Seminars and Colloquia by Series

A KAM-like theorem for normally hyperbolic quasi-periodic tori leading to efficient algorithms

Series
CDSNS Colloquium
Time
Monday, February 3, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles05
Speaker
Marta Canadell – Univ. of Barcelona
We present a KAM-like theorem for the existence of quasi-periodic tori with a prescribed Diophantine rotation for a discrete family of dynamical system. The theorem is stated in an a posteriori format, so it can be used to validate numerical computations. The method of proof provides an efficient algorithm for computing quasi-periodic tori. We also present implementations of the algorithm, illustrating them throught several examples and observing different mechanisms of breakdown of qp invariant tori. This is a joint work with Alex Haro.

Particle Physics and Cosmology from Almost Commutative Manifolds

Series
Math Physics Seminar
Time
Friday, January 31, 2014 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Mairi Sakellariadou – King's College, Physics –

Please Note: This is a joint Seminar School of Mathematics and Center of Relativistic Astrophysics, Georgia Tech

The unification of the four fundamental forces remains one of the most important issues in theoretical particle physics. In this talk, I will first give a short introduction to Non-Commutative Spectral Geometry, a bottom-up approach that unifies the (successful) Standard Model of high energy physics with Einstein's General theory of Relativity. The model is build upon almost-commutative spaces and I will discuss the physical implications of the choice of such manifolds. I will show that even though the unification has been obtained only at the classical level, the doubling of the algebra may incorporate the seeds of quantisation. I will then briefly review the particle physics phenomenology and highlight open issues and current proposals. In the last part of my talk, I will explore consequences of the Gravitational-Higgs part of the spectral action formulated within such almost-commutative manifolds. In particular, I will study modifications of the Friedmann equation, propagation of gravitational waves and the onset of inflation. I will show how current measurements (Gravity Probe, pulsars, and torsion balance) can constrain free parameters of the model. I will conclude with a short discussion on open questions. Download the POSTER

The generalized cycle-cocycle reversal system for partial graph orientations

Series
Combinatorics Seminar
Time
Friday, January 31, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Spencer Backman – Georgia Tech
We introduce a discrete dynamical system on the set of partial orientations of a graph, which generalizes Gioan's cycle-cocycle reversal system. We explain how this setup allows for a new interpretation of the linear equivalence of divisors on graphs (chip-firing), and a new proof of Baker and Norine's combinatorial Riemann Roch formula. Fundamental connections to the max-flow min-cut theorem will be highlighted.

Dynamics of ferromagnets: averaging methods, bifurcation diagrams, and thermal noise effects

Series
Job Candidate Talk
Time
Friday, January 31, 2014 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Katherine Newhall – Courant Institute
Driving nanomagnets by spin-polarized currents offers exciting prospects in magnetoelectronics, but the response of the magnet to such currents remains poorly understood. For a single domain ferromagnet, I will show that an averaged equation describing the diffusion of energy on a graph captures the low-damping dynamics of these systems. In particular, I compute the mean times of thermally assisted magnetization reversals in the finite temperature system, giving explicit expressions for the effective energy barriers conjectured to exist. I will then outline the problem of extending the analysis to spatially non-uniform magnets, leading to a transition state theory for infinite dimensional Hamiltonian systems.

Rescheduled for March 12: Spatial epidemic models: lattice differential equation analysis of wave and droplet-like behavior

Series
Mathematical Biology Seminar
Time
Friday, January 31, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chi-Jen Wang – Iowa State
Spatially discrete stochastic models have been implemented to analyze cooperative behavior in a variety of biological, ecological, sociological, physical, and chemical systems. In these models, species of different types, or individuals in different states, reside at the sites of a periodic spatial grid. These sites change or switch state according to specific rules (reflecting birth or death, migration, infection, etc.) In this talk, we consider a spatial epidemic model where a population of sick or healthy individual resides on an infinite square lattice. Sick individuals spontaneously recover at rate *p*, and healthy individual become infected at rate O(1) if they have two or more sick neighbors. As *p* increases, the model exhibits a discontinuous transition from an infected to an all healthy state. Relative stability of the two states is assessed by exploring the propagation of planar interfaces separating them (i.e., planar waves of infection or recovery). We find that the condition for equistability or coexistence of the two states (i.e., stationarity of the interface) depends on orientation of the interface. We also explore the evolution of droplet-like configurations (e.g., an infected region embedded in an all healthy state). We analyze this stochastic model by applying truncation approximations to the exact master equations describing the evolution of spatially non-uniform states. We thereby obtain a set of discrete (or lattice) reaction-diffusion type equations amenable to numerical analysis.

Graphs, Knots, and Algebras

Series
ACO Distinguished Lecture
Time
Tuesday, January 28, 2014 - 16:30 for 1 hour (actually 50 minutes)
Location
Clough Commons Room 152
Speaker
Alexander Schrijver – University of Amsterdam and CWI Amsterdam

Please Note: SHORT BIO: Alexander Schrijver is Professor of Mathematics at the University of Amsterdam and researcher at the Center for Mathematics and Computer Science (CWI) in Amsterdam. His research focuses on discrete mathematics and optimization, in particular on applying methods from fundamental mathematics. He is the author of four books, including 'Theory of Linear and Integer Programming' and 'Combinatorial Optimization - Polyhedra and Efficiency'. He received Fulkerson Prizes in 1982 and 2003, Lanchester Prizes in 1987 and 2004, a Dantzig Prize in 2003, a Spinoza Prize in 2005, a Von Neumann Theory Prize in 2006, and an Edelman Award in 2008. He is a member of the Royal Netherlands Academy of Arts and Sciences since 1995 and of three foreign academies, received honorary doctorates from the Universities of Waterloo and Budapest, and was knighted by the Dutch Queen in 2005.

Many graph invariants can be described as 'partition functions' (in the sense of de la Harpe and Jones). In the talk we give an introduction to this and we present characterizations of such partition functions among all graph invariants. We show how similar methods describe knot invariants and give rise to varieties parametrizing all partition functions. We relate this to the Vassiliev knot invariants, and show that its Lie algebra weight systems are precisely those weight systems that are 'reflection positive'. The talk will be introductory and does not assume any specific knowledge on graphs, knots, or algebras.

$L^2$-geometry of diffeomorphism groups and the equations of hydrodynamics

Series
PDE Seminar
Time
Tuesday, January 28, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gerard Misiolek – University of Notre Dame
In 1966 V. Arnold observed that solutions to the Euler equations of incompressible fluids can be viewed as geodesics of the kinetic energy metric on the group of volume-preserving diffeomorphisms. This introduced Riemannian geometric methods into the study of ideal fluids. I will first review this approach and then describe results on the structure of singularities of the associated exponential map and (time premitting) related recent developments.

Extremal Eigenvalue Problems in Optics, Geometry, and Data Analysis

Series
Job Candidate Talk
Time
Tuesday, January 28, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Braxton Osting – UCLA, Math
Since Lord Rayleigh conjectured that the disk should minimize the first eigenvalue of the Laplace-Dirichlet operator among all shapes of equal area more than a century ago, extremal eigenvalue problems have been an active research topic. In this talk, I'll demonstrate how extremal eigenvalue problems arise in a variety of contexts, including optics, geometry, and data analysis, and present some recent analytical and computational results in these areas. One of the results I'll discuss is a new graph partitioning method where the optimality criterion is given by the sum of the Dirichlet energies of the partition components. With intuition gained from an analogous continuous problem, we introduce a rearrangement algorithm, which we show to converge in a finite number of iterations to a local minimum of a relaxed objective function. The method compares well to state-of-the-art approaches when applied to clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations.

Comparing the slice and ribbon genera of knots via braided surfaces

Series
Geometry Topology Seminar
Time
Monday, January 27, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark Hughes – SUNY, Stony Brook
In this talk I will discuss bounds on the slice genus of aknot coming from it's representation as a braid closure, starting withthe slice-Bennequin inequality. From there I will use surfacebraiding techniques of Rudolph and Kamada to exhibit a new lower boundon the ribbon genus of a knot, given some knowledge about what slicesurfaces it bounds.

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