# All Seminars

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Title: Primality Testing and Integer Factorization Using Elliptic Curves
Defense: Master's Defense
Speaker: Andrew Wilson of Emory University
Contact: Andrew Wilson, andrew.wilson@emory.edu
Date: 2017-04-06 at 4:15PM
Venue: E406
Abstract:
Testing integers for primality and factoring large integers is an extremely important subject for our daily lives. Every time we use a credit card to make online purchases we are relying on the difficulty of factoring large integers for the security of our personal information. Similar encryption methods are used by governments around the world to protect their classi ed information, stressing the importance of the subject of primality testing and factoring algorithms to both personal and national security. Elementary number theory has been a key tool in the foundation of primality testing and factoring algorithms, speci fically the work of Euler and Fermat, whose developments on modular arithmetic give us key tools that we still use today in the more complex primality tests and factoring methods. More recently people have used deeper ideas from geometry, namely elliptic curves, to develop faster tests and algorithms. In this thesis we continue this trend, and develop new primality tests that utilize previous theory of elliptic curves over nite elds. The primary point is that the points on these curves form a special group, which breaks down when working over Z/NZ, when N is not prime. Our theorems make use of the work of Kubert, Hasse, Mazur, and many more to yield a primality test that gives no false positives.
Title: Rank of matrices with few distinct entries
Seminar: Combinatorics
Speaker: Boris Bukh of Carnegie Mellon University
Contact: Dwight Duffus, dwight@mathcs.emory.edu
Date: 2017-04-05 at 4:00PM
Venue: W303
Abstract:
Many applications of the linear algebra method to combinatorics rely on bounds on ranks of matrices with few distinct entries and constant diagonal. In this talk, I will explain some of these applications. I will also present a classification of sets \textit{L} for which no low-rank matrix with entries in \textit{L} exists.
Title: Generalized Cross Validation for Ill-Posed Inverse Problems
Defense: Honors
Speaker: Hanyong Wu of Emory University
Contact: Hanyong Wu,
Date: 2017-04-05 at 5:00PM
Venue: W306
Abstract:
In this thesis, we will introduce two popular regularization tools for ill- posed linear inverse problem, truncated singular value decomposition and Tikhonov regularization. After that we will implement them with the gener- alized cross validation (GCV) method to choose regularization parameters. We consider in particular problems that have noise in the measured data, noise in the matrix, and noise in both the measured data and the matrix. Numerical experiments are used to test the GCV method for each of these noise models.
Title: An Algorithm for Numerically Computing Preimages of the $j$-invariant
Defense: Masters
Speaker: Ethan Alwaise of Emory University
Contact: Ethan Alwaise, ethan.alwaise@emory.edu
Date: 2017-04-05 at 5:30PM
Venue: E408
Abstract:
Here we explore the problem of numerically computing preimages of the $j$-invariant. We present an algorithm based on studying the asymptotics of the Fourier coefficients of the logarithmic derivative of $j(\tau)$. We use recent work of Bringmann et al., which gives asymptotics for the Fourier coefficients of divisor modular forms, to identify the real and imaginary parts of the preimage.
Title: Equivariant analogs of the arithmetic of del Pezzo surfaces
Seminar: Algebra
Speaker: Alexander Duncan of University of South Carolina
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2017-04-04 at 5:00PM
Venue: W306
Abstract:
Given an algebraic variety X over a non-closed field, one might ask if X is rational, is unirational, has a rational point, has a Zariski-dense set of rational points, or has a 0-cycle of degree 1. All of these properties have equivariant'' generalizations to the case where the variety has an action of algebraic group G. The corresponding properties are interesting even when the base field is algebraically closed. Moreover, one can exploit this connection to establish geometric facts using arithmetic methods and vice versa. I will outline this correspondence with an emphasis on del Pezzo surfaces. In particular, I will completely characterize the equivariant analogs of the above properties for del Pezzo surfaces of degree greater than or equal to 3 over the complex numbers. I will also discuss some partial results for degrees 1 and 2 that, despite being about complex surfaces, have arithmetic ramifications
Title: Zero-Cycles on Torsors under Linear Algebraic Groups
Defense: Dissertation
Speaker: Reed Sarney of Emory University
Contact: Reed Sarney, rlgordo@emory.edu
Date: 2017-04-03 at 1:00PM
Venue: W303
Abstract:
Let $k$ be a field, let $G$ be a smooth connected linear algebraic group over $k$, and let $X$ be a $G$-torsor. Totaro asked: if $X$ admits a zero-cycle of degree $d$, does $X$ have a closed {\'e}tale point of degree dividing $d$? We give a positive answer in two cases: \begin{enumerate} \item $G$ is an algebraic torus of rank $\leq 2$ and $\textup{ch}(k)$ is arbitrary, and \item $G$ is an absolutely simple adjoint group of type $A_1$ or $A_{2n}$ and $\textup{ch}(k) \neq 2$. \end{enumerate} We also present the first known examples where Totaro's question has a negative answer.
Title: Rank-Favorable Bounds for Rational Points on Superelliptic Curves
Defense: Master's
Speaker: Noam Kantor of Emory University
Contact: Noam Kantor, noam.kantor@emory.edu
Date: 2017-04-03 at 3:00PM
Venue: MSC: N304
Abstract:
Let $C$ be a curve of genus at least two, and let $r$ be the rank of the rational points on its Jacobian. Under mild hypotheses on $r$, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on $C$ by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on $r$: when $r$ is small, there should be fewer points on $C$. In a 2013 paper, Stoll established such a rank-favorable" bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results to superelliptic curves. We also discuss a possible strategy for proving a rank-favorable bound for arbitrary curves based on the metrized complexes of Amini and Baker. Our results have stark implications for bounding the number of rational points on a curve, since $r$ is expected to be small for most" curves.
Title: Scalable Computational pathology: From Interactive to Deep Learning
Defense: Dissertation
Speaker: Michael Nalisnik of Emory University
Contact: Lee Cooper, lee.cooper@emory.edu
Date: 2017-03-30 at 10:00AM
Venue: W306
Abstract:
Advances in microscopy imaging and genomics have created an explosion of patient data in the pathology domain. Whole-slide images of histologic sections contain rich information describing the diverse cellular elements of tissue microenvironments. These images capture, in high resolution, the visual cues that have been the basis of pathologic diagnosis for over a century. Each whole-slide image contains billions of pixels and up to a million or more microanatomic objects whose appearances hold important prognostic information. Combining this information with genomic and clinical data provides insight into disease biology and patient outcomes. Yet, due to the size and complexity of the data, the software tools needed to allow scientists and clinicians to extract insight from these resources are non-existent or limited. Additionally, current methods utilizing humans is highly subjective and not repeatable. This work aims to address these shortcomings with a set of open-source computational pathology tools which aim to provide scalable, objective and repeatable classification of histologic entities such as cell nuclei.\\ \\ We first present a comprehensive interactive machine learning framework for assembling training sets for the classification of histologic objects within whole-slide images. The system provides a complete infrastructure capable of managing the terabytes worth of images, object features, annotations and metadata in real-time. Active learning algorithms are employed to allow the user and system to work together in an intuitive manner, allowing the efficient selection of samples from unlabeled pools of objects numbering in the hundreds of millions. We demonstrate how the system can be used to phenotype microvascular structures in gliomas to predict survival, and to explore the molecular pathways associated with these phenotypes. Quantitative metrics are developed to describe these structures.\\ \\ We also present a scalable, high-throughput, deep convolutional learning framework for the classification of histologic objects is presented. Due to its use of representation learning, the framework does not require the images to be segmented, instead learning optimal task-specific features in an unbiased manner. Addressing scalability, the graph-based, parallel architecture of the framework allows for the processing of large whole-slide image archives consisting of hundreds of slides and hundreds of millions of histologic objects. We explore the efficacy of various deep convolutional network architectures and demonstrate the system's capabilities classifying cell nuclei in lower grade gliomas.
Title: The Artin-Schreier Theorem in Galois Theory
Defense: Honors Thesis
Speaker: Yining Cheng of Emory University
Contact: TBA
Date: 2017-03-30 at 1:00PM
Venue: W303
Abstract:
We first list and state some basic definitions and theorems of the Galois theory of finite extensions, as well as state and prove the Kummer theory and the Artin-Schreier extensions as prerequisites. The main part of this thesis is the proof of the Artin-Schreier Theorem, which states that an algebraic closed field having finite extension with its subfield F has degree at most two and F must have characteristic 0. After the proof, we will discuss the applications for the Artin-Schreier Theorem.
Title: Theory for New Machine Learning Problems and Applications
Seminar: N/A
Speaker: Yingyu Liang of Princeton University
Contact: James Lu, jlu@mathcs.emory.edu
Date: 2017-03-30 at 4:00PM
Venue: W201
Abstract:
Machine learning has recently achieved great empirical success. This comes along with new challenges, such as sophisticated models that lack rigorous analysis, simple algorithms with practical success on hard optimization problems, and handling large scale datasets under resource constraints. In this talk, I will present some of my work in addressing such challenges.\\ \\This first part of the talk focuses on learning semantic representations for text data. Recent advances in natural language processing build upon the approach of embedding words as low dimensional vectors. The fundamental observation that empirically justifies this approach is that these vectors can capture semantic relations. A probabilistic model for generating text is proposed to mathematically explain this observation and existing popular embedding algorithms. It also reveals surprising connections to classical notions such as Pointwise Mutual Information, and allows to design novel, simple, and practical algorithms for applications such as sentence embedding.\\ \\In the second part, I will describe my work on distributed unsupervised learning over large-scale data distributed over different locations. For the prototypical tasks clustering, Principal Component Analysis (PCA), and kernel PCA, I will present algorithms that have provable guarantees on solution quality, communication cost nearly optimal in key parameters, and strong empirical performance. \\ \\Bio: Yingyu Liang is an associate research scholar in the Computer Science Department at Princeton University. His research interests are providing rigorous analysis for machine learning models and designing efficient algorithms for applications. He received a B.S. in 2008 and an M.S. in 2010 in Computer Science from Tsinghua University, and a Ph.D. degree in Computer Science from Georgia Institute of Technology in 2014. He was a postdoctoral researcher in 2014-2016 in the Computer Science Department at Princeton University.