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Series: Graph Theory Seminar

Györi and Lovasz independently proved that a k-connected graph can be partitioned into k subgraphs, with each subgraph connected, containing a prescribed vertex, and with a prescribed
vertex count. Lovasz used topological methods, while Györi found a purely graph theoretical approach. Chen et al. later generalized the topological proof to graphs with weighted
vertices, where the subgraphs have prescribed weight sum rather than vertex count. The weighted result was recently proven using Györi's approach by Chandran et al. We will use the
Györi approach to generalize the weighted result slightly further. Joint work with Robin Thomas.

Series: School of Mathematics Colloquium

This talk is about the structure theory of measure-preserving systems: transformations of a finite measure space that preserve the measure. Many important examples arise from stationary processes in probability, and simplest among these are the i.i.d. processes. In ergodic theory, i.i.d. processes are called Bernoulli shifts. Some of the main results of ergodic theory concern an invariant of systems called their entropy, which turns out to be intimately related to the existence of `structure preserving' maps from a general system to Bernoulli shifts. I will give an overview of this area and its history, ending with a recent advance in this direction. A measure-preserving system has the weak Pinsker property if it can be split, in a natural sense, into a direct product of a Bernoulli shift and a system of arbitrarily low entropy. The recent result is that all ergodic measure-preserving systems have this property. This talk will assume graduate-level real analysis and measure theory, and familiarity with the basic language of random variables. Past exposure to entropy, measure-theoretic probability or ergodic theory will be helpful, but not essential.

Wednesday, April 18, 2018 - 14:10 ,
Location: Skiles 006 ,
Sarah Davis ,
GaTech ,
Organizer: Anubhav Mukherjee

The theorem of Dehn-Nielsen-Baer says the extended mapping class group is isomorphic to the outer automorphism group of the fundamental group of a surface. This theorem is a beautiful example of the interconnection between purely topological and purely algebraic concepts. This talk will discuss the background of the theorem and give a sketch of the proof.

Series: Analysis Seminar

We discuss the probability that a continuous stationary Gaussian process on whose spectral measure vanishes in a neighborhood of the origin stays non-negative on an interval of long interval. Joint work with Naomi Feldheim, Ohad Feldheim, Fedor Nazarov, and Shahaf Nitzan

Series: PDE Seminar

Epitaxial growth is an important physical process for forming solid films or other nano-structures. It occurs as atoms, deposited from above, adsorb and diffuse on a crystal surface. Modeling the rates that atoms hop and break bonds leads in the continuum limit to degenerate 4th-order PDE that involve exponential nonlinearity and the p-Laplacian with p=1, for example. We discuss a number of analytical results for such models, some of which involve subgradient dynamics for Radon measure solutions.

Series: Geometry Topology Seminar

Augmentations and exact Lagrangian fillings are closely related. However, not all the augmentations of a Legendrian knot come from embedded exact Lagrangian fillings. In this talk, we show that all the augmentations come from possibly immersed exact Lagrangian fillings. In particular, let ∑ be an immersed exact Lagrangian filling of a Legendrian knot in $J^1(M)$ and suppose it can be lifted to an embedded Legendrian L in J^1(R \times M). For any augmentation of L, we associate an induced augmentation of the Legendrian knot, whose homotopy class only depends on the compactly supported Legendrian isotopy type of L and the homotopy class of its augmentation of L. This is a joint work with Dan Rutherford.

Series: Geometry Topology Seminar

Based on the known examples, it had been conjectured that all L-space knots in S3 are strongly invertible. We show this conjecture is false by constructing large families of asymmetric hyperbolic knots in S3 that admit a non-trivial surgery to the double branched cover of an alternating link. The construction easily adapts to produce such knots in any lens space, including S1xS2. This is joint work with John Luecke.

Monday, April 16, 2018 - 13:55 ,
Location: Skiles 005 ,
Xiuyuan Cheng ,
Duke University ,
xiuyuan.cheng@duke.edu ,
Organizer: Wenjing Liao

Filters in a Convolutional Neural Network
(CNN) contain model parameters learned from enormous amounts of data.
The properties of convolutional filters in a trained network directly
affect the quality of the data representation being produced. In this
talk, we introduce a framework for decomposing convolutional filters
over a truncated expansion under pre-fixed bases, where the expansion coefficients are learned from data. Such a structure not only reduces the number of trainable parameters and computation load but
also explicitly imposes filter regularity by bases truncation. Apart
from maintaining prediction accuracy across image classification
datasets, the decomposed-filter CNN also produces a stable
representation with respect to input variations, which is proved under generic assumptions on the basis expansion. Joint work with Qiang Qiu, Robert Calderbank, and Guillermo Sapiro.

Series: CDSNS Colloquium

Transition State Theory describes how a reactive system crosses an energy barrier that is marked by a saddle point of the potential energy. The transition from the reactant to the product side of the barrier is regulated by a system of invariant manifolds that separate trajectories with qualitatively different behaviour.

The situation becomes more complex if there are more than two reaction channels, or possible outcomes of the reaction. Indeed, the monkey saddle potential, with three channels, is known to exhibit chaotic dynamics at any energy. We investigate the boundaries between initial conditions with different outcomes in an attempt to obtain a qualitative and quantitative description of the relevant invariant structures.

TBA

Series: GT-MAP Seminars

There are 5 short presentations in this mini-workshop. Please go to http://gtmap.gatech.edu or http://gtmap.gatech.edu/events/mini-workshop-mathematics-and-control for schedule, title and abstract.