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Title: TBA
Seminar: Algebra
Speaker: Frank Thorne of University of South Carolina
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-04-24 at 4:00PM
Venue: W304
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Abstract:
TBA
Title: TBA
Seminar: Algebra
Speaker: Brandon William of UC Berkeley
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-04-17 at 4:00PM
Venue: W304
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Abstract:
TBA
Title: TBA
Colloquium: Algebra
Speaker: K. Soundararajan of Stanford University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-04-12 at 4:00PM
Venue: W304
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Abstract:
TBA
Title: TBA
Colloquium: N/A
Speaker: Sherry Li of Lawrence Berkeley National Lab
Contact: Lar Ruthotto, lruthotto@emory.edu
Date: 2018-04-05 at 4:00PM
Venue: W201
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Abstract:
TBA
Title: TBA
Seminar: Algebra
Speaker: Jennifer Berg of Rice
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-04-03 at 4:00PM
Venue: W304
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Abstract:
TBA
Title: Patching and local-global principles for gerbes with an application to homogeneous spaces
Defense: Dissertation
Speaker: Bastian Haase of Emory University
Contact: Bastian Haase, bastian.haase@emory.edu
Date: 2018-04-02 at 4:00PM
Venue: W302
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Abstract:
Starting in 2009, Harbater and Hartmann introduced a new patching setup for semi-global fields, establishing a patching framework for vector spaces, central simple algebras, quadratic forms and other algebraic structures. In subsequent work with Krashen, the patching framework was refined and extended to torsors and certain Galois cohomology groups. After describing this framework, we will discuss an extension of the patching equivalence to bitorsors and gerbes. Building up on these results, we then proceed to derive a characterisation of a local- global principle for gerbes and bitorsors in terms of factorization. These results can be expressed in the form of a Mayer-Vietoris sequence in non-abelian hypercohomology with values in the crossed-module $G->Aut(G)$. After proving the local-global principle for certain bitorsors and gerbes using the characterization mentioned above, we conclude with an application on rational points for homogeneous spaces.
Title: TBA
Seminar: Algebra
Speaker: Nathan Kaplan of UC Irvine
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-03-27 at 4:00PM
Venue: W304
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Abstract:
TBA
Title: On Spanning Trees with few Branch Vertices
Defense: Dissertation
Speaker: Warren Shull of Emory University
Contact: Warren Shull, warren.edward.shull@emory.edu
Date: 2018-03-05 at 4:00PM
Venue: W301
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Abstract:
Hamiltonian paths, which are a special kind of spanning tree, have long been of interest in graph theory and are notoriously hard to compute. One notable feature of a Hamiltonian path is that all its vertices have degree two in the path. In a tree, we call vertices of degree at least three \emph{branch vertices}. If a connected graph has no Hamiltonian path, we can still look for spanning trees that come "close," in particular by having few branch vertices (since a Hamiltonian path would have none). \bigskip A conjecture of Matsuda, Ozeki, and Yamashita posits that, for any positive integer $k$, a connected claw-free $n$-vertex graph $G$ must contain either a spanning tree with at most $k$ branch vertices or an independent set of $2k+3$ vertices whose degrees add up to at most $n-3$. In other words, $G$ has this spanning tree whenever $\sigma_{2k+3}(G)\geq n-2$. We prove this conjecture, which was known to be sharp.
Title: On Cycles, Chorded Cycles, and Degree Conditions
Defense: Dissertation
Speaker: Ariel Keller of Emory University
Contact: Ariel Keller, ariel.keller@emory.edu
Date: 2018-03-01 at 3:00PM
Venue: MSC N301
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Abstract:
Sufficient conditions to imply the existence of certain substructures in a graph are of considerable interest in extremal graph theory, and conditions that guarantee a large set of cycles or chorded cycles are a recurring theme. This dissertation explores different degree sum conditions that are sufficient for finding a large set of vertex-disjoint cycles or a large set of vertex-disjoint chorded cycles in a graph. \vskip.1in For an integer $t\ge 1$, let $\sigma_t (G)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We first prove that if a graph $G$ has order at least $7k+1$ and degree sum condition $\sigma_4(G)\ge 8k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. Then, we consider an equivalent condition for chorded cycles, proving that if $G$ has order at least $11k+7$ and $\sigma_4(G)\ge 12k-3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint chorded cycles. We prove that the degree sum condition in each result is sharp. Finally, we conjecture generalized degree sum conditions on $\sigma_t(G)$ for $t\ge 2$ sufficient to imply that $G$ contains $k$ vertex-disjoint cycles for $k \ge 2$ and $k$ vertex-disjoint chorded cycles for $k \ge 2$. This is joint work with Ronald J. Gould and Kazuhide Hirohata.
Title: Counting points, counting fields, and heights on stacks
Colloquium: Algebra
Speaker: Jordan Ellenberg of University of Wisconsin-Madison
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2018-02-27 at 4:00PM
Venue: W303
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Abstract:
The basic objects of algebraic number theory are number fields, and the basic invariant of a number field is its discriminant, which in some sense measures its arithmetic complexity. A basic finiteness result is that there are only finitely many degree-$d$ number fields of discriminant at most $X$; more generally, for any fixed global field $K$, there are only finitely many degree-$d$ extensions $L/K$ whose discriminant has norm at most $X$. (The classical case is where $K = \mathbb{Q}$.) \\ When a set is finite, we greedily ask if we can compute its cardinality. Write $N_d(K,X)$ for the number of degree-$d$ extensions of $K$ with discriminant at most $d$. A folklore conjecture holds that $N_d(K,X)$ is on order $c_d X$. In the case $K = \mathbb{Q}$, this is easy for $d=2$, a theorem of Davenport and Heilbronn for $d=3$, a much harder theorem of Bhargava for $d=4$ and 5, and completely out of reach for $d > 5$. More generally, one can ask about extensions with a specified Galois group $G$; in this case, a conjecture of Malle holds that the asymptotic growth is on order $X^a (\log X)^b$ for specified constants $a,b$. \\ I'll talk about two recent results on this old problem: \\ 1) (joint with TriThang Tran and Craig Westerland) We prove that $N_d(\mathbb{F}_q(t),X)) < c_{\epsilon} X^{1+\epsilon}$ for all $d$, and similarly prove Malle’s conjecture ``up to epsilon" — this is much more than is known in the number field case, and relies on a new upper bound for the cohomology of Hurwitz spaces coming from quantum shuffle algebras: https://arxiv.org/abs/1701.04541 \\ 2) (joint with Matt Satriano and David Zureick-Brown) The form of Malle's conjecture is very reminiscent of the Batyrev-Manin conjecture, which says that the number of rational points of height at most $X$ on a Batyrev-Manin variety also grows like $X^a (\log X)^b$ for specified constants $a,b$. What’s more, an extension of $\mathbb{Q}$ with Galois group $G$ is a rational point on a Deligne--Mumford stack called $BG$, the classifying stack of $G$. A natural reaction is to say “the two conjectures is the same; to count number fields is just to count points on the stack BG with bounded height?” The problem: there is no definition of the height of a rational point on a stack. I'll explain what we think the right definition is, and explain how it suggests a heuristic which has both the Malle conjecture and the Batyrev--Manin conjecture as special cases.