Seminars and Colloquia by Series

Galois-equivariant and motivic homotopy

Series
Geometry Topology Seminar
Time
Monday, December 9, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kyle Ormsby – MIT
For a group G, stable G-equivariant homotopy theory studies (the stabilizations of) topological spaces with a G-action up to G-homotopy. For a field k, stable motivic homotopy theory studies varieties over k up to (a stable notion of) homotopy where the affine line plays the role of the unit interval. When L/k is a finite Galois extension with Galois group G, there is a functor F from the G-equivariant stable homotopy category to the stable motivic homotopy category of k. If k is the complex numbers (or any algebraically closed characteristic 0 field) and L=k (so G is trivial), then Marc Levine has shown that F is full and faithful. If k is the real numbers (or any real closed field) and L=k[i], we show that F is again full and faithful, i.e., that there is a "copy" of stable C_2-equivariant homotopy theory inside of the stable motivic homotopy category of R. We will explore computational implications of this theorem.This is a report on joint work with Jeremiah Heller.

The Jones polynomial as Euler characteristic

Series
School of Mathematics Colloquium
Time
Friday, December 6, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Lipshitz – University of North Carolina, Chapel Hill

Please Note: Kickoff of the Tech Topology Conference from December 6-8, 2013. For complete details see ttc.gatech.edu

We will start by defining the Jones polynomial of a knot and talking about some of its classical applications to knot theory. We will then define a fancier version ("categorification") of the Jones polynomial, called Khovanov homology and mention some of its applications. We will conclude by talking about a further refinement, a Khovanov homotopy type, sketch some of the ideas behind its construction, and mention some applications. (This last part is joint work with Sucharit Sarkar.) At least the first half of the talk should be accessible to non-topologists.

Interlacing Families and Kadison--Singer

Series
ACO Colloquium
Time
Friday, December 6, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Adam Marcus – Crisply.com and Yale Unversity
We will outline the proof that gives a positive solution to Weaver's conjecture $KS_2$. That is, we will show that any isotropic collection of vectors whose outer products sum to twice the identity can be partitioned into two parts such that each part is a small distance from the identity. The distance will depend on the maximum length of the vectors in the collection but not the dimension (the two requirements necessary for Weaver's reduction to a solution of Kadison--Singer). This will include introducing a new technique for establishing the existence of certain combinatorial objects that we call the "Method of Interlacing Polynomials." This talk is intended to be accessible by a general mathematics audience, and represents joint work with Dan Spielman and Nikhil Srivastava.

Distributions of Angles in Random Packing on Spheres

Series
Stochastics Seminar
Time
Thursday, December 5, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tiefeng Jiang – University of Minnesota
We study the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in p-dimensional spaces as the number of points n goes to infinity, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that ``all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and connections with some open problems in physics and mathematics are also discussed. This is a joint work with Tony Cai and Jianqing Fan.

Out-of-equilibrium dynamics for the nonlinear Schroedinger equation: From energy cascades to weak turbulence

Series
Job Candidate Talk
Time
Thursday, December 5, 2013 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zaher Hani – Courant Institute, New York University
Out-of-equilibrium dynamics are a characteristic feature of the long-time behavior of nonlinear dispersive equations on bounded domains. This is partly due to the fact that dispersion does not translate into decay in this setting (in contrast to the case of unbounded domains like $R^d$). In this talk, we will take the cubic nonlinear Schroedinger equation as our model, and discuss some aspects of its out-of-equilibrium dynamics, from energy cascades (i.e. migration of energy from low to high frequencies) to weak turbulence.

Non-lifting of a subgroup of the mapping class group

Series
Geometry Topology Student Seminar
Time
Wednesday, December 4, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Robert Krone – Georgia Tech
The mapping class group of a surface is a quotient of the group of orientation preserving diffeomorphisms. However the mapping class group generally can't be lifted to the group of diffeomorphisms, and even many subgroups can't be lifted. Given a surface S of genus at least 2 and a marked point z, the fundamental group of S naturally injects to a subgroup of MCG(S,z). I will present a result of Bestvina-Church-Souto that this subgroup can't be lifted to the diffeomorphisms fixing z.

Some properties of a variational model for the reconstruction of occluded boundaries

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, December 4, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Riccardo March – Istituto per le Applicazioni del Calcolo "Mauro Picone" of C.N.R and University of Rome
We consider a variational model for image segmentation which takes into account the occlusions between different objects. The model consists in minimizing a functional which depends on: (i) a partition (segmentation) of the image domain constituted by partially overlapping regions; (ii) a piecewise constant function which gives information about the visible portions of objects; (iii) a piecewise constant function which constitutes an approximation of a given image. The geometric part of the energy functional depends on the curvature of the boundaries of the overlapping regions. Some variational properties of the model are discussed with the aim of investigating the reconstruction capabilities of occluded boundaries of shapes. Joint work with Giovanni Bellettini.

Dimension of Planar Posets

Series
Research Horizons Seminar
Time
Wednesday, December 4, 2013 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Tom Trotter – School of Math
Answering a question of R. Stanley, we show that for each t ≥1, there is a least positive integer f(t) so that a planar poset with t minimal elements has dimension at most f(t). In particular, we show that f(t) ≤ 2t + 1 and that this inequality is tight for t=1 and t=2. For larger values of t, we can only show that f(t) ≥ t+3. This research is joint work with Georgia Tech graduate student Ruidong Wang.

Self-Diffusion and Cross-Diffusion Equations: $W^{1,p}$-Estimates and Global Existence of Smooth Solutions

Series
PDE Seminar
Time
Tuesday, December 3, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tuoc V. Phan – University of Tennessee, Knoxville
We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension.We obtain this result by deriving global $W^{1,p}$-estimates of Calder\'{o}n-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation techniquetogether with a new two-parameter scaling argument.The talk is based on my joint work with Luan Hoang (Texas Tech University) and Truyen Nguyen (University of Akron)

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