All Seminars

Title: Geometric regularity theory for diffusive processes and their intrinsic free boundaries
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
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
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
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.
Title: Radiation Fields for Wave Equations
Colloquium: Analysis and Differential Geometry
Speaker: Dean Baskin of TAMU
Contact: David Borthwick, dborthw@emory.edu
Date: 2019-02-07 at 4:00PM
Venue: MSC W303
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Abstract:
Radiation fields are rescaled limits of solutions of wave equations that capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space. I will assume no prior familiarity with PDEs.
Title: Pair correlation estimates for the zeros of the zeta-function via semidefinite programming
Seminar: Algebra
Speaker: David de Laat of Emory University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-02-05 at 4:00PM
Venue: MSC W201
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Abstract:
In this talk I will explain how semidefinite programming (a convex optimization technique generalizing linear programming) can be used to obtain improved bounds for quantities related to the distribution of the nontrivial zeros of zeta. I will show how this connects to the sphere packing problem and speculate about future improvements. No prior knowledge about convex optimization is assumed.
Title: An arithmetic count of the lines meeting four lines in P3
Seminar: Algebra
Speaker: Padma Srinivasan of Georgia Tech
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-01-29 at 4:00PM
Venue: MSC W201
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Abstract:
We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k, this enrichment counts the number of lines meeting four lines defined over k in P3, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A1 -homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.
Title: On The Structure of Unique Shortest Paths in Graphs
Seminar: Combinatorics
Speaker: Greg Bodwin of The Georgia Institute of Technology
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2019-01-28 at 4:00PM
Venue: MSC E408
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Abstract:
Let P be a system of unique shortest paths through a graph with real edge weights (i.e. a finite metric). An obvious fact is that P is "consistent," meaning that no two of these paths can intersect each other, split apart, and then intersect again later. But is that all? Can any consistent path system be realized as unique shortest paths in some graph? Or are there more forbidden combinatorial intersection patterns out there to be found?\\ \\ In this talk, we will characterize exactly which path systems can or can't be realized as unique shortest paths in some graph by giving a complete list of new forbidden intersection patterns. Our characterization theorem is based on a new connection between graph metrics and certain boundary operators used in some recent graph homology theories. This connection also leads to a principled topological understanding of some of the popular algebraic tricks currently used in the literature on shortest paths.
Title: Selmer groups, ranks of elliptic curves, and applications
Seminar: Algebra
Speaker: Ari Shnidman of Boston College
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-01-22 at 4:00PM
Venue: MSC W201
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Abstract:
I'll discuss some forthcoming results on Selmer groups in twist families of elliptic curves. In work with Lemke Oliver, we bound the average size of the 2-Selmer group in quadratic twist families, when E[2](Q) = 0. This bounds the average Mordell-Weil rank in such families. I'll also discuss work with Alpoge and Bhargava on Selmer groups of sextic twists of elliptic curves, with an application to a question about cubic fields.
Title: The number of Gallai colorings
Seminar: Combinatorics
Speaker: Jie Han of University of Rhode Island
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2019-01-18 at 4:00PM
Venue: MSC W301
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Abstract:
An edge coloring of the complete graph Kn is called a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle in which all three edges have distinct colors. Given a set of k colors and integer n, we are interested in the number of Gallai colorings of Kn with colors from the given set. In particular, we show that for k at most exponential in n, namely, k < 2^n/4300, almost all Gallai colorings use at most 2 colors. Interestingly, this statement fails for k > 2^n/2. This is joint work with Josefran O. Bastos and Fabricio S. Benevides (University of Ceara, Brazil).