Sarah Schott : Computational Complexity
- Graduate/Faculty Seminar ( 247 Views )What does it mean for a problem to be in P, or NP? What is NP completeness? These are questions, among others, that I hope to answer in my talk on computational complexity. Computational complexity is a branch of theoretical computer science dealing with analysis of algorithms. I hope to make it as accessible as possible, with no prior knowledge of algorithms and running times.
Dino J. Lorenzini : Linear algebra: my lack, your luck?
- Graduate/Faculty Seminar ( 117 Views )Given a (n x n)-matrix M over a commutative integral domain R, one can try to associate to it a diagonal matrix called the Smith Normal Form of M. This can be done when R is the ring of integers, or the polynomial ring F[x] over a field F, and various applications of the existence of the Smith Normal Form are discussed in matrix theory. Which commutative integral domains R have the property that every matrix with coefficients in R admits a Smith Normal form? This is a very old question, as for instance Wedderburn in 1915 already discussed the case where R is the ring of holomorphic functions. I will review all necessary concepts, and discuss several easily stated open problems in this circle of ideas.
Lea Renner : Left-Ordered Groups
- Graduate/Faculty Seminar ( 107 Views )Group theory is one of the basic topics of abstract algebra and therefore probably well-known. In this talk, we are going to introduce left-orders on groups and expand the fundamental theorem of homomorphisms from groups to ordered groups. We will see some examples of left-ordered groups that show different levels of orderability and, time permitting, formulate a theorem of A. Rhemtulla which discusses the existence of torsion-free groups without any order.
Curtis Porter : CRash CouRse in CR Geometry
- Graduate/Faculty Seminar,Uploaded Videos ( 2140 Views )CR geometry studies real hypersurfaces in complex vector spaces and their generalizations, CR manifolds. In many cases of interest to complex analysis and PDE, CR manifolds can be considered ``curved versions" of homogeneous spaces according to Elie Cartan’s generalization of Klein’s Erlangen program. Which homogeneous space is the ``flat model" of a CR manifold depends on the Levi form, a tensor named after a mathematician who used it to characterize boundaries of pseudoconvex domains. As in the analytic setting, the Levi form plays a central role in the geometry of CR manifolds, which we explore in relation to their homogeneous models.
Lihan Wang : Approximation of Correctors and Multipoles in Random Elliptic Media
- Graduate/Faculty Seminar ( 182 Views )We consider the whole-space decaying solution of second-order elliptic PDE in divergence form with space dimension d=3, where the coefficient field is a realization of a stationary, uniformly elliptic, unit range ensemble of random field, and the right-hand-side is deterministic and compactly supported in a ball of size \ell. Given the coefficient field in a large box of size L much larger than \ell, we are interested in an algorithm to compute the gradient of the solution with the "best" artificial boundary condition on the domain of size L which describes the correct long-range multipole behavior. We want to show that, with high probability, our algorithm reaches the CLT-type lower bound of error. Joint work with Jianfeng Lu and Felix Otto.
Chris O'Neill : Matroids, and How to Make Your Proofs Multitask
- Graduate/Faculty Seminar ( 125 Views )What do vector arrangements, discrete graphs, and perfect matchings have in common? These seemingly unrelated objects (and many others) have a very similar underlying structure, known as a matroid. As a result, studying matroids allows you to simultaneously study many different objects from all over mathematics. In addition, many properties and constructions from these various objects, such as loops, duals, bases, cycles, rank, polynomial invariants, and minors (subgraphs), generalize naturally to matroids. In this talk, we will give a general definition of a matroid, and motivate their study by examining some of these constructions in detail. The only prerequisite for this talk is basic linear algebra.
Shahar Kovalsky : Bending cubes with optimization and computational geometry
- Graduate/Faculty Seminar ( 116 Views )Problems in computational geometry, such as parameterizing a surface or computing shape deformation under geometric constraints, pose various challenges. I will give brief overview of related problems, highlighting the link between discrete differential geometry, optimization and computer graphics. Then, we will see how convex optimization can be used to approximate a specific class of geometric problems, that include shape mapping, bending a cube (https://youtu.be/iOwPGG5-54Q) and perhaps matching lemur teeth.
Alexander Watson : Wave-packet dynamics in locally periodic media with a focus on the effects of Bloch band degeneracies
- Graduate/Faculty Seminar ( 104 Views )We study the dynamics of waves in media with a local periodic structure which varies adiabatically (over many periods of the periodic lattice) across the medium. We focus in particular on the case where symmetries of the periodic structure lead to degeneracies in the Bloch band dispersion surface. An example of such symmetry-induced degeneracies are the `Dirac points of media with `honeycomb lattice symmetry, such as graphene. Our results are as follows: (1) A systematic and rigorous derivation of the `anomalous velocity of wave-packets due to the Bloch bands Berry curvature. The Berry curvature is large near to degeneracies, where it takes the form of a monopole. We also derive terms which do not appear in the works of Niu et al. which describe a `field-particle coupling effect between the evolution of observables associated with the wave-packet and the evolution of the wave packet envelope. These terms are of the same order as the anomalous velocity. (2) Restricting to one spatial dimension, the derivation of the precise dynamics when a wave-packet is incident on a Bloch band degeneracy. In particular we derive the probability of an inter-band transition and show that our result is consistent with an appropriately interpreted Landau-Zener formula. I will present these results for solutions of a model Schr\{o}dinger equation; extending our results to systems described by Maxwell's equations is the subject of ongoing work. This is joint work with Michael Weinstein and Jianfeng Lu.
Mauro Maggioni : A primer on wavelets and their applications
- Graduate/Faculty Seminar ( 110 Views )Wavelets are widely used in signal processing (e.g. analysis of sounds and music) and imaging, for tasks such as denoising and compression (ever wondered how jpeg works?). In harmonic analysis they have been used to understand and solve problems involving integral operators motivated by PDEs. In numerical PDEs they lead to fast algorithms for solving certain types of integral equations and PDEs. I will give a gentle introduction to wavelets and some of their motivating applications, accompanied by live demos. If time allows, I will discuss shortcomings and how they have been addressed in more recent developments and generalizations.
Guangliang Chen : Modeling High Dimensional Data by Linear/Nonlinear Models
- Graduate/Faculty Seminar ( 109 Views )Nowadays researchers encounter high dimensional data that arises in a variety of forms such as digital images, videos, and hyperspectral images. How to efficiently and effectively modeling such data sets has become an active research topic. A common model is to approximate such data by a mixture of affine subspaces. In this talk I will present a fast and accurate algorithm that can solve this problem in full generality, addressing both theoretical and applied issues. If time permits, I will also talk about the use of nonlinear models. This is joint work with Mauro Maggioni.
Jianfeng Lu : Surface hopping: Mystery and opportunities for mathematicians
- Graduate/Faculty Seminar ( 185 Views )Surface hopping is a very popular approach in theoretical chemistry for mixed quantum-classical dynamics. Yes, the above sentence looks scary. Let us start over again ... We will examine from a mathematical point of view how stochastic trajectories can be used to approximate solutions to a Schrodinger equation (which is different from what Feynman did). Besides some applications in chemistry, this is a nice topic since it combines ideas from asymptotic analysis, applied probability, and applied harmonic analysis. The only background assumed in this talk is "separation of variables" (and of course some PDEs where separation of variables is applied to).
Yuriy Mileyko : Enter Skeleton: a brief overview of skeletal structures
- Graduate/Faculty Seminar ( 180 Views )Skeletal structures, such as medial axis and curve skeleton, are a particular class of shape descriptors. They have numerous applications in shape recognition, shape retrieval, animation, morphing, registration, and virtual navigation. This talk will give a brief overview of the medial axis and the curve skeleton. The focus will be on the properties of the two objects crucial to applications. We shall show that the rigorous mathematical definition of the medial axis has allowed for an extensive and successful study of such properties. The curve skeleton, on the other hand, is typically defined by the set of properties it has to possess. As a result, numerous methods for computing the curve skeleton have been proposed, each providing mostly experimental verification of the required properties. If time permits, I will mention my work on defining shape skeleta via persistent homology, thus providing a powerful platform for investigating their properties.
Benoit Charbonneau : Gauge theory and modern problems in geometry
- Graduate/Faculty Seminar ( 112 Views )I will survey some modern questions in geometry that were solved or that could be solved using tools of gauge theory. This talk should be accessible to first year grad students, and of interest to anyone who is curious about what happens in the field of geometry.
Miles M. Crosskey : Mathematics in Magic
- Graduate/Faculty Seminar ( 204 Views )Many simple card tricks rely on mathematical principles and logic. I will be talking about some of these tricks, and the interesting ideas behind them. Hopefully I will have time to show you two or three tricks, and the proof to how they work. I will be using work from Mathematical Magic by Diaconis and Graham. The exciting thing about these tricks is they do not rely upon sleight of hand, and come out looking stunning nonetheless.
Liz Munch : Failure Filtrations and Coverage of Fenced Sensor Networks: An Application of Computational Topology
- Graduate/Faculty Seminar ( 131 Views )Although originally formed as an esoteric field of study, in the last few decades Algebraic Topology has emerged as a vastly applicable field. In this talk we will discuss the basics of Computational Topology and an application to one such coverage problem in sensor networks which even involves a little probability. This talk will be accessible to anyone who enjoys doing math via lots and lots of pictures.