All Seminars

Title: Increasing paths in edge-ordered hypergraphs
Defense: Dissertation
Speaker: Bradley Elliott of Emory University
Contact: Bradley Elliott, bradley.elliott@emory.edu
Date: 2020-03-27 at 3:30PM
Venue: https://emory.zoom.us/j/345080312
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Abstract:
Abstract: In this defense, we will see many variations on a classic problem of ordering the vertices or edges of a graph and determining the length of "increasing" paths in the graph. For finite graphs, the vertex-ordering problem is completely solved, and there has been recent progress on the edge-ordering problem. Here, we also provide upper bounds for the edge-ordering problem with respect to complete hypergraphs and Steiner triple systems. We also prove the hypergraph version of the vertex-ordering problem: every vertex-ordered hypergraph H has an increasing path of at least $chi(H)-1$ edges, where $chi(H)$ is the chromatic number of $H$.\\ \\ For countably infinite graphs, a similar problem is studied, asking which graphs contain an infinite increasing path regardless of how their vertices or edges are ordered. Here we answer such questions for hypergraphs. For example, we show that the condition that a hypergraph contains a subhypergraph with all infinite degrees is equivalent to the condition that any vertex-ordering permits an infinite increasing path. We prove a similar result for edge-orderings. In addition, we find an equivalent condition for a graph to have the property that any vertex-ordering permits a path of arbitrary finite length. Finally, we study related problems for orderings by all integers $Z$ (instead of just positive integers $N$). For example, we show that for every countable graph, there is an ordering of its edges by $Z$ that forbids infinite increasing paths.
Title: Local-global principles for norm one tori over semi-global fields.
Defense: Dissertation
Speaker: Sumit Chandra Mishra of Emory University
Contact: David Zureick-Brown, DAVID.M.BROWN.JR@GMAIL.COM
Date: 2020-03-24 at 4:00PM
Venue: https://emory.zoom.us/j/382949597
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Abstract:
Let K be a complete discretely valued field with residue field k (e.g. k((t)) ). Let n be an integer coprime to char(k). Let F = K(x) be the rational function field in one variable over F and L/F be any Galois extension of degree n. Suppose that either k is algebraically closed or k is finite field containing a primitive nth root of unity. Then we show that an element in F? is a norm from the extension L/F if and only if it is a norm from the corresponding extensions over the completions of F at all discrete valuations of F. We also prove that such a local-global principle holds for product of norms from cyclic extensions of prime degree if k is algebraically closed.
Title: Deep Learning with Graph Structured Data: Methods, Theory, and Applications
Seminar: Numerical Analysis and Scientific Computing
Speaker: Jie Chen of MIT-IBM Watson AI Lab, IBM Research
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2020-03-06 at 2:00PM
Venue: MSC W201
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Abstract:
Graphs are universal representations of pairwise relationship. With the rise of deep learning that demonstrates promising parameterizations of functions on Euclidean and regularly structured data (e.g., images and sequences), natural interests seek extensions of neural networks for irregularly structured data, including notably, graphs. This talk aims at painting a global picture of the emerging research on graph deep learning and inspiring novel directions. The speaker will share his recent research on modeling, computation, and applications of graph neural networks. Whereas modeling network architectures under different learning settings draws major interests in the field, understanding the capacity and limits of these networks attracts increasing attention. Moreover, efficient training and inference with large graphs or large collections of graphs need to address challenges beyond those of usual neural networks with regularly structured data. Of separate interest is the learning of a hidden graph structure if objects or variables interact, the subject of which interfaces with causality in machine learning. Last but not least, graphs admit numerous interesting applications, among which the speaker touches drug design, cryptocurrency forensics, cybersecurity, and power systems.
Title: Harry Potter's Cloak via Transformation Optics
Colloquium: Combinatorics
Speaker: Gunther Ulhmann of University of Washington
Contact: David Borthwick, dborthw@emory.edu
Date: 2020-03-05 at 5:00PM
Venue: Oxford Road Building Presentation Room (311)
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Abstract:
Can we make objects invisible? This has been a subject of human fascination for millennia in Greek mythology, movies, science fiction, etc., including the legend of Perseus versus Medusa and the more recent Star Trek and Harry Potter. In the last two decades or so there have been several scientific proposals to achieve invisibility. We will introduce in a non-technical fashion one of them, the so-called "transformation optics," that has received the most attention in the scientific literature.
Title: Connections between mock modular forms and vertex operator algebras
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
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
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
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
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
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.