All Seminars

Title: Athens-Atlanta joint Number Theory Seminar
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
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
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
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
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
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
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
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.
Title: Spanning subgraphs in uniformly dense and inseparable graphs
Seminar: Combinatorics
Speaker: Mathias Schacht of The University of Hamburg and Yale University
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2019-09-06 at 4:00PM
Venue: MSC W301
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Abstract:
We consider sufficient conditions for the existence of k-th powers of Hamiltonian cycles in n-vertex graphs G with minimum degree cn for arbitrarily small c >0 . About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that c=k/k+1 suffices for large n. For smaller values of c the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density d>0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least c, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalises recent results of Staden and Treglown.
Title: Modular linear differential equations
Seminar: Algebra
Speaker: Kiyokazu Nagatomo of Osaka University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-09-03 at 4:00PM
Venue: MSC W303
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Abstract:
The most naive definition of \textit{modular linear differential equations} (MLDEs) would be linear differential equations whose space of solutions are invariant under the weight $k$ slash action of $\Gamma_1=SL_2(\mathbb{Z})$, for some $k$. Then under an analytic condition for coefficients functions and the Wronskians of a~basis of the space of solutions of equations, we have (obvious) expressions of MLDEs as: \[ L(f) \,=\,\mathfrak{d}_k^n(f)+\sum_{i=2}^nP_{2i}\mathfrak{d}_k^{n-i}(f) \] where $P_{2i}$ is a modular form of of weight $2i$ on $SL_2(\mathbb{Z})$ and $\mathfrak{d}_k(f)$ is the \textit{Serre derivative}. (We could replace $\Gamma$ by a Fuchsian subgroup of $SL_2(\mathbb{R})$ and allow the modular forms $P_{2i}$ to be meromorphic.) However, the iterated Serre derivative $\mathfrak{d}_k^n(f)$ (called a ``higher Serre derivation'' because as an operator it preserves modulality) is very complicated since it involves the Eisenstein series $E_2$. MLDEs, of course, can be given in the form % \[ % \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f)\quad\text{where $D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}$.} % \] \[ \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f) \] where \[ D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}. \] Then it is not easy to know if the equation above is an MLDE except the fact that $Q_i$ are quasimodular forms. Very recently, Y.~Sakai and D.~Zagier (my collaborators) found formulas of $\mathsf{L}(f)$ by using the Rankin--Cohen products between $f$ and $g_i$. This is a modular form of weight $2i$, which is a linear function of the differential of~$Q_{j}$. Moreover, there are \textit{inversion formulas} which express $Q_i$ as a linear function of the derivatives of $g_{j}$. The most important fact is that the order $n$ and $n-1$ parts are equal to the so-called higher Serre derivative in the sense of Kaneko and Koike, where the group is $\Gamma_1$. (This holds for any Fuchsian group.) \\ Finally, the most important nature of my talk is that I will use a \textbf{blackboard} instead of \textbf{slides}ss.