# Seminars archive

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 Upcoming Seminars Tue02/27/20184:00pm Colloquium: AlgebraCounting points, counting fields, and heights on stacksJordan Ellenberg, University of Wisconsin-MadisonContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Show 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 $K = Q$.) \\ 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 $K = Q$, 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 $N_d(F_q(t),X)) < c_{\eps} X^{1+\eps}$ 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 $Q$ 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. Thu03/01/20183:00pm Defense: DissertationOn Cycles, Chorded Cycles, and Degree ConditionsAriel Keller, Emory UniversityContact: Ariel Keller, ariel.keller@emory.eduVenue: MSC N301Download printable flyer (PDF, 50.4 kB)Show abstractSufficient 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. Tue03/27/20184:00pm Seminar: AlgebraTitle to be announcedNathan Kaplan, UC IrvineContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.2 kB) Mon04/02/20184:00pm Defense: DissertationPatching and local-global principles for gerbes with an application to homogeneous spacesBastian Haase, Emory UniversityContact: Bastian Haase, bastian.haase@emory.eduVenue: Mathematics and Science Center, Room W302Download printable flyer (PDF, 44.9 kB)Show abstractStarting 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. Tue04/03/20184:00pm Seminar: AlgebraTitle to be announcedJennifer Berg, RiceContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.4 kB) Thu04/05/20184:00pm ColloquiumTitle to be announcedSherry Li, Lawrence Berkeley National LabContact: Lar Ruthotto, lruthotto@emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 19.3 kB) Thu04/12/20184:00pm Colloquium: AlgebraTitle to be announcedK. Soundararajan, Stanford UniversityContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.7 kB) Tue04/17/20184:00pm Seminar: AlgebraTitle to be announcedBrandon William, UC BerkeleyContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.1 kB) Tue04/24/20184:00pm Seminar: AlgebraTitle to be announcedFrank Thorne, University of South CarolinaContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 19.3 kB) Past Seminars Wed10/10/20123:00pm Seminar: AlgebraInfinitesimal Deformation Theory and Grothendieck TopologiesJonathan Wise, CU BoulderContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.4 kB)Show abstractTo probe the infinitesimal structure of a moduli space of geometric objects, one seeks to understand families of those objects over "fat points". Understanding such families frequently yields a great deal of information about the moduli space. Remarkably, these deformation problems tend to admit cohomological solutions of a common form: obstructions in H 2, deformations in H 1, and automorphisms in H 0. I will offer an explanation for this common form, coming from some exotic Grothendieck topologies. We will see how this point of view works in several examples. No prior knowledge about Grothendieck topologies or deformation theory will be assumed. Fri10/05/20124:00pm ColloquiumKLR conjecture in Sparse Random GraphsMathias Schacht, University of HamburgContact: Vojtech Rodl, rodl@mathcs.emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 38.2 kB)Show abstractThe KLR conjecture of Kohayakawa, Luczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G(n,p) satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most applications to random graphs. In particular, our result implies a number of recent probabilistic threshold results. We also discuss several further applications. This joint work with Conlon, Gowers, and Samotij. Wed10/03/20123:00pm Seminar: AlgebraPeriods of Modular Forms and Identities between Eisenstein SeriesWissam Raji, American University of BeirutContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.1 kB)Show abstractBorisov and Gunnels observed in 2001 that certain linear relations between products of two holomorphic weight 1 Eisenstein series had the same structure as the relations between periods of modular forms. We give a conceptual reason for this and for similar phenomena in all weights. This involves an unconventional way of expanding the Rankin-Selberg convolution of a cusp form with an Eisenstein series. (Joint with Kamal Khuri-Makdisi). Thu09/27/20124:00pm ColloquiumOptimal partitions of measuresGershon Wolansky, Technion - Israel Institute of TechnologyContact: Professor Vladimir Oliker, oliker@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 54.2 kB)Show abstractLet $X$ be a probability measure space and $\psi_1....\psi_N$ measurable, real valued functions on $X$. Consider all possible partitions of $X$ into $N$ disjoint subdomains $X_i$ on which $\int_{X_i}\psi_i$ are prescribed. I'll address the question of characterizing the set $(m_1,,,m_N) \in \mathbb{R}^N$ for which there exists a partition $X_1, \ldots X_N$ of $X$ satisfying $\int_{X_i}\psi_i= m_i$ and discuss some optimization problems on this set of partitions. The relation of this problem to semi-discrete version of optimal mass transportation is discussed, as well as applications to game theory. Wed09/26/20123:00pm Seminar: Algebra and Number TheorySyzygies and Boij--Soederberg TheoryDaniel Erman, University of MichiganContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 36.8 kB)Show abstractFor a system of polynomial equations, it has long been known that the relations (or syzygies) among the polynomials provide powerful insights into the properties and invariants of the corresponding projective varieties. Boij--Soederberg Theory provides a powerful perspective on syzygies, and in particular reveals a surprising duality between syzygies and cohomology of vector bundles. I will describe some new results on this duality and on the properties of syzygies. Fri09/21/20123:00pm Defense: DissertationModeling Rich Interactions for Web Search Intent Inference, Ranking and EvaluationQi Guo, Emory UniversityContact: Qi Guo, qguo3@emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 40.3 kB)Show abstractThe thesis aims to enable more intelligent Web search engines by modeling and interpreting a wide range of searcher interactions, including queries, clicks, time, and finer grained interactions such as mouse cursor movements and scrolling behavior (or pinching, zooming and sliding with a touch screen). The thesis spans three key areas in Web search, namely, understanding information needs, ranking result documents, and evaluating search experience.\\ \\ First, the thesis developed techniques for inferring the immediate searcher information needs through mining the rich interactions and context in a search session. The developed techniques improve the prediction of general search intent, commercial search intent and future ad clickthrough over the state-of-the-art methods that only exploit query and click signals.\\ \\ Second, the thesis developed techniques for estimating document relevance to improve search result ranking. The Post-Click Behavior (PCB) relevance prediction model was introduced, which focuses on estimating the "intrinsic" document relevance from a rich set of fine-grained interactions on the viewed result documents in a search session, outperforming the state-of-theart methods that are based on the time information.\\ \\ Third, the thesis developed techniques for automatically evaluating search experience or search engine performance at different levels. The first level is the query-level, where techniques for predicting query performance were developed, enabling evaluation and diagnostic for particular queries or query classes. The second level is the session-level, where techniques for predicting search success were developed, which include a principled framework to study Web search success, and fine-grained interaction models that improve prediction accuracies for both desktop and mobile settings. The third level is the level of using multiple search engines, where the developed techniques focus on understanding and predicting the rationales of engine switching in a search session. Wed09/19/20123:00pm Seminar: Algebra and Number TheoryAlmost harmonic Maass forms and Kac-Wakimoto charactersAmanda Folsom, Contact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 33.4 kB)Show abstractWe address a question Kac, and explain the modular properties of certain characters due to Kac and Wakimoto pertaining to $sl(m|n)^$, where $n$ is a positive integer. We prove that these characters are essentially holomorphic parts of new automorphic objects we call "almost harmonic Maass forms," which generalize both weak Maass forms and almost holomorphic modular forms. By new methods involving meromorphic Jacobi forms, this generalizes prior works of Bringmann-Ono and Bringmann-Folsom, which treat only the case $n=1$. This is joint work with Kathrin Bringmann (University of Cologne). Fri09/14/20124:00pm ColloquiumMinimum Degree and Disjoint Cycles in Generalized Claw-free GraphsRalph Faudree, The University of MemphisContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 50.4 kB)Show abstractFor $s \geq 3$ a graph is $K_{1,s}$-free, if it does not contain an induced subgraph isomorphic to $K_{1,s}$. For $s = 3$, such graphs are called claw-free graphs. Results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions will be be discussed, such as if $G$ is claw-free of sufficiently large order $n = 3k$ with $\delta (G) \geq n/2$, then $G$ contains $k$ disjoint triangles. Also, the extension of results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to $K_{1,s}$-free graphs for $s > 3$ will be presented. These results will be used to prove the existence of minimum degree conditions that imply the existence of powers Hamiltonian cycle in generalized claw-free graphs. Fri09/14/20124:00pm Seminar: CombinatoricsMinimum Degree and Disjoint Cycles in Generalized Claw-free GraphsRalph Faudree, The University of MemphisContact: Dwight Duffus, dwight@math.cs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 50.2 kB)Show abstractFor $s \geq 3$ a graph is $K_{1,s}$-free, if it does not contain an induced subgraph isomorphic to $K_{1,s}$. For $s = 3$, such graphs are called claw-free graphs. Results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions will be be discussed, such as if $G$ is claw-free of sufficiently large order $n = 3k$ with $\delta (G) \geq n/2$, then $G$ contains $k$ disjoint triangles. Also, the extension of results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to $K_{1,s}$-free graphs for $s > 3$ will be presented. These results will be used to prove the existence of minimum degree conditions that imply the existence of powers Hamiltonian cycle in generalized claw-free graphs. Fri09/14/20123:00pm Seminar: Computer Science and InformaticsEstimation of a semiparametric mixture of regression models: application to the ChipMIX modelPierre Vandekerkhove, Visiting professor, Department of Materials Science and Engineering, Georgia Institute of TechnologyContact: Alfredo Tirado-Ramos, atirado@emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 38.5 kB)Show abstractIn this talk we will present a new mixture of regression models which is a generalization of the semiparametric two-component mixture model, which can be used as an exploratory and complementary tool for the modeling of two-color ChIP-chip datasets. An implementation and numerical performances for this method is discussed, using several simulated datasets and one real microarray dataset, the ChipMIX model.