All Seminars

Title: Energy and equilibrium for granular materials
Seminar: Analysis and Differential Geometry
Speaker: John McCuan of Georgia Institute of Technology
Contact: Vladimir Oliker, oliker@emory.edu
Date: 2018-10-16 at 4:00PM
Venue: MSC W303
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Abstract:
We consider a model for multi-grain clusters of material on a substrate involving scaled surface energies. It is shown that natural equilibrium conditions are not sufficient to ensure an energy equilibrium in certain cases. Theoretical and numerical results suggest the possibility of a gravity driven granular scale. This is joint work with Vadim Derdach, Amy Novick-Cohen, and Ray Treinen.
Title: Inductive Methods for Counting Number Fields
Seminar: Algebra
Speaker: Jiuya Wang of Duke University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-10-02 at 4:00PM
Venue: MSC W301
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Abstract:
We propose general frameworks to inductively counting number fields. By applying these methods, we prove the asymptotic distribution for extensions with Galois groups in the form of direct product or wreath product. For both way of inductions, the key ingredients are uniform estimates on the number of number fields with certain conditions. By unifying the approaches, we extend the framework to a more general set up and prove results for more general type of products. This will involve my thesis and in progress work with Robert J. Lemke Oliver and Melanie Matchett Wood.
Title: On the Erdos-Gyarfas distinct distances problem with local constraints
Seminar: Combinatorics
Speaker: Cosmin Pohoata of The California Institute of Technology
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2018-10-01 at 4:00PM
Venue: MSC E408
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Abstract:
In 1946 Erdos asked to determine or estimate the minimum number of distinct distances determined by an n-element planar point set V. He showed that a square integer lattice determines \Theta(n/\sqrt{log n}) distinct distances, and conjectured that any n-element point set determines at least n^{1−o(1)} distinct distances. In 2010-2015, Guth and Katz answered Erdos’s question by proving that any n-element planar point set determines at least \Omega(n/log n) distinct distances. In this talk, we consider a variant of this problem by Erdos and Gyarfas. For integers n, p, q with p \geq q \geq 2, determine the minimum number D(n,p,q) of distinct distances determined by a planar n-element point set V with the property that any p points from V determine at least q distinct distance. In a recent paper, Fox, Pach and Suk prove that when q = {p \choose 2} - p + 6, D(n,p,q) is always at least n^{8/7 - o(1)}. We will discuss a recent improvement of their result and some new bounds for a related (graph theoretic) Ramsey problem of Erdos and Shelah which arise. This is joint work with Adam Sheffer.
Title: Local-to-Global Extensions for Wildly Ramified Covers of Curves
Seminar: Algebra
Speaker: Renee Bell of University of Pennsylvania
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-09-25 at 4:00PM
Venue: MSC W301
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Abstract:
Given a Galois cover of curves $X \to Y$ with Galois group $G$ which is totally ramified at a point $x$ and unramified elsewhere, restriction to the punctured formal neighborhood of $x$ induces a Galois extension of Laurent series rings $k((u))/k((t))$. If we fix a base curve $Y$, we can ask when a Galois extension of Laurent series rings comes from a global cover of $Y$ in this way. Harbater proved that over a separably closed field, every Laurent series extension comes from a global cover for any base curve if $G$ is a $p$-group, and he gave a condition for the uniqueness of such an extension. Using a generalization of Artin--Schreier theory to non-abelian $p$-groups, we characterize the curves $Y$ for which this extension property holds and for which it is unique up to isomorphism, but over a more general ground field.
Title: Signed tropicalization of semialgebraic sets
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
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
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
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
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
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.