# 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 W303Download printable flyer (PDF, 59.6 kB)Show abstractThe 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. 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 Tue04/03/20122:30pm Seminar: Dissertation DefenseProblems on Sidon sets of integersSangjune Lee, Emory UniversityContact: Sangjune Lee, slee242@emory.eduVenue: Mathematics and Science Center, Room W304Download printable flyer (PDF, 66.5 kB)Show abstractA set~$A$ of non-negative integers is a \textit{Sidon set} if all the sums~$a_1+a_2$, with~$a_1\leq a_2$ and~$a_1$,~$a_2\in A$, are distinct. In this dissertation, we deal with three results on Sidon sets: two results are about finite Sidon sets in $[n]=\{0,1,\cdots, n-1\}$ and the last one is about infinite Sidon sets in $\mathbb{N}$ (the set of natural numbers). \\ \\ First, we consider the problem of Cameron--Erd\H{o}s estimating the number of Sidon sets in $[n]$. We obtain an upper bound $2^{c\sqrt{n}}$ on the number of Sidon sets which is sharp with the previous lower bound up to a constant factor in the exponent. \\ \\ Next, we study the maximum size of Sidon sets contained in sparse random sets $R\subset [n]$. Let~$R=[n]_m$ be a uniformly chosen, random $m$-element subset of~$[n]$. Let $F([n]_m)=\max\{|S|\colon S\subset[n]_m\hbox{ is Sidon}\}$. Fix a constant~$0\leq a\leq1$ and suppose~$m=(1+o(1))n^a$. We show that there is a constant $b=b(a)$ for which~$F([n]_m)=n^{b+o(1)}$ almost surely and we determine $b=b(a)$. Surprisingly, between two points $a=1/3$ and $a=2/3$, the function~$b=b(a)$ is constant. \\ \\ Next, we deal with infinite Sidon sets in sparse random subsets of $\mathbb{N}$. Fix $0<\delta\leq 1$, and let $R=R_{\delta}$ be the set obtained by choosing each element $i\subset\mathbb{N}$ independently with probability $i^{-1+\delta}$. We show that for every $0<\delta\leq 2/3$ there exists a constant $c=c(\delta)$ such that a random set $R$ satisfies the following with probability 1: \begin{itemize} \item Every Sidon set $S\subset R$ satisfies that $|S\cap [n]|\leq n^{c+o(1)}$ for every sufficiently large $n$. \item There exists a large Sidon set $S\subset R$ such that $|S\cap [n]| \geq n^{c+o(1)}$ for every sufficiently large $n$. \end{itemize} Fri03/30/20124:00pm Seminar: CombinatoricsOn highly connected monochromatic subgraphsTomasz Luczak, Emory University and Adam Mickiewicz UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 98.1 kB) Fri03/30/20124:00pm Defense: DissertationQuasi Isometric Properties of Graph Braid GroupsPraphat Fernandes, Emory UniversityContact: Praphat Fernandes, pxferna@emory.eduVenue: Mathematics and Science Center, Room W201Download printable flyer (PDF, 45 kB)Show abstractIn my thesis I initiate the study of the quasi-isometric properties of the 2 dimensional graph braid groups. I do this by studying the behaviour of flats in the geometric model spaces of the graph braid groups, which happen to be CAT(0) cube complexes. I define a quasi-isometric invariant of these graph braid groups called the intersection complex. In certain cases it is possible to calculate the dimension of this intersection complex from the underlying graph of the graph braid group. And I use the dimension of the intersection complex to prove that the family of graph braid groups $B_2(K_n)$ are quasi-isometrically distinct for all $n$. I also show that the dimension of the intersection complex for a graph braid group takes on every possible non-negative integer value. Thu03/29/20126:00pm EUMMA EventRiemann's zeros and the rhythm of the primesDavid Borthwick, Emory UniversityContact: Erin Nagle, erin@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 22.2 kB) Thu03/29/20123:00pm ColloquiumMonkey fieldsBrian Conrad, StanfordContact: Skip Garibaldi, skip@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 34.8 kB)Show abstractJoseph Ritt spent his entire career working with the real and complex fields, and he reportedly referred to fields of positive characteristic as "monkey fields". The development of algebraic geometry and number theory during the second half of the 20th century showed the tremendous usefulness of the so-called monkey fields even in the service of problems whose formulation only involves fields of characteristic 0. Sometimes the implications go in the other direction, using results in characteristic 0 to prove theorems over finite fields. We illustrate both directions of this interaction. Wed03/28/20124:00pm Seminar: Algebra and Number TheoryCM liftingBrian Conrad, StanfordContact: Skip Garibaldi, skip@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 35.7 kB)Show abstractThe classification of isogeny classes of simple abelian varieties over finite fields by Honda and Tate rests on the remarkable fact that, up to a finite ground field extension and isogeny, such abelian varieties admit lifts to CM abelian varieties in characteristic 0. Building on this, Tate proved that every abelian variety over a finite field is "of CM type". But this leaves open the question of whether characteristic-0 CM lifting can be done without introducing an isogeny or ground field extension. There are several precise versions of such a refined CM lifting question, and after reviewing some basics in CM theory I will formulate such problems and discuss positive and negative answers (and examples). This is joint work with C-L. Chai and F. Oort. Mon03/26/201212:00pm Defense: DissertationPrivacy Preserving Medical Data PublishingJames Gardner, Emory UniversityContact: James Gardner, jgardn3@emory.eduVenue: Mathematics and Science Center, Room E406Download printable flyer (PDF, 38.9 kB)Show abstractThere is an increasing need for sharing of medical information for public health research. Data custodians and honest brokers have an ethical and legal requirement to protect the privacy of individuals when publishing medical datasets. This dissertation presents an end-to-end Health Information DE-identification (HIDE) system and framework that promotes and enables privacy preserving medical data publishing of textual, structured, and aggregated statistics gleaned from electronic health records (EHRs). This work reviews existing de-identification systems, personal health information (PHI) detection, record anonymization, and differential privacy of multi-dimensional data. HIDE integrates several state-of-the-art algorithms into a unified system for privacy preserving medical data publishing. The system has been applied to a variety of real-world and academic medical datasets. The main contributions of HIDE include: 1) a conceptual framework and software system for anonymizing heterogeneous health data, 2) an adaptation and evaluation of information extraction techniques and modification of sampling techniques for protected health information (PHI) and sensitive information extraction in health data, and 3) applications and extension of privacy techniques to provide privacy preserving publishing options to medical data custodians, including de-identified record release with weak privacy and multidimensional statistical data release with strong privacy. Tue03/20/20124:00pm Defense: DissertationSome Mathematical Problems in Design of Free-Form Mirrors and LensesHasan Palta, Emory UniversityContact: Hasan Palta, hpalta@emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 39.1 kB)Show abstractIn this dissertation, we investigate several optics-related problems. The problems discussed in Chapters 1, 2, and 3 are concerned with the determination of surfaces reshaping collimated beams of light to obtain a priori given intensities on prescribed target sets. In optics, such transformations are performed by lenses and/or mirrors whose shapes need to be determined in order to satisfy the application requirements. These are inverse problems, which in analytical formulations lead to nonlinear partial differential equations of Monge-Amp\`{e}re type. In Chapter 4, we present several different designs of radiant energy concentrators. Our goal in these designs is to obtain a device that can capture solar rays with maximal efficiency. Fri03/16/20124:00pm Seminar: CombinatoricsMulti-commodity distribution using PageRankPaul Horn, Harvard UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 37.7 kB)Show abstractDiscontent breaks out on a graph!  Unhappiness, in the form of demand for various commodities spreads  according to a variation of the contact process beginning with some initial seed.  We wish to schedule shipments  of goods in order to ensure that demand (and hence unhappiness) is squelched.  On the other hand, shipments  are expensive so we wish to limit the total amount of shipments we make and only ship to 'important' vertices.   In this talk, we investigate a scheme which guarantees that all demand is met, and hence the contact process  dies out, quickly (with high probability).  When not all vertices are sent shipments, we get bounds on the  'escape probability' in terms of PageRank (and when there are multiple commodities, we get better bounds in  terms of a vectorized version of PageRank). Fri03/09/20124:00pm Seminar: CombinatoricsThe Lagrangian of a hypergraph and its application to extremal problemsYuejian Peng, Indiana State UniversityContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.2 kB)Show abstractIn 1965 Motzkin and Straus established a connection between the maximum clique number and the Lagrangian of a graph, and provided a new proof of Turan's theorem. This new proof aroused interest in the study of Lagrangians of hypergraphs. In the 1980's, Frankl and Rodl disproved the well-known jumping constant conjecture of Erdos by using Lagrangians of hypergraphs as a tool. We present more applications of Lagrangians of hypergraphs in determining non-jumping numbers of hypergraphs. We also present some Motzkin-Straus type results for hypergraphs