All Seminars
| Title: Local-global principles for torsors over arithmetic curves |
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| Seminar: Algebra and Number Theory |
| Speaker: David Harbater of University of Pennsylvania |
| Contact: R. Parimala, parimala@mathcs.emory.edu |
| Date: 2012-04-13 at 3:00PM |
| Venue: W304 |
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| Abstract: This talk, on joint work with Julia Hartmann and Daniel Krashen, concerns local-global principles over function fields of curves that are defined over a complete discretely valued field. Classically, one studies such principles over number fields, or over function fields of curves defined over a finite field. In that situation, if $G$ is an algebraic group, one can ask whether a $G$-torsor (principal homogeneous space for $G$) over the field must be trivial whenever it is locally trivial. This does not always hold, but the obstruction is always finite if $G$ is a linear algebraic group. This talk will study the analogous question in our situation. Applications include results about quadratic forms and central simple algebras. |
| Title: Modular and Lexical Matchings in the Middle Levels Graph |
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| Defense: Masters Thesis |
| Speaker: Kevin Wingfield of Emory University |
| Contact: Kevin Wingfield, kwingf2@emory.edu |
| Date: 2012-04-11 at 4:00PM |
| Venue: W302 |
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| Abstract: Classes of explicitly defined matchings in the middle levels bipartite graph induced by the Boolean lattice are investigated. The original motivation for an investigation into the Hamiltonicity of the middle levels bipartite graph came from the conjecture of Havel. Here, we collect some known results and present some new observations that indicate that, when disjoint, the 2-factors obtained from taking the union of parts of these matchings always contain short cycles. |
| Title: Option pricing models: Black-Scholes or hyperbolic Levy process? |
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| Defense: Honors thesis |
| Speaker: Chen Chen of Emory University |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2012-04-10 at 4:00PM |
| Venue: W304 |
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| Abstract: This thesis is an investigation into two option pricing models: widely-used Black-Scholes model and a modification of it, the hyperbolic Levy model. First, we have a detailed discussion about the celebrated Black-Scholes model. However, clearly there are many deficiencies in the Black-Scholes assumptions. In order to refine the Black-Scholes model, Eberlein and Keller (1995) introduced the hyperbolic Levy motion and claimed that the new model can provide a better valuation of derivative securities. We perform several statistical tests and show that the hyperbolic distributions can be well fitted to the financial data. This observation suggests us to replace the geometric Brownian motion in the Black-Scholes model by the hyperbolic Levy process and build the hyperbolic Levy pricing model. After an introduction into the Levy process theory, we attempt to numerically calculate the value of options according to the hyperbolic Levy model. |
| Title: Medical Devices Cybersecurity |
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| Seminar: Computer Science |
| Speaker: Kevin Fu of University of Massachusetts Amherst |
| Contact: Li Xiong, lxiong@mathcs.emory.edu |
| Date: 2012-04-06 at 3:00PM |
| Venue: MSC W301 |
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| Abstract: The Institute of Medicine commissioned my 2011 report on the role of trustworthy software in the context of U.S. medical device regulation. This talk will provide a glimpse into the risks, benefits, and regulatory issues for innovation of trustworthy medical device software. Today, it would be difficult to find medical device technology that does not critically depend on computer software. The technology enables patients to lead more normal and healthy lives. However, medical devices that rely on software (e.g., drug infusion pumps, linear accelerators) continue to injure or kill patients in preventable ways– despite the lessons learned from the tragic radiation incidents of the Therac-25 era. The lack of trustworthy medical device software leads to shortfalls in properties such as safety, effectiveness, dependability, reliability, usability, security, and privacy. Come learn a bit about the science, technology, and policy that shapes medical device software. Bio: Kevin Fu is an Associate Professor of Computer Science and adjunct Associate Professor of Electrical \& Computer Engineering at the University of Massachusetts Amherst. Prof. Fu makes embedded computer systems smarter: better security and safety, reduced energy consumption, faster performance. His most recent contributions on trustworthy medical devices and computational RFIDs appear in computer science and medical conferences and journals. The research is featured in critical articles by the NYT, WSJ, and NPR. Prof. Fu served as a visiting scientist at the Food \& Drug Administration, the Beth Israel Deaconess Medical Center of Harvard Medical School, and MIT CSAIL. He is a member of the NIST Information Security and Privacy Advisory Board. Prof. Fu received a Sloan Research Fellowship, NSF CAREER award, and best paper awards from various academic silos of computing. He was named MIT Technology Review TR35 Innovator of the Year. Prof. Fu received his Ph.D. in EECS from MIT when his research pertained to secure storage and web authentication. He also holds a certificate of achievement in artisanal bread making from the French Culinary Institute. |
| Title: Lines, Incidences, and a Conjecture of Solymosi |
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| Seminar: Combinatorics |
| Speaker: Albert Bush of Georgia Tech |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2012-04-06 at 4:00PM |
| Venue: W306 |
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| Abstract: Given any n points in the plane, the celebrated Szemeredi-Trotter theorem gives bounds on the number of lines that can each hit at least k points. J. Solymosi conjectured a significantly tighter bound with the stronger condition that the points be a grid and the lines be in general position -- no parallel lines, and no three lines meet at a single point. Using methods of Elekes as well as Borenstein and Croot, we prove Solymosi's conjecture. This is joint work with Gagik Amirkhanyan, Ernie Croot, and Chris Pryby. |
| Title: Problems on Sidon sets of integers |
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| Seminar: Dissertation Defense |
| Speaker: Sangjune Lee of Emory University |
| Contact: Sangjune Lee, slee242@emory.edu |
| Date: 2012-04-03 at 2:30PM |
| Venue: W304 |
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| Abstract: A 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} |
| Title: Topics in Ramsey Theory |
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| Defense: Dissertation |
| Speaker: Domingos Dellamonica Jr. of Emory University |
| Contact: Domingos Dellamonica Jr., ddellam@emory.edu |
| Date: 2012-04-03 at 4:00PM |
| Venue: W304 |
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| Abstract: In this thesis we discuss two results in Ramsey Theory.\\ \\ Result I: the size-Ramsey number of a graph $H$ is the smallest number of edges a graph $G$ must have in order to force a monochromatic copy of $H$ in every $2$-coloring of the edges of $G$. In 1990, Beck studied the size-Ramsey number of trees: he introduced a tree invariant $\beta(\cdot)$, and proved that the size-Ramsey number of a tree $T$ is at least $\beta(T)/4$. Moreover, Beck showed an upper bound for this number involving $\beta(T)$, and further conjectured that the size-Ramsey number of any tree~$T$ is of order $\beta(T)$. We answer his conjecture affirmatively. Our proof uses the expansion properties of random bipartite graphs.\\ \\ Result II: We prove the following metric Ramsey theorem. For any connected graph $G$ endowed with a linear order on its vertex set, there exists a graph $R$ such that in every coloring of the $t$-sets of vertices of $R$ it is possible to find a copy $G'$ of $G$ inside $R$ satisfying the following two properties:\\ \begin{itemize} \item the distance between any two vertices $x, y \in V(G')$ in the graph $R$ is the same as their distance within $G'$; \item the color of each $t$-set in $G'$ depends only on the graph-distance metric induced in $G'$ by the ordered $t$-set. \end{itemize} |
| Title: On highly connected monochromatic subgraphs |
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| Seminar: Combinatorics |
| Speaker: Tomasz Luczak of Emory University and Adam Mickiewicz University |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2012-03-30 at 4:00PM |
| Venue: W306 |
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| Abstract: |
| Title: Quasi Isometric Properties of Graph Braid Groups |
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| Defense: Dissertation |
| Speaker: Praphat Fernandes of Emory University |
| Contact: Praphat Fernandes, pxferna@emory.edu |
| Date: 2012-03-30 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: In 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. |
| Title: Monkey fields |
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| Colloquium: N/A |
| Speaker: Brian Conrad of Stanford |
| Contact: Skip Garibaldi, skip@mathcs.emory.edu |
| Date: 2012-03-29 at 3:00PM |
| Venue: W306 |
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| Abstract: Joseph 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. |