All Seminars
| Title: Signed tropicalization of semialgebraic sets |
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| Seminar: Algebra |
| Speaker: Philipp Jell of Georgia Tech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-09-18 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Tropicalzation of algebraic varieties has been proven to be a powerful tool in complex, real and arithmetic geometry. Different methods to tropicalize are shown to be equivalent by the tropical fundamental theorem. I will explain this theorem and report on joint work in progress with Claus Scheiderer and Josephine Yu. In this work we generalize the fundamental theorem to so call "signed tropicalizations" of semialgebraic sets and define the signed non-archimedean analytification of a semialgebraic set. |
| Title: Moonshine for Finite Groups |
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| Seminar: Algebra |
| Speaker: Madeline Locus Dawsey of Emory University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-09-11 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: {\it Weak moonshine} for a finite group $G$ is the phenomenon where an infinite dimensional graded $G$-module $$V_G=\bigoplus_{n\gg-\infty}V_G(n)$$ has the property that its trace functions, known as McKay-Thompson series, are modular functions. Recent work by Dehority, Gonzalez, Vafa, and Van Peski established that weak moonshine holds for every finite group. Since weak moonshine only relies on character tables, which are not isomorphism class invariants, non-isomorphic groups can have the same McKay-Thompson series. We address this problem by extending weak moonshine to arbitrary width $s\in\mathbb{Z}^+$. For each $1\leq r\leq s$ and each irreducible character $\chi_i$, we employ Frobenius' $r$-character extension $\chi_i^{(r)} \colon G^{(r)}\rightarrow\mathbb{C}$ to define McKay-Thompson series of $V_G^{(r)}:=V_G\times\cdots\times V_G$ ($r$ copies) for each $r$-tuple in $G^{(r)}:=G\times\cdots\times G$ ($r$ copies). These series are modular functions. We find that {\it complete} width 3 weak moonshine always determines a group up to isomorphism. Furthermore, we establish orthogonality relations for the Frobenius $r$-characters, which dictate the compatibility of the extension of weak moonshine for $V_G$ to width $s$ weak moonshine. |
| Title: Research Spotlights |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Alessandro Veneziani and Yuanzhe Xi of Emory University |
| Contact: Lars Ruthotto, lruthotto@emory.edul |
| Date: 2018-09-07 at 2:00PM |
| Venue: MSC N302 |
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| Abstract: The scientific computing group at Emory welcomes all for the second round of research spotlights. This week, Dr. Veneziani and Dr. Xi will present their groups’ work. Dr. Veneziani will give an overview of his work on numerical partial differential equations and their impact on medical decision-making. Dr. Xi will present new and ongoing work in high-performance computing for numerical linear algebra with applications in physics and machine learning. These high-level talks will not be too technical, and faculty and students working in other but related fields are encouraged to attend. |
| Title: Research Spotlights |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: James Nagy and Lars Ruthotto of Emory University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2018-08-31 at 2:00PM |
| Venue: MSC N302 |
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| Abstract: The scientific computing group at Emory will kick off the new academic year by short overviews of the faculty's ongoing research. This week, Dr. Nagy and Dr. Ruthotto will be in the spotlight. Dr. Nagy will give his overview of his group's efforts aiming at developing more efficient numerical linear algebra techniques for large-scale image processing. Dr. Ruthotto will present recent advances and open problems at the interface between PDEs, optimization, and machine learning. These high-level talks will not be too technical and faculty and students working in other but related fields are encouraged to attend. |
| Title: Efficient Solvers for Nonlinear Problems in Imaging |
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| Defense: Dissertation |
| Speaker: James L Herring of Emory University |
| Contact: James Herring, james.lincoln.herring@emory.edu |
| Date: 2018-05-16 at 3:00PM |
| Venue: MSC W301 |
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| Abstract: Nonlinear inverse problems arise in numerous imaging applications, and solving them is often difficult due to ill-posedness and high computational cost. In this work, we introduce tailored solvers for several nonlinear inverse problems in imaging within a Gauss-Newton optimization framework.\\ \\ We develop a linearize and project (LAP) method for a class of nonlinear problems with two (or more) sets of coupled variables. At each iteration of the Gauss-Newton optimization, LAP linearizes the residual around the current iterate, eliminates one block of variables via a projection, and solves the resulting reduced dimensional problem for the Gauss-Newton step. The method is best suited for problems where the subproblem associated with one set of variables is comparatively well-posed or easy to solve. LAP supports iterative, direct, and hybrid regularization and supports element-wise bound constraints on all the blocks of variables. This offers various options for incorporating prior knowledge of a desired solution. We demonstrate the advantages of these characteristics with several numerical experiments. We test LAP for two and three dimensional problems in super resolution and MRI motion correction, two separable nonlinear least squares problems that are linear in one block of variables and nonlinear in the other. We also use LAP for image registration subject to local rigidity constraints, a problem that is nonlinear in all sets of variables. These two classes of problems demonstrate the utility and flexibility LAP method.\\ \\ We also implement an efficient Gauss-Newton optimization scheme for the problem of phase recovery in bispectral imaging, a univariate nonlinear inverse problem. Using a fixed approximate Hessian, matrix-reordering, and stored matrix factors, we accelerate the Gauss-Newton step solve, resulting in a second-order optimization method which outperforms first-order methods in terms of cost per iteration and solution quality. |
| Title: Computational and Predictive Models for Brain Imaging Studies |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Yi Hong of The University of Georgia |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2018-05-04 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: Uncovering anatomical changes over time is important in understanding brain development, aging, and disease progression. Data for these studies, image and shape time series, have complex structures and are best treated as elements of non-Euclidean spaces. In this talk, I present our non-Euclidean models for image and shape regression to estimate the time-varying trend of a population by generalizing Euclidean regression and to predict a subject-specific trend by integrating image geometry with deep neural networks. I also introduce a complementary segmentation network that preprocesses image scans and accurately extracts the brain from both normal and pathological images. Our experimental results demonstrated the promise of our models in the study of normal brain aging and Alzheimer’s disease. |
| Title: Optimization Methods for Training Neural Networks |
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| Colloquium: Computational Mathematics |
| Speaker: Jorge Nocedal of Northwestern University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2018-04-27 at 3:00PM |
| Venue: MSC E208 |
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| Abstract: Most high-dimensional nonconvex optimization problems cannot be solved to optimality. It has been observed, however, that deep neural networks have a benign geometry that permits standard optimization methods to find acceptable solutions. However, solution times can be exorbitant. In addition, not all minimizers of the neural network loss functions are equally desirable, as some lead to prediction systems with better generalization properties than others. In this talk we discuss classical and new optimization methods in the light of these observations, and conclude with some open questions. BIO: Jorge Nocedal is the Walter P. Murphy Professor in the Department of Industrial Engineering and Management Sciences at Northwestern University. His research is in optimization, both deterministic and stochastic, and with emphasis on very large-scale problems. His current work is driven by applications in machine learning. He is a SIAM Fellow, was awarded the 2012 George B. Dantzig Prize, and the 2017 Von Neumann Theory Prize for contributions to theory and algorithms of optimization. |
| Title: Lattice Point Counting and Arithmetic Statistics |
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| Seminar: Algebra |
| Speaker: Frank Thorne of University of South Carolina |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2018-04-24 at 4:00PM |
| Venue: W304 |
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| Abstract: The Gauss Circle Problem asks how many lattice points are contained in a circle centered at the origin or radius R. A simple geometric argument establishes that this count is approximated by the area $\pi R^2$, with an error bounded by the perimeter $O(R)$. \\ ``Arithmetic statistics" is about arithmetic objects -- number fields, ideal class groups, and so on. Bhargava and many others have recently proved spectacular theorems by parametrizing such objects in terms of lattice points, and then using geometry to counting the lattice points. \\ Meanwhile, harmonic analysts have long known that you can do better than an error of $O(R)$ in Gauss's circle problem. I will describe a program to import such improvements into arithmetic statistics, and give an overview of the number theoretic results we hope to obtain. \\ This is ongoing joint work with Theresa Anderson and Takashi Taniguchi. |
| Title: A proof of a conjecture of Erd\H{o}s et al. about subgraphs of minimum degree k |
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| Seminar: Combinatorics |
| Speaker: Lisa Sauermann of Stanford University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2018-04-23 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Erd\H{o}s, Faudree, Rousseau and Schelp observed the following fact for every fixed integer $k \geq 2$: Every graph on $n \geq k-1$ vertices with at least $(k-1)(n-k+2)+(k-2)(k-3)/2$ edges contains a subgraph with minimum degree at least k. However, there are examples in which the whole graph is the only such subgraph. Erdos et al. conjectured that having just one more edge implies the existence of a subgraph on at most $(1-\epsilon_k)n$ vertices with minimum degree at least $k$, where $\epsilon_k>0$ depends only on $k$. In this talk, we will sketch a proof of this conjecture. The proof relies on ideas from a paper of Mousset, Noever and $\check{S}kori\acute{c}$. We will discuss these ideas and how they can be extended to give a proof of the full conjecture. |
| Title: Data Warehousing and Ensemble Learning of Omics Data |
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| Graduate Student Seminar: Computer Science |
| Speaker: Xiaobo Sun of Emory University |
| Contact: TBA |
| Date: 2018-04-20 at 1:00PM |
| Venue: Room GCR311 of Department of Biostatistics |
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| Abstract: The development and application of high-throughput genomics technologies has resulted in massive quantities of diverse omics data that continue to accumulate rapidly. These rich datasets offer unprecedented and exciting opportunities to address long standing questions in biomedical research. However, our ability to explore and query the content of diverse omics data is very limited. Existing dataset search tools rely almost exclusively on the metadata. A text-based query for gene name(s) does not work well on datasets where the vast majority of their content is numeric. To overcome this barrier, we have developed Omicseq, a novel web-based platform that facilitates the easy interrogation of omics datasets holistically, beyond just metadata to improve “findability”. The core component of Omicseq is trackRank, a novel algorithm for ranking omics datasets that fully uses the numerical content of the dataset to determine relevance to the query entity. The Omicseq system is supported by a scalable and elastic, NoSQL database that hosts a large collection of processed omics datasets. In the front end, a simple, web-based interface allows users to enter queries and instantly receive search results as a list of ranked datasets deemed to be the most relevant. Omicseq is freely available at http://www.omicseq.org. |