All Seminars
| 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). |
| Title: Stability and applications of quadrilaterals |
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| Seminar: Combinatorics |
| Speaker: Jie Ma of The University of Science and Technology of China |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-09-30 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: A famous theorem of Furedi states that for any integer $q \geq 15$, any $C_4$-free graph on $q^2+q+1$ vertices has at most $q(q+1)^2/2$ edges. It is well-known that this bound is tight for infinitely many integers $q$, by polarity graphs constructed from finite projective planes. In this talk, we will present a stability result of Furedi's theorem and then discuss its applications on extremal numbers of $C_4$. Joint work with Jialin He and Tianchi Yang. |
| Title: Local Immunodeficiency: Minimal Network and Stability |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Longmei Shu of Emory University |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2019-09-27 at 2:00PM |
| Venue: MSC W303 |
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| Abstract: Cooperation between different kinds of viruses, or cross-immunoreactivity, has been observed in many diseases. Instead of a one-to-one relationship between viruses and their corresponding antibodies, viruses work together. In particular, some diseases display a phenomenon where certain viruses sacrifice themselves, taking all the fire from the immune system while some other viruses stay invisible to the immune system. The fact that some viruses are protected from the immune system is called local immunodeficiency. A new math model has been developed to describe such cooperation in the viral population growth using a relationship network. Numerical simulation has already produced promising results. I analyzed some simple cases theoretically to find the smallest relationship network that has a stable and robust local immunodeficiency. |
| Title: Athens-Atlanta joint Number Theory Seminar |
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| Seminar: Algebra |
| Speaker: Jennifer Balakrishnan and Dimitris Koukoulopo of Boston U. and U. Montreal |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-09-24 at 4:00PM |
| Venue: TBA |
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| Abstract: Talks will be at the University of Georgia \\ \textbf{Jennifer Balakrishnan} (Boston University), 4:00 \\ A tale of three curves \\ We will describe variants of the Chabauty--Coleman method and quadratic Chabauty to determine rational points on curves. In so doing, we will highlight some recent examples where the techniques have been used: this includes a problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk. \\ \textbf{Dimitris Koukoulopoulos} (U. Montreal), 5:15 \\ On the Duffin-Schaeffer conjecture \\ Let S be a sequence of integers. We wish to understand how well we can approximate a ``typical'' real number using reduced fractions whose denominator lies in S. To this end, we associate to each q in S an acceptable error $\delta_q$>0. When is it true that almost all real numbers (in the Lebesgue sense) admit an infinite number of reduced rational approximations a/q, q in S, within distance $\delta_q$? In 1941, Duffin and Schaeffer proposed a simple criterion to decided whether this is case: they conjectured that the answer to the above question is affirmative precisely when the series $\sum_{q\in S} \phi(q)\delta_q$ diverges, where phi(q) denotes Euler's totient function. Otherwise, the set of ``approximable'' real numbers has null measure. In this talk, I will present recent joint work with James Maynard that settles the conjecture of Duffin and Schaeffer. |
| Title: Techniques for High-Performance Construction of Fock Matrices |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Hua Huang of Georgia Institute of Technology |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2019-09-20 at 2:00PM |
| Venue: MSC W303 |
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| Abstract: This work presents techniques for high performance Fock matrix construction when using Gaussian basis sets. Three main techniques are considered. (1) To calculate electron repulsion integrals, we demonstrate batching together the calculation of multiple shell quartets of the same angular momentum class so that the calculation of large sets of primitive integrals can be efficiently vectorized. (2) For multithreaded summation of entries into the Fock matrix, we investigate using a combination of atomic operations and thread-local copies of the Fock matrix. (3) For distributed memory parallel computers, we present a globally-accessible matrix class for accessing distributed Fock and density matrices. The new matrix class introduces a batched mode for remote memory access that can reduce synchronization cost. The techniques are implemented in an open-source software library called GTFock. |
| Title: Local-global principles for norms over semi-global fields |
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| Seminar: Algebra |
| Speaker: Sumit Chandra Mishra of Emory University |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-09-17 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Let $K$ be a complete discretely valued field with residue field $\kappa$. Let $F$ be a function field in one variable over $K$ and $\mathcal{X}$ a regular proper model of $F$ with reduced special fibre $X$ a union of regular curves with normal crossings. Suppose that the graph associated to $\mathcal{X}$ is a tree (e.g. $F = K(t)$). Let $L/F$ be a Galois extension of degree $n$ with Galois group $G$ and $n$ coprime to char$(\kappa)$. Suppose that $\kappa$ is algebraically closed field or a finite field containing a primitive $n^{\rm th}$ root of unity. Then we show that an element in $F^*$ is a norm from the extension $L/F$ if it is a norm from the extensions $L\otimes_F F_\nu/F_\nu$ for all discrete valuations $\nu$ of $F$. |