All Seminars
| Title: Modules for subgroups of $M_{24}$ with meromorphic trace functions |
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| Seminar: Algebra |
| Speaker: Lea Beneish of Emory |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-03-19 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: We give vertex operator algebra constructions of infinite-dimensional graded modules for certain subgroups of $M_{24}$. We begin by proving the existence of a module for $M_{24}$, whose trace functions are weight two quasimodular forms. Existence of this module implies certain divisibility conditions on the number of $\mathbb{F}_p$ points on Jacobians of modular curves. We write similar expressions which we show are trace functions for modules of cyclic groups with arbitrary prime order. These expressions can be modified so we can give a vertex operator algebra construction for these modules. However, this modification comes at the expense of any relationship to Jacobians of modular curves. By adding a term to the quasimodular forms, we obtain meromorphic Jacobi forms and prove the existence of a module for $M_{24}$ with these trace functions. For certain subgroups of $M_{24}$, we give a vertex operator algebra construction for a module with these trace functions. In particular, these module constructions give an explicit realization of the relationship between the trace functions and divisibility conditions the number of $\mathbb{F}_p$ points on Jacobians of modular curves. |
| Title: On the relativistic Landau equation |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Maja Taskovic of University of Pennsylvania |
| Contact: David Borthwick, dborthw@emory.edu |
| Date: 2019-02-28 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In kinetic theory, a large system of particles is described by the particle density function. The Landau equation, derived by Landau in 1936, is one such example. It models a dilute hot plasma with fast moving particles that interact via Coulomb interactions. This model does not include the effects of Einstein’s theory of special relativity. However, when particle velocities are close to the speed of light, which happens frequently in a hot plasma, then relativistic effects become important. These effects are captured by the relativistic Landau equation, which was derived by Budker and Beliaev in 1956. We study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. The difficulty of the problem lies in the extreme complexity of the kernel in the relativistic collision operator. We present a new decomposition of such kernel. This is then used to prove the global Entropy dissipation estimate, the propagation of any polynomial moment for a weak solution, and the existence of a true weak solution for a large class of initial data. This is joint work with Robert M. Strain. |
| Title: A new approach to bounding $L$-functions |
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| Seminar: Algebra |
| Speaker: Jesse Thorner of Stanford |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-02-26 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: An $L$-function is a type of generating function with multiplicative structure which arises from either an arithmetic-geometric object (like a number field, elliptic curve, abelian variety) or an automorphic form. The Riemann zeta function $\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ is the prototypical example of an $L$-function. While $L$-functions might appear to be an esoteric and special topic in number theory, time and again it has turned out that the crux of a problem lies in the theory of these functions. Many equidistribution problems in number theory rely on one's ability to accurately bound the size of $L$-functions; optimal bounds arise from the (unproven!) Riemann Hypothesis for $\zeta(s)$ and its extensions to other $L$-functions. I will discuss some motivating equidistribution problems along with recent work (joint with K. Soundararajan) which produces new bounds for $L$-functions by proving a suitable "statistical approximation" to the (extended) Riemann Hypothesis. |
| Title: A pair-degree condition for Hamiltonian cycles in 3-graphs |
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| Seminar: Combinatorics |
| Speaker: Bjarne Schuelke of The University of Hamburg |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-02-25 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: For graphs, the fundamental theorem by Dirac ensuring the existence of Hamiltonian cycles in a graph G with large minimum degree was generalised by Chvatal to a characterisation of those degree sequences that force a graph to have a Hamiltonian cycle. After a Dirac-like result was proved for 3-uniform hypergraphs by Rodl, Rucinski, and Szemeredi, we will discuss a first step towards a more general characterisation of pair degree matrices of 3-uniform hypergraphs that force Hamiltonicity. The presented result can be seen as a 3-uniform analogue of a result on graphs by Posa that is more general than Dirac's and is generalised by Chvatal's theorem. In particular we will prove that for each c > 0 there exists an n such that the following holds: If H is a 3-uniform hypergraph with vertex set {1,...,n} and d(i,j) > min { (i+j)/2, n/2 } + cn holds for all pairs of vertices, then H contains a tight Hamilton cycle. |
| Title: Filtering techniques for eigenvalue problems |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Yousef Saad of University of Minnesota |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2019-02-22 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: The solution of large symmetric eigenvalue problems is central to applications ranging from electronic structure calculations to the study of vibrations in mechanical systems. A few of these applications require the computation of a large number of eigenvalues and associated eigenvectors. For example, when dealing with excited states in quantum mechanics, it is not uncommon to seek a few tens of thousands of eigenvalues of matrices of sizes in the tens of millions. In such situations it is imperative to resort to `spectrum slicing' strategies, i.e., strategies that extract slices of the spectrum independently. The presentation will discuss a few techniques in this category, namely those based on a combination of filtering (polynomial, rational) and standard projection methods (Lanczos, subspace iteration). Filtering consists of computing eigenvalues and vectors of a matrix of the form $B=f(A)$, where $f$ is typically a polynomial or rational function. With the mapping $f$ the wanted eigenvalues of the original matrix are transformed in such a way that they become easier to extract. This particular area blends ideas from approximation theory with standard matrix algorithms. The presentation will emphasize rational filtering and will discuss some recent work on nonlinear eigenvalue problems. |
| Title: Turan-type problems for bipartite graphs and hypergraphs |
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| Colloquium: Combinatorics |
| Speaker: Liana Yepremyan of University of Oxford |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-02-22 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: A central problem of extremal combinatorics is to determine the Turan number of a given graph or a hypergraph F, i.e. the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. For graphs, the asymptotic answer to this question is given by the celebrated Erdos-Stone theorem. However, it is wide open for bipartite graphs and for hypergraphs. For bipartite graphs, a beautiful conjecture of Erdos and Simonovits says that every rational number between 1 and 2 must appear as an exponent of some Turan number. Despite decades of effort, there are only few families for which this conjecture is known to be true. We will discuss some recent progress on this as well as related so-called supersaturation problems. For hypergraphs, since the problem was introduced over sixty years ago, it has only been solved for relatively few hypergraphs F. Many of these results were found very recently by means of the stability method, which has brought new life to research in a challenging area. We will discuss a variation of this method utilizing the Lagrangian function (we call it local stability method) which gives a generic unified approach for obtaining exact Turan numbers from asymptotic results and allowed us to enlarge the list of known Turan numbers of hypergraphs, in particular solving a conjecture of Frankl and Furedi from the 80's. Various parts of the work are joint with Sergey Norin, Adam Bene Watts, Tao Jiang and Jie Ma. |
| Title: Inverse problems for nonlinear hyperbolic equations |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Yiran Wang of Stanford University |
| Contact: David Borthwick, dborthw@emory.edu |
| Date: 2019-02-21 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: We consider the inverse problem of determining coefficients of nonlinear hyperbolic equations from measurements of wave responses. The problem has wide applications for example in general relativity (the Einstein equations) and seismology (the elastic wave equations). It is known that the nonlinear interaction of waves could generate new responses and such interactions have been studied using plane waves in the literature. In this talk, we analyze the nonlinear response and give a precise characterization using microlocal methods, and show how nonlinearity helps us to solve the inverse problem. |
| Title: Indefinite theta functions and quantum modular forms of higher depth |
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| Seminar: Algebra |
| Speaker: Jonas Kaszian of Mathematics Institute University of Cologne |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-02-21 at 5:00PM |
| Venue: MSC E408 |
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| Abstract: In the last years, the understanding of indefinite theta functions has advanced through the work of multiple authors, exhibiting them as mock modular forms of higher depth. We will present results (joint with Kathrin Bringmann and Antun Milas) where we used the completions of indefinite theta functions to show quantum modular properties of certain higher rank false theta functions (appearing naturally in representation theory). |
| Title: Topics in the analytic theory of L-functions and harmonic Maass forms. |
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| Defense: Dissertation |
| Speaker: Ian Wagner of Emory University |
| Contact: Ian Wagner, ian.wagner@emory.edu |
| Date: 2019-02-19 at 8:30AM |
| Venue: MSC E408 |
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| Abstract: This thesis presents several new results in the theory of $L-functions$, modular forms, and harmonic Maass forms. We prove a general congruence result for mixed weight modular forms using facts about direct products of Galois representations. As an application we prove explicit congruences for the conjugacy growth series of wreath products of finite groups and finitary permutations groups. We continue studying the $p-adic$ properties of modular forms and begin to answer a question of Mazur's about the existence of an eigencurve for harmonic Maass forms by constructing two infinite familes of harmonic Maass Hecke eigenforms, and then assemble these forms to produce $p-adic$ Hecke eigenlines. We also study the hyperbolicity of doubly infinite families of polynomials related to the partition function and general L-functions. As a result we prove when the partition function satisfies the higher Turan inequalities and provide evidence for the Generalized Riemann Hypothesis for suitable L-functions. We also show that these L-functions satisfy the Gaussian Unitary Ensemble random matrix model in derivative aspect. Finally, we study the recent connection between sphere packing and energy optimization and modular forms. We construct a number of infinite families of Schwartz functions using modular forms, which are eigenfunctions of the Fourier transform. |
| Title: Partitions, Prime Numbers, and Moonshine |
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| Defense: Dissertation |
| Speaker: Madeline Dawsey of Emory University |
| Contact: Madeline Dawsey, madeline.locus@emory.edu |
| Date: 2019-02-19 at 10:00AM |
| Venue: MSC E408 |
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| Abstract: Abstract: We prove new number theoretic results in combinatorics, analytic number theory, and representation theory. In particular, in combinatorics we prove conjectured inequalities regarding the Andrews $spt-function$ by effectively estimating $spt(n)$ using new methods from the theory of quadratic forms. We also provide recursion relations for the coefficients of conjugacy growth series for wreath products of finitary permutation groups, which provide some measure of the algebraic complexity of these groups. In analytic number theory, we reproduce the Chebotarev density of certain subsets of prime numbers through a restricted infinite sum involving the Mobius function. Finally, we refine the theory of moonshine so that the modular forms associated to the representation theory of finite groups are in fact group isomorphism invariants. We obtain this “higher width moonshine” for all finite groups by employing the classical Frobenius r-characters, which we prove satisfy necessary orthogonality relations. |