All Seminars
| Title: Eigenvalues of the Laplace Operator on Quantum Graphs |
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| Defense: Dissertation |
| Speaker: Haozhe Yu of Emory University |
| Contact: Haozhe Yu, haozhe.yu@emory.edu |
| Date: 2023-06-20 at 3:00PM |
| Venue: MSC W303 |
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| Abstract: This thesis focuses on estimates of eigenvalues on compact quantum graphs. On quantum graphs with all standard vertex condition, we prove an upper bound of eigenvalues based on the Davies inequality. We also prove some improvements of known upper bounds for eigenvalue gaps and ratios for metric trees. We finally establish a lower bound of eigenvalue gaps based on the idea of the weighted Cheeger constant on graphs with at least one Dirichlet vertex. |
| Title: Direct and inverse problems for elastic dislocations in geophysics |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Anna L. Mazzucato of Penn State University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2023-04-21 at 11:00AM |
| Venue: MSC W303 |
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| Abstract: I will discuss a model for dislocations in an elastic medium, modeling faults in the Earth's crust. The direct problem consists in solving a non-standard boundary value/interface problem for in-homogeneous, possibly anisotropic linear elasticity with piecewise-Lipschitz coefficients. The non-linear inverse problem consists in determining the fault surface and slip vector from displacement measurements made at the surface. In applications, these come from GPS arrays and satellite interferometry. We establish uniqueness for the inverse problem under some geometric conditions, using unique continuation results for systems. We also derive a shape derivative formula for an iterative reconstruction algorithm. This is joint work with Andrea Aspri (Milan University, Italy), Elena Beretta (NYU-Abu Dhabi), and Maarten de Hoop (Rice). |
| Title: Abhyankar’s Conjecture and the Inverse Galois Problem |
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| Seminar: Algebra |
| Speaker: Jim Phillips of Seton Hall University |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-04-18 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Abhyankar’s Conjecture qualifies which finite groups occur as Galois groups of certain covers of curves in positive characteristic, thus providing an affirmative answer to a type of inverse Galois problem. In this talk, I will give an overview of the history of the conjecture and describe some current work that addresses a related question: given a cover of the projective line in the situation of Abhyankar’s Conjecture with one branch point, what can be said of the associated ramification jumps? |
| Title: Speeding up Krylov subspace methods for matrix functions via randomization |
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| Seminar: CODES@emory |
| Speaker: Alice Cortinovis of Stanford University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-04-13 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: In this talk we consider the computation of the action of a matrix function f(A), such as the matrix exponential or the matrix square root, on a vector b. For a general matrix A, this can be done by computing the compression of A onto a suitable Krylov subspace. Such compression is usually computed by forming an orthonormal basis of the Krylov subspace using the Arnoldi method. In this talk, we propose to compute (non-orthonormal) bases in a faster way and to use a fast randomized algorithm for least-squares problems to compute the compression of A onto the Krylov subspace. We present some numerical examples which show that our algorithms can be faster than the standard Arnoldi method while achieving comparable accuracy. |
| Title: Recovery of the nonlinearity from the modified scattering map |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Gong Chen of Georgia Institute of Technology |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2023-04-13 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We consider the problem of recovering the nonlinearity in a nonlinear Schrödinger equation from scattering data, a problem for which there is a relatively large literature. We consider a new situation in which the equation does not admit standard scattering, but instead features the modified scattering behavior with logarithmic phase correction. We prove that even in this case, the modified scattering data suffices to determine the unknown nonlinearity. |
| Title: High-Definition Hedgehogs |
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| Seminar: Combinatorics |
| Speaker: Eoin Peter Hurley of |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-04-11 at 4:00PM |
| Venue: MSC E408 |
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| Abstract: Suppose I want to edge colour a complete hypergraph so that I avoid a fixed hypergraph H as a monochromatic subgraph. How large can my complete graph be? And how does the answer depend on the number of colours I use? Certainly I need to use 2 colours for the question to be non-trivial, but what if I use 3 or 5 or 173 colours, does this help? In the case of graphs, the answer is “not much”, but, in a breakthrough paper "Hedgehogs Are Not ColorBlind", Conlon, Fox and Rödl show that for a certain family of hypergraphs, called hedgehogs, it makes an exponential difference! This observation has ramifications for some of the most important questions in quantitative Ramsey theory. Further, Conlon, Fox and Rödl asked, can we replace the exponential difference by a difference of a tower of arbitrary height? We answer this in the affirmative, showing that Hedgehogs see in high definition. In this talk, I will discuss the main ideas and consequences of the result, including some surprising conjectures. Joint work with Quentin Dubroff, António Girão and Corrine Yap. |
| Title: Local data of elliptic curves under quadratic twists and isogeny |
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| Seminar: Algebra |
| Speaker: Manami Roy of Fordham University |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-04-11 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: For a minimal proper regular model of a rational elliptic curve E at a prime p, we can compute the local data at p, which includes the special fiber of the minimal model (i.e., Néron type), the exponent appearing at the prime p in the of the conductor of E, and the local Tamagawa number at p. We will discuss how the Kodaira-Néron types and the local Tamagawa numbers of rational elliptic curves change over isogeny graphs. To answer this question, we examine how local data of rational elliptic curves change under quadratic twists. Our aim is to answer an open problem on how the Kodaira-Néron types and the local Tamagawa numbers of isogenous rational elliptic curves with wild ramification change under 2- or 3-isogeny. This is an ongoing project with Alex Barrios, Nandita Sahajpal, Darwin Tallana, Bella Tobin, and Hanneke Wiersema. |
| Title: The Calderon Problem: 40 Years Later |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Gunther Uhlmann of University of Washington |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2023-04-07 at 10:00AM |
| Venue: MSC W303 |
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| Abstract: Calderon's inverse problem asks whether one can determine the conductivity of a medium by making voltage and current measurements at the boundary. This question arises in several areas of applications including medical imaging and geophysics. I will report on some of the progress that has been made on this problem since Calderon proposed it, including recent developments on similar problems for nonlinear equations and nonlocal operators. |
| Title: Range Restricted GMRES methods and some experimental considerations |
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| Seminar: CODES@emory |
| Speaker: Lucas Onisk of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-04-06 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: The right-hand side vector of linear discrete ill-posed problems that can occur in science and engineering applications often represents an experimental measurement that is contaminated by error. The solution to these problems is typically very sensitive to this error. Previous works have shown that error propagation into the computed solution may be reduced by using specially designed iterative methods that allow the user to select the subspace in which the approximate solution is computed. Since the dimension of this subspace is often quite small, its choice is important for the quality of the computed solution. Iterative methods that modify the Generalized Minimal RESidual (GMRES) and block GMRES methods for the solution of appropriate linear systems of equations will be discussed, as well as experimental considerations. |
| Title: On Pisier type problems |
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| Defense: Dissertation |
| Speaker: Marcelo Sales of Emory University |
| Contact: Marcelo Sales, marcelo.tadeu.sales@emory.edu |
| Date: 2023-04-05 at 4:00PM |
| Venue: MSC N301 |
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| Abstract: A subset $A$ of integers is \textit{free} if for every two distinct subsets $B, B'\subset A$ we have \[\sum_{b\in B}b\neq \sum_{b'\in B'} b'.\] Pisier asked if, for every subset $A$ of integers, the following two statements are equivalent:\\ \\ (1) $A$ is a union of finitely many free sets.\\ (2) There exists $\epsilon>0$ such that every finite subset $B\subset A$ contains a free subset $C\subset B$ with $|C|\geq \epsilon |B|$.\\ \\ In a more general framework, the Pisier question can be seen as the problem of determining if statements (1) and (2) are equivalent for subsets of a given structure with the prescribed property. We study the problem for several structures including $B_h$-sets, arithmetic progressions, independent sets in hypergraphs, and configurations in the euclidean space. |