Dalhousie University Mathematics Colloquium, 2011/12

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Abstracts

Abstract: A Coxeter group is an abstraction of a group generated by reflections. Coxeter groups are important objects of study in a number of areas of Algebra, Combinatorics, Geometry and Topology, and also have applications in other areas of Mathematics.

An important tool for studying Coxeter groups is a partial order associated to each Coxeter group, called the Bruhat order, because of its connection to the Bruhat decomposition of certain kinds of algebraic groups.

The Bruhat order has a number of remarkable properties, one of which is the fact that the (elementwise) product of two principal downsets in the Bruhat order is another principal downset. In this talk, I will show that this property, in combination with a few other simple properties of the Bruhat order, serves to characterise Coxeter groups. That is:

Theorem: If G is a group, and we have a partial order on G such that:

1. The identity is the smallest element in the partial order
2. Inverse preserves the order.
3. Elements which cover the identity in the partial order generate G as a group.
4. The product of two principal downsets is another principal downset.

Persi Diacconis (Stanford): "The Search for Randomness"

DVD of the 6th annual Chelluri Lecture, Cornell University.

Abstract: I will look at some of our most primitive examples of random phenomena: tossing a coin, rolling a roulette ball, and shuffling cards. In all cases, physics, math, and experiments show that, while randomness is possible, usually we are lazy and things are not so random after all. Connections to statistical modeling and the link between mathematics and the real world are drawn.

Format: We will show the DVD of the 6th annual Chelluri Lecture, given in April 2011 at Cornell University in memory of Sastri's son Thyagaraju (Raju) Chelluri. The Chelluri Lecture series is offered in memory of Thyagaraju (Raju) Chelluri, a brilliant student, gifted scholar, and wonderful human being who graduated magna cum laude in mathematics from Cornell in 1999 and was awarded a Ph.D. posthumously from Rutgers University in 2004.

Scott Rodney (Cape Breton University): "Degenerate Sobolev Spaces and Existence of Weak Solutions to a Family of Degenerate Elliptic Dirichlet Problems"

Abstract: In recent years, the Monge-Ampère equation has been linked to families of quasilinear degenerate elliptic PDEs of second order. As a result, a greater understanding of the Monge-Ampère equation can be gained through a regularity theory for degenerate elliptic equations. This talk will introduce the notion of degenerate Sobolev spaces attached to such a regularity theory. It will be demonstrated that a Sobolev inequality adapted to a degenerate quadratic form leads to existence results for Dirichlet problems attached to second order linear PDEs in divergence form.

Francis Valiquette (Dalhousie University): "Inductive moving frames"

Abstract: The theory of moving frames is a powerful tool for studying geometric properties of submanifolds under a Lie group action G. At the core of the method, one of the main steps consists of constructing the invariants of the group action. In the first part of the talk, I will explain how these invariants can be obtained using the theory of equivariant moving frames recently developed by Fels and Olver.

In the second part of the talk I will assume that the Lie group G is a subgroup of some larger group H and show how to streamline the construct a moving frame for H using the moving frame of G. As a byproduct, we will be able to effortlessly express the invariants of H in terms of the invariants of G.

Theodore Kolokolnikov (Dalhousie University): "Particle interaction models of biological aggregation"

Abstract: Animal groups often form striking aggregation patterns. Examples include from schools of fish to locust swarms, to patterns in bacterial cultures. In this talk, we discuss a very simple model of swarming based on pairwise particle interactions with short-range attraction and long-range repulsion, which can lead to very complex and intriguing patterns in two or three dimensions. Depending on the relative strengths of attraction and repulsion, a multitude of various patterns are observed, from nearly-constant density swarms to annular solutions, to complex N-fold symmetry patterns.

We show that annular-type patterns may form if short-range repulsion is sufficiently weak. A Turing-type analysis of such annular states further reveals a wealth of possible instabilities which often lead to complicated and beautiful patterns. We also classify two-ring, annular and triangular patterns which arise when the ring becomes unstable. In three dimensions, we classify the stability of a sphere using spherical harmonics.

Finally, we consider the inverse problem: given a target pattern, how to custom-design the interaction force to obtain said pattern as a steady state.

Galia Dafni (Concordia): "Hardy spaces old and new"

Abstract: The theory of Hardy spaces started out in complex variables and is still an important area of function theory in the unit disk. We will review the classical definitions and continue on to discuss real Hardy spaces, which have become an integral part of the field of harmonic analysis, in particular as it relates to the boundedness of singular integral operators. This theory also has connections to PDE, via duality with the space BMO (of functions of bounded mean oscillation) and through versions of the div-curl lemma, used in compensated compactness. Most recently, special Hardy spaces have been designed to fit various boundary value-problems, as well as to serve in non-Euclidean settings.

Alan Lindsay (University of Arizona): "Mathematical Modeling of Micro-Electro Mechanical Systems (MEMS)"

Abstract: A Micro electro-mechanical systems (MEMS) capacitor is a microscopic device consisting of two plates held opposite one other. The lower plate is immobile while the upper plate is fixed along its edges but free to deflect in the presence of an electric potential towards the lower plate. The deflecting upper plate may reach a stable equilibrium, however, if the applied potential exceeds a threshold, known as the pull-in voltage, the upper plate will touchdown on the lower plate. This event is crucial for the operation of certain devices (e.g., switches) but will compromise the utility of others (e.g., sensors). This loss of a stable equilibrium is known as the pull-in instability and mathematical modeling of its onset and consequences is the topic of this talk.

When certain physical assumptions are applied, the deflection of the upper surface can be modeled as a fourth order PDE with a singular non-linearity whose solutions are studied. It is shown that the model captures the pull-in instability of the device and provides a prediction of the pull-in voltage. The bifurcation structure of the equilibrium equations are analyzed and shown to contain some very interesting structures. When the pull-in voltage is exceeded, it is demonstrated that the device may touchdown on multiple isolated points or on a continuous set of points which are predicted for certain domains by means of asymptotic expansions. This could potentially allow a MEMS device to perform very exotic tasks.

Peter van Heijster (Boston University): "Destabilizing of stationary spots in a planar three-component FitzHugh-Nagumo equation"

Abstract: In this talk, I will analyze the destabilization of a stable stationary radially symmetric spot arising in a specific planar three-component FitzHugh-Nagumo equation. In particular, I am interested in the bifurcation of the stationary spot to a traveling spot. As it turns out, there is a competition between this drift bifurcation and several other Hopf bifurcations. We formally determine asymptotic conditions for these bifurcations using singular perturbation techniques and weakly nonlinear analysis. We also check our asymptotic results using AUTO and a direct solver. We obtain perfect agreement between both numerical methods and the asymptotic results. This is joint work with Björn Sandstede.

Mahya Ghandehari (Dalhousie University): "Geometric graphs and their limits"

Abstract: The study of large finite graphs and their growing sequences have become a very important topic in combinatorics and computer science. One approach to study such sequences is to identify graphs with elements of a metric space, and investigate the limiting object which may be represented by a symmetric function on [0,1]. Such graph limits may be used in understanding the properties of certain sequences of graphs. In this talk, we will study sequences of geometric graphs and obtain specific properties for the limit of such sequences.

This talk is based on joint work with Huda Chuang, Matt Hurshman, Jeannette Janssen, and Nauzer Kalyaniwalla.

Huaichun Wang (Dalhousie University): "Modeling Heterogeneous Processes of Protein Evolution for Inferring Phylogenies"

Joint Mathematics and Statistics Colloquium.

Abstract: This talk focuses on statistical modeling of protein evolution for inferring organismal phylogenies. After a general introduction to maximum likelihood and evolutionary models in phylogenetic inference, I will first discuss our work on modeling the covarion process of protein evolution. The covarion process is one that rates of evolution at sequence sites change in different parts of the tree. This is distinct from the rates-variation-across-sites (RAS) process that evolutionary rates are constant across the tree. The changing rates of evolution along the tree under the covarion process are formulated as a Markov model of rate switching between different rate classes. The models, implemented in our new phylogenetic inference package (PROCOV), gave significantly better likelihoods than using the RAS model for all 23 protein data sets we tested. In one case the covarion models were found to support a different optimal topology than the RAS model, highlighting the importance of covarion modeling.

Current phylogenetic models, including the ones implemented in PROCOV, usually assume a stationary distribution of amino acid frequencies that are averaged across sites. However, many sites in a protein sequence alignment have specific structural/functional properties requiring specific amino acid types. The frequency distribution of amino acids at these sites is vastly different from the average frequency distribution. In the next part of the talk I will discuss our new phylogenetic mixture model that takes into account the site-specific amino acid frequencies and implemented it in QmmRAxML for phylogenetic inference. Specifically, it takes an amino acid substitution matrix (e.g., JTT), an overall amino acid frequency vector (from JTT or the sequence alignment of interest), and several precomputed site-specific amino acid frequency vectors, evaluates the likelihoods under the combined models. It further uses an Expectation Maximization algorithm to optimize the weights associated with the different amino acid frequency vectors. We applied the program to analyze 19 protein data sets and found the amino acid frequency mixture model achieved significant better likelihoods than the non-mixture model. We further found the mixture model support different optimal topology than the non-mixture model on several data sets.

Mahdi Shafiei (Dalhousie University): "Bayesian inference of metabolic divergence among microbial communities"

Joint Mathematics and Statistics Colloquium.

Abstract: In this talk, I will present a hierarchical mixed-membership model for metabolic networks. The gut microbiota is believed to have 10 times bacteria than the number of human cells. With the rapidly growing availability of high-throughput sequencing and metagenomic data, some studies focus on characterizing the functional diversity and profile of gut microbial communities. Metagenomic sequences from a sample can be mapped to molecular reactions composing a metabolic network. Nodes on this network are chemical compounds and edges are reactions connecting one or more substrates to one or more products. The abundance of reactions in the network can vary for each sample.

The proposed model assumes that each network is a mixture of several microbiome types. This enables the model to provide a mixed-membership clustering of metabolic networks and the corresponding samples. Each microbiome type is then defined as a mixture of metabolic subsystems that are shared across all networks. A metabolic subsystem is a sub-network composed of a number of reactions. These subsystems are identified based on the network topology and the observed abundance of different reactions across metabolic networks constructed for samples. The parameters of the model are estimated using collapsed Gibbs sampling by integrating out some parameters of the model. We show that the subsystems discovered by our model provide valuable insights into the functionally differentiating components of a microbiome.

Peter LeFanu Lumsdaine (Dalhousie University): "The Hopf fibration, via constructive logic"

Abstract: The Hopf fibration is a beautiful and well-studied object of algebraic topology: a non-trivial fiber bundle S3 → S2, with fiber S1. Many constructions exist; probably the most popular is algebraic, using complex numbers. Recent work in constructive logic (“homotopy type theory”) prompted the idea of investigating inductive definitions for spaces. Just as we usually characterise e.g. the natural numbers with an inductive definition, one can similarly characterise many spaces (up to homotopy-equivalence) by inductive definitions: the circle S1, for instance, is generated by a basepoint and a path from the basepoint to itself. This is especially fruitful for defining fibrations over such spaces. I will sketch how the Hopf fibration S1 → S3 → S2, and its little brother S0 → S1 → S1, can be constructed using the inductive characterisations — with no complex numbers required!

Alexander S. Green (Dalhousie University): "Dependent Types and the Curry-Howard Isomorphism"

Abstract: The Curry-Howard Isomorphism gives a direct relationship between systems of formal logic, and computational calculi. Speaking at a more general level, it gives a correspondence between computer programs and proofs. In this talk, we shall first take a brief look at the correspondence between the simply typed lambda-calculus and minimal propositional logic, and then go on to see how dependent types correspond to first-order logic. The talk will finish with a look at some examples of proofs that are written as programs in a dependently-typed language, leading to a very brief look at a formal proof of the Four-Colour theorem.

Miklos Bartha (Memorial University): "Feedback, trace, and iteration in computer science"

Abstract: Feedback, trace, and iteration are closely related concepts, which have been studied intensively through various models in mathematics and computer science. While iteration is primarily used as a fixed-point operation in algebraic theories, feedback and trace have a more general scope covering all symmetric (braided) monoidal categories. The talk will introduce these operations in a simple algebraic language, and identify the basic equational axioms valid in the structures built around them. Special attention will be given to the algebra (category) of finite sets and relations, which can be viewed in two separate traced monoidal settings. Both of these structures play a crucial role in the theory of computing, and their counterparts in the category of finite dimensional Hilbert spaces give rise to an interesting quantum computation model.

Simona Paoli (University of Leicester): "From homotopy to higher categories"

Abstract: Several notions in higher category theory arose from questions and structures in homotopy theory. This is particularly the case for the problem of modelling the building blocks of topological spaces, called n-types, using algebraic and categorical language. In this talk I will survey some of the main ideas involved in the evolution from homotopy to higher categories. This talk will not assume any specialist knowledge of homotopy theory nor higher category theory.

Tobias Müller (University of Utrecht): "First order logic and random (geometric) graphs"

Abstract: We say that a graph property is first order expressible if it can be written as a logic sentence using the universal and existential quantifiers with variables ranging over the nodes of the graph, the usual connectives AND, OR, NOT, parentheses and the relations = and ~, where x ~ y means that x and y share an edge. For example, the property that G contains a triangle can be written as

Exists x,y,z : (x ~ y) AND (x ~ z) AND (y ~ z).

Starting from the sixties, first order expressible properties have been studied extensively on the most commonly studied model of random graphs, the Erdos-Renyi model. A number of very attractive and surprising results have been obtained, and by now we have a fairly full description of the behaviour of first order expressible properties on this model.

The Gilbert model of random graphs is obtained as follows. We take n points uniformly at random from the d-dimensional unit torus, and join two points by an edge if and only their distance is at most r.

After a brief overview of some of the most important work on first order expressible properties of random graphs, I will discuss joint work with S. Haber which tells a nearly complete story on first order expressible properties of the Gilbert random graph model. In particular we settle conjectures of McColm and of Agarwal-Spencer. (Joint with S. Haber)

About the speaker:

Tobias Mueller is an assistant professor at the mathematical institute of Utrecht University. Before joining Utrecht University, he was a researcher at the Centrum Wiskunde & Informatica (CWI), where he still spends approximately one day per week. Prior to that he held postdoctoral positions in Eindhoven and in Tel Aviv. He did his doctorate in Oxford under the supervision of Colin McDiarmid.

Piotr Rudnicki (University of Alberta): "Mizar and its applications"

Abstract: Mizar is a proof-checking system used to build the Mizar Mathematical Library (MML) - a long term project aiming at building a comprehensive library of formalized mathematical knowledge.

The main goal for the design of Mizar (an effort led by Dr. Andrzej Trybulec) has been to create a formal system close to the mathematical vernacular used in main-stream mathematical publications. An additional requirement was that the language be simple enough to enable computerized processing, in particular mechanical verification of correctness. The continual development of Mizar has resulted in a language, software for checking the correctness of texts written in it, numerous utility programs, a centrally maintained library of mathematics, all of which are available on the Internet (http://mizar.org). The logic of Mizar is classical, the proofs are written in a natural deduction style and like almost all of mathematics it is based on ZF set theory beefed up with axiom of arbitrarily large, strongly inaccessible cardinals. Mizar definitions allow to introduce new constructors for types, algebraic structures, terms, adjectives, and atomic formulae. The Mizar language and the checking software evolve; the evolution is driven by the growing library.

This talk offers general information on the Mizar system, its history, its current state and its future. I would be delighted to meet with anyone interested in more detail.

Ray McLenaghan (University of Waterloo): "On the quasi-linearization of the equations of motion of a mechanical system"

Abstract: It is shown that a necessary and sufficient condition for the quasi-linearization of the Lagrange equations of motion of a conservative holonomic mechanical system is that the configuration space admits a local transitive isometry group. This condition is satisfied if the space is locally a symmetric space. Examples illustrating this criterion are discussed.

Srinivasa Vardhan (Courant Institute): "Random Graphs"

Abstract: We consider a dense random graph with N vertices. The edge between a pair of vertices can be either present or absent. The probability is p for any edge to be present and independently so for different edges. We can count the number of triangles or other finite graphs that occur as subgraphs in our large graph. Although there is a law of large numbers that provides an estimate of the count, one can have graphs where the count is off. This is a rare event with small probability. The question is to understand how small this probability is and what it implies for the random graphs. For example, if we have more triangles than normal, do we necessarily have more edges?

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