal processing has created a great deal of excitement. In the Eighties and Nineties, much effort was given to developing representation systems, such as wavelets, in which a class of signals could be sparsely represented. In this decade, compressed sensing has studied how one might efficiently acquire a signal which already has a sparse representation. It is now well established that if a signal is sparse (most of its entries are essentially zero), one may acquire all the information in the signal from fantastically fewer measurements than previously thought necessary. In this talk, I will give an overview of compressed sensing, focusing on the va- riety of mathematics which has been brought together to establish these important results. Specifically, I will outline ideas from the theory of polytopes, random matrix theory, and the theory of large deviations as applied to compressed sensing. I will very briefly mention ideas from convex optimization, geometric functional analysis, Fourier analysis, and number theory which have also established important results in compressed sensing. In the end, phase transitions will be used to compare these results.
Let be a bounded, pseudoconvex domain of finite type with smooth boundary. We assume further that the Levi form of is diagonalizable. In this talk, we will discuss background and recent progress of the -Neumann problem on . Then we discuss the ``possible'' optimal estimates of the solution. Using the result of this problem, we obtain solving operator for inhomogeneous Cauchy-Riemann equation in . Here is a given (0,1)-form.
The FC-center of a group G is the characteristic subgroup F of all elements whose conjugacy class is finite. If G=F, then G is called an FC-group. We show that a compact group G is an FC-group if and only if its center Z(G) is open (that is, G is center by finite) if and only if its commutator subgroup is finite (that is, G is finite by commutative). Now let G be a compact group and let p denote the Haar measure of the set of all pairs (x,y) in GxG for which [x,y]=1; this is the probability that two randomly picked elements commute. We prove that p>0 if and only if the FC-center F of G is open and so has finite index. If these conditions are satisfied, then Z(F) is a characteristic normal abelian open subgroup of G and G is abelian by finite. This is joint work with Francesco G. Russo.
Although widely used in practice, the performance of the popular network clustering technique called "modularity maximization" is not well understood when applied to networks with unknown modular structure. In this talk, I'll show that precisely in the case we want it to perform the best---that is, on modular networks---the modularity function Q exhibits extreme degeneracies, in which the global maximum is hidden among an exponential number of high-modularity solutions. Further, these degenerate solutions can be structurally very dissimilar, suggesting that any particular high- modularity partition, or statistical summary of its structure, should not be taken as representative of the other degenerate solutions. These results partly explain why so many heuristics do well at finding high-modularity partitions and why different heuristics can disagree on the modular composition the same network. I'll conclude with some forward-looking thoughts about the general problem of identifying network modules from connectivity data alone, and the likelihood of circumventing this degeneracy problem.
Paul Adrian Dirac invented a delightful topological puzzle involving a pair of scissors and two loops of string. Take a pair of scissors and two pieces of string, each about 10 feet long. Pass each string through the handle of the scissors, and then tie the ends to make a loop. Stand on the loops so that when you raise the scissors to the level of your face the loops will produce four untwisted strands of cord running from scissors to the floor. Hold the scissors vertically, pointing towards the ceiling, and then give scissors a full turn of 360 degrees (in either direction) around a vertical axis. This will, of course, twist the cords. It is not possible to untwist the cords without rotating the scissors in any way. You will be able to alter the way the string is tangled, but no amount of manipulation of the strings can bring the structure back to its original state. After convincing yourself that the task is impossible, go back to the original untangled position. Now give the scissors two full turns (720 degrees) in either direction. Believe it or not, it is now possible to return the scissors and the strings to their original state without rotating the scissors in any further way! The talk will focus on a mathematical explanation of this phenomenon involving advanced mathematics - Homotopy theory. No knowledge of this theory will be assumed. All necessary definitions and details will be given.
Are there spaces X, such that every fibration with base X or with fibre X is a trivial fibration? This curios question will be used as an excuse to explain some classical results on the classification of fibrations. The answer will then be given by some explicit computations of classifying spaces.
This talk will take an historical/mathematical perspective to the Riemann zeta function zeta(s) and the famed Riemann Hypothesis (RH), generally considered the most important and probably most difficult unsolved question in mathematics. Riemann wrote a single paper of just 8 pages on the subject, in 1859. The RH is stated there but it is not at all clear from this paper that
(a) Riemann thought it was important,
Some sixty odd years after his death, answers to these questions became clearer, thanks to an exhaustive two year study of Riemann's unpublished notes, the "Nachlass", by Karl Ludwig Siegel. In this talk we shall examine the methods that Riemann had likely used to study the zeros of zeta(s), compute a few zeros ourselves by these very same methods, and if time permits mention briefly some of the major developments since the time of Siegel's study including some work of the author.
For updates and corrections, contact Peter Selinger.