All Seminars
| 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. |
| Title: Graph limit approach in extremal combinatorics |
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| Colloquium: Combinatorics |
| Speaker: Jan Volec of University of Hamburg and Emory University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-02-18 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Over the last decade, different theories of combinatorial limits have emerged, and attracted substantial attention. In the first part of my talk, I will present the main ideas behind these techniques and describe their applications to problems in extremal combinatorics. Next, as a particular example of the techniques, I will focus on a conjecture of Kelly, Kuhn and Osthus on the degree thresholds of oriented cycles. |
| Title: Graphs and Hypercubes |
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| Colloquium: Combinatorics |
| Speaker: Eoin Long of University of Oxford |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-02-15 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Graphs and hypercubes are two of the most widely studied structures in Combinatorics. While interesting in their own right, this study is also motivated by their broad applicability, arising naturally in many areas including Discrete Geometry, Probability and Computer Science. In this talk I will discuss a number of questions and results concerning both of these structures, with an emphasis on interactions between the two. This will include topics in finite set theory, isoperimetric inequalities and Ramsey theory. |
| Title: Geometric regularity theory for diffusive processes and their intrinsic free boundaries |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Eduardo Teixeira of University of Central Florida |
| Contact: David Borthwick, dborthw@emory.edu |
| Date: 2019-02-12 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Diffusion is a phenomenon associated with averaging, spreading, or balancing of quantities in a given process. These are innate trends in several fields of natural sciences, and this is why diffusion is such a popular concept across disciplines. When models involve sharp changes in the parameters that describe them, free boundaries and interfaces are formed and the mathematical treatment of such problems becomes rather more involved. Throughout the last 40 years or so, a robust mathematical theory has been developed to investigate diffusive phenomena presenting free boundaries. Methods, ideas, and insights originating from different fields of research merged together as to produce a comprehensive geometric regularity theory for free boundary problem, and in this talk I will provide a panoramic overview of such endeavor. Recently, it has been observed that even ordinary diffusive models, i.e. the ones with no concrete free boundaries, carry in their intrinsic geometry a sort of “artificial” or “transcendental” or, if you prefer, "non-physical” free boundaries. This radical new approach to the analysis of nonlinear PDEs has led to a plethora of unanticipated results and I will discuss some of these achievements. |
| Title: Berkovich Spaces and Dual Complexes of Degenerations |
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| Seminar: Algebra |
| Speaker: Enrica Mazzon of Imperial College |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-02-12 at 5:00PM |
| Venue: MSC W201 |
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| Abstract: In the late nineteen-nineties Berkovich developed a new approach to non-archimedean analytic geometry. This theory has quickly found many applications in algebraic and arithmetic geometry. In particular it turned out that there are strong connections between Berkovich spaces and the birational geometry of varieties. \\ In this talk, I will introduce the central objects of this theory: degeneration of varieties, dual complexes and essential skeletons. As an application, I will explain how the non-archimedean approach applies to the study of some degenerations of hyper-Kahler varieties, giving new results in accordance with the predictions of mirror symmetry. This is joint work with Morgan Brown. |
| Title: Fun with Mac Lane valuations |
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| Seminar: Algebra |
| Speaker: Andrew Obus of Baruch college, CUNY |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-02-12 at 6:00PM |
| Venue: MSC W201 |
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| Abstract: Mac Lane's technique of ``inductive valuations" is over 80 years old, but has only recently been used to attack problems about arithmetic surfaces. We will give an explicit, hands-on introduction to the theory, requiring little background beyond the definition of a non-archimedean valuation. We will then outline how this theory is helpful for resolving ``weak wild" quotient singularities of arithmetic surfaces, as well as for proving conductor-discriminant inequalities for higher genus curves. The first project is joint work with Stefan Wewers, and the second is joint work with Padmavathi Srinivasan. |
| Title: Random Ramsey Theorems |
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| Colloquium: Combinatorics |
| Speaker: Rajko Nenadov of ETH Zurich |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-02-11 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We say that a graph G is Ramsey for a graph H if in every colouring of the edges of G with red and blue we can find a monochromatic copy of H, that is a copy with all edges having the same colour. Perhaps surprising at first, Ramsey's theorem states that for every graph H there is a graph G which is Ramsey for H. This existence statement raises many further questions: how many vertices or edges such G needs to have, can it share some structural properties with H such as girth or the clique number, is there a choice for G which always gives an induced monochromatic copy of H and, if yes, then how many vertices such G needs to have? Many of these questions can be fully resolved using random graphs and quite often the random graphs give the best known quantitative bounds. Even though the first major results on Ramsey properties of random graphs date back to the beginning of the '90s and the work of Rödl and Ruci?ski, many questions and conjectures remain open. In this talk we will review some of the recent progress on these questions and discuss current challenges. |