All Seminars
| Title: Connections between mock modular forms and vertex operator algebras |
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| Defense: Number Theory |
| Speaker: Lea Beneish of Emory University |
| Contact: David Zureick-Brown, DAVID.M.BROWN.JR@GMAIL.COM |
| Date: 2020-03-03 at 3:00PM |
| Venue: MSC W303 |
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| Abstract: The results in this dissertation come in two flavors, first we aim to strengthen the analogy between monstrous and umbral moonshine using vertex operator algebras, and second we derive structural results on vertex operator algebras using mock modular forms. \\ Towards strengthening the analogy between umbral and monstrous moonshine, we reframe Mathieu moonshine by repackaging the Mathieu moonshine mock modular forms in a few different ways, verifying the existence of corresponding modules, and giving various applications including connections with arithmetic. We produce vertex operator algebra constructions of some of these modules. \\ Using results from orbifold theory and from the theory of mock modular forms, we derive new structural results on vertex operator algebras. In joint work with Victor Manuel Aricheta, we study the asymptotic structure sequences of $G$-modules where $G$ are finite automorphism groups of certain vertex operator algebras (in particular this holds for umbral moonshine modules). And in joint work with Michael Mertens, we use Weierstrass mock modular forms to relate a dimension formula for certain vertex operator algebras to the arithmetic of modular curves. |
| Title: Non-Archimedean and Tropical Techniques in Arithmetic Geometry |
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| Defense: Algebra |
| Speaker: Jackson Morrow of Emory University |
| Contact: David Zureick-Brown, DAVID.M.BROWN.JR@GMAIL.COM |
| Date: 2020-03-03 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Let $K$ be a number field, and let $C/K$ be a curve of genus $g \geq 2$. In 1983, Faltings famously proved that the set $C(K)$ of $K$-rational points is finite. Given this, several questions naturally arise: \begin{enumerate} \item How does this finite quantity $\#C(K)$ varies in families of curves? \item What is the analogous result for degree $d>1$ points on $C$? \item What can be said about a higher dimensional variant of Faltings result? \end{enumerate} In this thesis, we will prove several results related to the above questions. \\ In joint with with J.~Gunther, we prove, under a technical assumption, that for each positive integer $d > 1$, there exists a number $B_d$ such that for each $g > d$, a positive proportion of odd hyperelliptic curves of genus $g$ over $\mathbb{Q}$ have at most $B_d$ ``unexpected'' points of degree $d$. Furthermore, we may take $B_2 = 24$ and $B_3 = 114$. \\ Our other results concern the strong Green--Griffiths--Lang--Vojta conjecture, which is the higher dimensional version of Faltings theorem (ne\'e the Mordell conjecture). More precisely, we prove the strong non-Archimedean Green--Griffiths--Lang--Vojta conjecture for closed subvarieties of semi-abelian varieties and for projective surfaces admitting a dominant morphism to an elliptic curve. \\ Time permitting, we will introduce a new construction of the non-Archimedean Kobayashi pseudo-metric for a Berkovich analytic space $X$ and provide evidence that our definition is the ``correct'' one. In particular, if this pseudo-metric is an actual metric on $X$, then it defines the Berkovich analytic topology and $X$ does not admit a non-constant morphism from any analytic tori. |
| Title: Global mild solutions of the Landau and non-cutoff Boltzmann equation |
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| Seminar: Algebra |
| Speaker: Robert M. Strain of University of Pennsylvania |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2020-02-28 at 2:00PM |
| Venue: MSC W201 |
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| Abstract: This paper proves the existence of small-amplitude global-in-time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well-known works (Guo, 2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, has still remained an open problem to obtain global solutions in an $L^\infty_{x,v}$ framework, similar to that in (Guo-2010), for the Boltzmann equation with cutoff in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity-diffusion-type collision operator in the non-cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In this work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus, or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the $L^\infty_T L^2_v$ norm, in velocity and time, of the distribution function is in the Wiener algebra $A(\Omega)$ in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large-time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. To the best of our knowledge these results may be the first ones to provide an elementary understanding of the existence theories for the Landau or non-cutoff Boltzmann equations in the situation where the spatial domain has a physical boundary. \\ \\This is a joint work with Renjun Duan (The Chinese University of Hong Kong), Shuangqian Liu (Jinan University) and Shota Sakamoto (Tohoku University). |
| Title: Symmetry of hypersurfaces with ordered mean curvature in one direction |
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| Colloquium: Analysis and Differential Geometry |
| Speaker: Yanyan Li of Rutgers University |
| Contact: Vladimir Oliker, oliker@emory.edu |
| Date: 2020-02-26 at 3:00PM |
| Venue: PAIS 220 |
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| Abstract: For a connected n-dimensional compact smooth hypersurface M without boundary embedded in $R^{n+1}$, a classical result of A.D. Aleksandrov shows that it must be a sphere if it has constant mean curvature. Nirenberg and I studied a one-directional analog of this result: if every pair of points $(x', a)$, $(x', b)$ in M with $a < b$ has ordered mean curvature $H(x', b)\le H(x', a)$, then M is symmetric about some hyperplane $x_{n+1} = c$ under some additional conditions.\\ \\ Our proof was done by the moving plane method and some variations of the Hopf Lemma. In a recent joint work with Xukai Yan and Yao Yao, we have obtained the symmetry of M under some weaker assumptions using a variational argument, giving a positive answer to a conjecture raised by Nirenberg and I in 2006. |
| Title: Linear and rational factorization of tropical polynomials |
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| Seminar: Algebra |
| Speaker: Bo Lin of Georgia Tech |
| Contact: David Zureick-Brown, DAVID.M.BROWN.JR@GMAIL.COM |
| Date: 2020-02-25 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this talk, we will introduce a rich class of tropical polynomials in n variables, which admit factorization and rational factorization into well-behaved factors. We present efficient algorithms of their factorizations with examples. Special families of these polynomials have appeared in economics, discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to open problems of interest in discrete geometry. |
| Title: Wave decay for star-shaped waveguides |
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| Seminar: Analysis and PDE |
| Speaker: Kiril Datchev of Purdue |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2020-02-20 at 2:30PM |
| Venue: MSC N306 |
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| Abstract: Let $\Omega \subset \mathbb R^d$ be an unbounded open set. We wish to understand how decay of solutions to the wave equation on $\Omega$ is related to the geometry of $\Omega$.\\ \\ When $\mathbb R^d \setminus \Omega$ is bounded, this is the celebrated obstacle scattering problem. Then a particularly favorable geometric assumption, going back to the original work of Morawetz, is that the obstacle is star shaped. We adapt this assumption to the study of waveguides, which are domains bounded in some directions and unbounded in others, such as tubes or wires. We prove sharp wave decay rates for various waveguides, including the example of a disk removed from a straight planar waveguide, that is to say $\Omega = ((-1,1) \times \mathbb R) \setminus D$, where $D$ is a closed disk contained in $(-1,1) \times \mathbb R$. Our results are based on establishing estimates and pole-free regions for the resolvent of the Laplacian near the continuous spectrum.\\ \\ This talk is based on joint work with Tanya Christiansen. |
| Title: A limit theorem in Optimal transportation theory |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Professor Gershon Wolansky of Israel Institute of Technology - Technion |
| Contact: Vladimir Oliker, oliker@emory.edu |
| Date: 2020-02-11 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: I’ll review the fundamental theory of optimal transportation and define the notion of conditional Wasserstein metric on the set of probability measures. If time permits I’ll discuss various applications to control, regularity of flows, dynamics, and design of optimal networks. |
| Title: Moduli spaces in computer vision |
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| Colloquium: Algebra and Number Theory |
| Speaker: Max Lieblich of University of Washington |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2020-02-10 at 2:30PM |
| Venue: Mathematics and Science Center: MSC E208 |
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| Abstract: Moduli theory is one of the cornerstones of algebraic geometry. The underlying idea of the theory is that, given a class of mathematical objects, one can often find a universal space parametrizing those objects, and the geometry of this space gives us insight into the objects being parametrized. After introducing moduli theory with some basic classical examples, I will discuss recent applications to computer vision. As it turns out, the roots of computer vision are tightly intertwined with classical projective geometry. I will present the early history and basic geometric problems of computer vision, and then I will talk about how modern methods give us deeper insight into these problems, including new understandings of core algorithms that are used billions of times a day all over the planet. |
| Title: The foundation of a matroid |
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| Colloquium: Algebra and Number Theory |
| Speaker: Matt Baker of Georgia Institute of Technology |
| Contact: David Zureick-Brown, david.zureick.brown@gmail.com |
| Date: 2020-02-05 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Originally introduced independently by Hassler Whitney and Takeo Nakasawa, matroids are a combinatorial way of axiomatizing the notion of linear independence in vector spaces. If $K$ is a field and $n$ is a positive integer, any linear subspace of $K^n$ gives rise to a matroid; such matroid are called \textbf{representable} over $K$. Given a matroid $M$, one can ask over which fields $M$ is representable. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the \textbf{universal partial field} of a matroid $M$, which governs the representations of $M$ over all partial fields. Unfortunately, most matroids (asymptotically 100\%, in fact) are not representable over any partial field, and in this case, the universal partial field gives no information. Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the \textbf{foundation} of a matroid. The foundation of $M$ is a type of algebraic object which we call a \textbf{pasture}; pastures include both hyperfields and partial fields. Pastures form a natural class of field-like objects within Lorscheid's theory of ordered blueprints, and they have desirable categorical properties (e.g., existence of products and coproducts) that make them a natural context in which to study algebraic invariants of matroids. The foundation of a matroid $M$ represents the functor taking a pasture $F$ to the set of rescaling equivalence classes of $F$-representations of $M$; in particular, $M$ is representable over a pasture $F$ if and only if there is a homomorphism from the foundation of $M$ to $F$. (In layman's terms, what we're trying to do is recast as much as possible of the theory of matroids and their representations in functorial ``Grothendieck-style'' algebraic geometry, with the goal of gaining new conceptual insights into various phenomena which were previously understood only through lengthy case-by-case analyses and ad hoc computations.) As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for ternary matroids (matroids representable over the field of three elements). The proof of this classification theorem relies crucially on Tutte's celebrated Homotopy Theorem. |
| Title: An introduction to counting curves arithmetically |
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| Colloquium: Algebra and Number Theory |
| Speaker: Jesse Kass of University of South Carolina |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2020-02-03 at 2:30PM |
| Venue: Mathematics and Science Center: MSC E208 |
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| Abstract: A long-standing program in algebraic geometry focuses on counting the number of curves in special configuration such as the lines on a cubic surface (27) or the number of conic curves tangent to 5 given conics (3264). While many important counting results have been proven purely in the language of algebraic geometry, a major modern discovery is that curve counts can often be interpreted in terms of algebraic topology and this topological perspective reveals unexpected properties. One problem in modern curve counting is that classical algebraic topology is only available when working over the real or complex numbers. A successful solution to this problem should produce curve counts over fields like the rational numbers in such a way as to record interesting arithmetic information. My talk will explain how to derive such counts using ideas from A1-homotopy theory. The talk will focus on joint work with Marc Levine, Jake Solomon, and Kirsten Wickelgren including a new result about lines on the cubic surface. |