All Seminars
| Title: Integers represented by positive-definite quadratic forms and Petersson inner products |
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| Seminar: Algebra |
| Speaker: Jeremy Rouse of Wake Forest University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-11-05 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: We give a survey of results about the problem of determining which integers are represented by a given quaternary quadratic form $Q$. A necessary condition for $Q(x_1,x_2,x_3,x_4)$ to represent $n$ is for the equation $Q(x_1,x_2,x_3,x_4) = n$ to have a solution with $x_1,x_2,x_3,x_4 \in Z_p$ for all $p$. But even when $n$ is sufficiently large, this is not sufficient for $Q$ to represent $n$. The form $Q$ is anisotropic at the prime $p$ if for $x_1,x_2,x_3,x_4 \in Z_p$, $Q(x_1,x_2,x_3,x_4) = 0$ implies that $x_1=x_2=x_3=x_4=0$. Suppose that $A$ is the Gram matrix for $Q$ and $D(Q) = \det(A)$. We show that if $n >> D(Q)^{6+\epsilon}$, $n$ is locally represented by $Q$, but $Q$ fails to represent $n$, then there is an anisotropic prime $p$ so that $p^2 | n$ and $np^{2k}$ is not represented by $Q$ for any $k \geq 1$. We give sharper results when $D(Q)$ is a fundamental discriminant and discuss applications to universality theorems like the 15 and 290 theorems of Bhargava and Hanke. |
| Title: Quasirandomness and hypergraph regularity, II |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of The University of Hamburg and Yale University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-11-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: This is a continuation of the last talk. We shall focus on the proof of the equivalence of DISC and DEV in 3-uniform hypergraphs. While the simple implication "DEV implies DISC" easily follows by appropriate applications of the Cauchy-Schwarz inequality, the opposite implication is based on the regularity method for hypergraphs. |
| Title: A refined Brill-Noether theory over Hurwitz spaces |
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| Seminar: Algebra |
| Speaker: Hannah Larson of Stanford Univeristy |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-10-29 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: The celebrated Brill-Noether theorem says that the space of degree $d$ maps of a general genus $g$ curve to $\mathbb{P}^r$ is irreducible. However, for special curves, this need not be the case. Indeed, for general $k$-gonal curves (degree $k$ covers of $\mathbb{P}^1$), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map $C \rightarrow \mathbb{P}^1$, using the splitting type of push forwards of line bundles to $\mathbb{P}^1$. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general $k$-gonal curves. |
| Title: Patient-Specific Modeling in Cardiac Electrophysiology: Parameter Estimation and Personalization |
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| Defense: Dissertation |
| Speaker: Alessandro Barone of Emory University |
| Contact: Alessandro Barone, alessandro.barone@emory.edu |
| Date: 2019-10-25 at 3:10PM |
| Venue: MSC W303 |
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| Abstract: Computational modeling in cardiac electrophysiology (EP) has long played a central role in the study of physio-pathological dynamics of electrical propagation. One of the most significant challenges to face is the translation process of numerical (in silico) investigations to clinical practice. In silico simulations can potentially impact the quality of cardiac arrhythmia therapy, reducing the risk of in vivo testing. However, the clinical use of virtual experiments is hindered by the need of customization of mathematical models to patient-specific data. The personalization process involves the fine tuning of many model parameters, that cannot be measured directly, via accurate and efficient data assimilation techniques. This work is particularly focused on the estimation of cardiac conductivities, crucial parameters of the Bidomain and Monodomain models – currently the most used mathematical descriptions of cardiac electrical behavior. This Thesis addresses the challenge described above yielding four main contributions. (1) We perform an extensive and thorough synthetic and experimental validation of the deterministic variational data assimilation method proposed by Yang and Veneziani in 2015 to retrieve conductivities from potential recordings. The results demonstrate that the procedure provides accurate space-dependent conductivity estimates that reproduce most of the observed dynamics. (2) The Proper Generalized Decomposition (PGD) reduced-order model technique is investigated for the first time in EP to improve the efficiency of the variational technique. Relying on the off-line/on-line paradigm and without the need of any preliminary knowledge of the high-fidelity solution, we show in 2D and 3D settings that the strategy enables nearly real-time estimation preserving reasonable accuracy. (3) With the goal of assessing the robustness of the results, we propose a statistical formulation of the estimation problem for Monodomain conductivities. Exploiting the computational convenience of the on-line PGD solution, the methodology allows a reliable quantification of the uncertainty of two-dimensional estimates. (4) Using a virtual personalized heart model efficiently reconstructed from high resolution MRI images and ECG data via a physics-based reduced-order model approach, we perform a preliminary study of the induction of ventricular electrical anomalies with respect to different conduction properties in view of optimizing arrhythmia treatments in silico. |
| Title: Isentropic Approximation |
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| Seminar: Analysis and PDEs |
| Speaker: Ronghua Pan of Georgia Institute of Technology |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2019-10-24 at 3:00PM |
| Venue: MSC E308A |
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| Abstract: In the study of compressible flows, the isentropic model was often used to replace the more complicated full system when the entropy is near a constant. This is based on the expectation that the corresponding isentropic model is a good approximation to the full system when the entropy is sufficiently close to the constant. We will discuss the mathematical justification of isentropic approximation in Euler flows and in Navier-Stokes-Fourier flows. This is based on the joint work with Y. Chen, J. Jia, and L. Tong. |
| Title: On the number of small prime power residues |
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| Seminar: Algebra |
| Speaker: Kubra Benli of University of Georgia |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-10-22 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Let $p$ be a prime number. For each positive integer $k\geq 2$, it is widely believed that the smallest prime that is a $k$th power residue modulo $p$ should be $O(p^{\epsilon})$, for any $\epsilon>0$. Elliott has proved that such a prime is at most $p^{\frac{k-1}{4}+\epsilon}$, for each $\epsilon>0$. In this talk we will discuss the distribution of the prime $k$th power residues modulo $p$ in the range $[1, p]$, with a more emphasis on the subrange $[1,p^{\frac{k-1}{4}+\epsilon}]$, for $\epsilon>0$. |
| Title: Asynchronous Iterative Methods for Solving Sparse Linear Systems |
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| Seminar: Computational Mathematics |
| Speaker: Jordi Wolfson-Pou of Georgia Institute of Technology |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2019-10-18 at 2:00PM |
| Venue: MSC W303 |
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| Abstract: Reducing synchronization in iterative methods for solving large sparse linear systems may become one of the most important goals for such solvers on exascale computers. Research in asynchronous iterative methods has primarily considered the asymptotic behavior of basic iterative methods, e.g., Jacobi. However, practical behavior of basic iterative methods has not been extensively studied, and little research has been done on asynchronous multigrid methods. In this talk, the transient behavior of asynchronous Jacobi is examined. A simplified model of asynchronous Jacobi is analyzed, and results from shared and distributed memory experiments are presented to support the analysis. Two important results are shown. First, if a process is slower than all others (delayed in its computation), asynchronous Jacobi can continue to reduce the residual, even if the number of delayed iterations is similar in value to the size of the matrix. This result demonstrates how useful asynchronous Jacobi can be on heterogeneous architectures or for problems with large load imbalances, where some processes can be significantly slower than others. Second, asynchronous Jacobi can converge when synchronous Jacobi does not, and the convergence rate of asynchronous Jacobi can increase with increased concurrency. This is an important result when considering the amount of concurrency in future exascale machines; removing synchronization points not only reduces overall wall-clock time on its own, but also can allow convergence in fewer iterations, which further reduces the overall execution time. Asynchronous multigrid methods are also examined in this talk. Models of asynchronous additive multigrid methods are introduced, and a parallel implementation of asynchronous multigrid is presented. Experimental results show that asynchronous multigrid can exhibit grid-size independent convergence and can be faster than classical multigrid in terms of solve wall-clock time. |
| Title: Derived Categories, Arithmetic, and Rationality Questions |
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| Seminar: Algebra |
| Speaker: Alicia Lamarche of University of South Carolina |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-10-08 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety $X$, to what extent can $D^b(X)$ be used as an invariant to answer rationality questions? In particular, what properties of $D^b(X)$ are implied by $X$ being rational, stably rational, or having a rational point? On the other hand, is there a property of $D^b(X)$ that that implies that $X$ is rational, stably rational, or has a rational point? \\ In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full \'etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin. |
| Title: Quasirandomness and hypergraph regularity |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of The University of Hamburg and Yale University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-10-04 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The regularity method was pioneered by Szemeredi for graphs and is an important tool in extremal combinatorics. Over the last two decades, several extensions to hypergraphs were developed which were based on seemingly different notions of quasirandom hypergraphs. We show that the concepts behind these approaches are essentially equivalent. This joint work with B. Nagle and V. Rodl. |
| Title: Lines on cubic surfaces |
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| Seminar: Algebra |
| Speaker: Eva Bayer Fluckinger of EPFL |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-10-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: The aim of this talk is to give a formula expressing the trace form associated with the 27 lines of a cubic surface \\ (joint with Jean-Pierre Serre). |