All Seminars
| 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 $F$. |
| Title: Total curvature and the isoperimetric inequality: Proof of the Cartan-Hadamard conjecture |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Mohammad Ghomi of Georgia Institute of Technology |
| Contact: Vladimir Oliker, oliker@emory.edu |
| Date: 2019-09-17 at 4:00PM |
| Venue: PAIS 220 |
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| Abstract: The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we show that this inequality also holds in spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller. The proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for the smooth approximation of the signed distance function, via inf-convolution and Reilly type formulas among other techniques. Immediate applications include sharp extensions of Sobolev and Faber-Krahn inequalities to spaces of nonpositive curvature. This is joint work with Joel Spruck. |
| Title: Structured Matrix Approximation by Separation and Hierarchy |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Difeng Cai of Emory University |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2019-09-13 at 2:00PM |
| Venue: MSC W303 |
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| Abstract: The past few years have seen the advent of big data, which brings unprecedented convenience to our daily life. Meanwhile, from a computational point of view, a central question arises amid the exploding amount of data: how to tame big data in an economic and efficient way. In the context of matrix computations, the question consists in the ability to handle large dense matrices. In this talk, I will first introduce data-sparse hierarchical representations for dense matrices. Then I will present recent development of a versatile algorithm called SMASH to operate dense matrices with optimal complexity in the most general setting. Various applications will be presented to demonstrate the advantage of SMASH over traditional approaches. |
| Title: Analytic representations of large discrete structures |
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| Seminar: Combinatorics |
| Speaker: Daniel Kral of Masaryk University and the University of Warwick |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-09-13 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The theory of combinatorial limits aims to provide analytic models representing large graphs and other discrete structures. Such analytic models have found applications in various areas of computer science and mathematics, for example, in relation to the study of large networks in computer science. We will provide a brief introduction to this rapidly developing area of combinatorics and we will then focus on several questions motivated by problems from extremal combinatorics and computer science. The two topics that we will particularly discuss include quasirandomness of discrete structures and a counterexample to a a conjecture of Lovasz, which was was one of the two most cited conjectures in the area and which informally says that optimal solutions to extremal graph theory problems can be made asymptotically unique by introducing finitely many additional constraints. |
| Title: Computing unit groups |
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| Seminar: Algebra |
| Speaker: Justin Chen of Georgia Tech |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-09-10 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: The group of units of a ring is one of the most basic, yet mysterious, invariants of the ring. Little is known about the structure of the unit group in general, much less explicit algorithms for computation, although the need for these do arise in applications such as tropical geometry. I will discuss some general questions about unit groups, and then specialize to the case of coordinate rings of classical algebraic varieties - in particular, describing explicit algorithms for computation in the case of smooth curves of low genus (rational and elliptic). This is based on joint work with Sameera Vemulapalli and Leon Zhang. |
| Title: A Step in the Right Dimension: Tensor Algebra and Applications |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Elizabeth Newman of Emory University |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2019-09-06 at 2:00PM |
| Venue: MSC W303 |
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| Abstract: As data have become more complex to reflect multi-way relationships in the real world, tensors have become essential to reveal latent content in multidimensional data. In this talk, we will focus on a tensor framework based on the M-product, a general class of tensor-tensor products which imposes algebraic structure in a high-dimensional space (Kilmer and Martin, 2011; Kernfeld et al., 2015). The induced M-product algebra inherits matrix-mimetic properties and offers provably optimal, compressed representations. To demonstrate the efficacy of working in an algebraic tensor framework, we will explore two applications: classifying data using tensor neural networks and forming sparse representations using tensor dictionaries. |