October 21: Keivan Mallahi-Karai (Jacobs University, Bremen), Minimal faithful representation of Chevalley groups over finite rings
Abstract: For a finite group G, denote by m_f(G) the minimum possible dimension of a faithful linear representation of G. I will talk about recent lower bound for m_f(G) for some Chevalley groups G over finite rings. It turns out to be asymptotically the same as in the results by Landazuri, Seitz and Zalesskii in the case of split Chevalley groups over finite fields. This is joint work with Mohammad Bardestani, Camelia Karimianpour, and Hadi Salmasian.
October 28: Kelsey Horan (CUNY Graduate Center), Deep Convolutional Neural Networks for Left Ventricle Segmentation
Abstract: Left Ventricle (LV) segmentation is very important to quantitative global and regional cardiac function analysis. Most of the current LV segmentation methods are based on Active Contour and Region Growing approaches which require manual initialization by the clinician. Therefore, a fully automated segmentation method truly satisfies physician needs. In this work we propose a Gabor-based Deep Convolutional Neural Network to automatically segment the LV wall in Magnetic Resonance Imaging (MRI). We also introduce a novel updating procedure for the network that takes into consideration the mathematical structure of the network parameters. Then, we propose alternative CNN parameters that are elements of algebraic groups.
This is joint work with Somayeh Molaei, Delaram Kahrobaei, Kayvan Najarian, and Brahmajee Nallamothu.
November 4: Nigel Boston (University of Wisconsin, Madison), Information Inequalities, Entropy Regions, and Groups
Abstract: Entropy regions are fundamental but poorly understood objects, central to computing network coding capacities, but also describable purely in terms of groups. Nan and I have used group theory to shed light on these mysterious regions.
November 11: Emina Soljanin (Rutgers University), (Secure) Linear Network Coding Multicast: A Theoretical Minimum and An Open Problem
Abstract: Network coding is an elegant mathematical technique introduced at the turn of the millennium to improve network throughput and performance. Consider a directed acyclic graph (network) where h source-nodes produce elements of some finite field (source symbols). Edges carry linear combinations of their parent node inputs. In turn, this implies that edges throughout the network carry linear combinations of the source symbols. The network coding multicast problem asks: How should nodes in such a network with N receivers combine their inputs to ensure that each h edges observed by a receiver carry independent combinations of the source symbols? Moreover, what is the minimum field size necessary to realize combinations with this property? The field size problem is completely solved in the case of two sources and arbitrarily many receivers, but in no other cases. This talk will show how graph theory and algebraic geometry have been instrumental in proving the two-source case and gaining some further insights.
December 2: Roy Lederman (Princeton University), A Representation Theory Perspective on Simultaneous Alignment and Classification with Applications in Cryo-EM
Abstract: Single particle Cryo-electron microscopy (EM) recently joined X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM does not require crystallization necessary for X-ray crystallography, and unlike NMR it is not limited to small size molecules. In single particle Cryo-EM the 3-D structure is determined from many noisy 2-D projection images of individual, ideally identical frozen-hydrated macromolecules whose orientations and positions are random and unknown. Cryo-EM has been named Method of the Year 2015 by the journal Nature Methods after recent advancements in detector technology led to a breakthrough in near-atomic resolution reconstruction of molecules whose structure cannot be obtained by other techniques.
One of the great opportunities in Cryo-EM is studying heterogeneous samples, containing two or more distinct types or conformations of molecules. Taking advantage of this opportunity presents an algorithmic challenge: determining both the class and orientation of each particle; this often requires an initial guess of the structures and iterative estimations of the structures and particle orientations and class labels.
We propose an algorithm for simultaneous alignment and classification with the goal of simultaneously classifying Cryo-EM images and aligning them within their respective classes. Our proposed approach can also be extended to the case of continuous heterogeneity.
December 9: Manhattan Algebra Day
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