# Seminars archive

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 Upcoming Seminars Tue09/25/20184:00pm Seminar: AlgebraTitle to be announcedRenee Bell, University of PennsylvaniaContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 19.8 kB) Mon10/01/20184:00pm Seminar: CombinatoricsOn the Erdos-Gyarfas distinct distances problem with local constraintsCosmin Pohoata, The California Institute of TechnologyContact: Dwight Duffus, dwightduffus@emory.eduVenue: Mathematics and Science Center, Room E408Download printable flyer (PDF, 115 kB)Show abstractIn 1946 Erdos asked to determine or estimate the minimum number of distinct distances determined by an n-element planar point set V. He showed that a square integer lattice determines \Theta(n/\sqrt{log n}) distinct distances, and conjectured that any n-element point set determines at least n^{1−o(1)} distinct distances. In 2010-2015, Guth and Katz answered Erdos’s question by proving that any n-element planar point set determines at least \Omega(n/log n) distinct distances. In this talk, we consider a variant of this problem by Erdos and Gyarfas. For integers n, p, q with p \geq q \geq 2, determine the minimum number D(n,p,q) of distinct distances determined by a planar n-element point set V with the property that any p points from V determine at least q distinct distance. In a recent paper, Fox, Pach and Suk prove that when q = {p \choose 2} - p + 6, D(n,p,q) is always at least n^{8/7 - o(1)}. We will discuss a recent improvement of their result and some new bounds for a related (graph theoretic) Ramsey problem of Erdos and Shelah which arise. This is joint work with Adam Sheffer. Tue10/16/20184:00pm Seminar: AlgebraTitle to be announcedEva Bayer Fluckinger, EPFLContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 20.3 kB) Tue10/23/20184:00pm Seminar: AlgebraJoint Athens-Atlanta Number Theory SeminarLarry Rolen and Bianca Viray, Vanderbilt and University of WashingtonContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 23.7 kB) Tue10/30/20184:00pm Seminar: AlgebraTitle to be announcedAnne Qu\'eguiner-Mathieu, ParisContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 19.6 kB) Tue11/27/20184:00pm Seminar: AlgebraTitle to be announcedNatalie Paquette, CaltechContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 19.3 kB) Past Seminars Wed02/27/20133:00pm Seminar: AlgebraCanonical Representatives for divisor classes on tropical curvesFarbod Shokrieh, Georgia TechContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.1 kB)Show abstractTropical curves are algebraic curves defined over the tropical semi-ring. They essentially carry the same information as metric graphs. There is a reasonable theory of divisors in this setting. For example, there is a tropical analogue of the Riemann-Roch theorem. The main technique in studying divisors on tropical curves is often to look for nice canonical representatives in linear equivalence classes. In this talk, we will describe various canonical representatives for divisor classes on tropical curves. We first revisit the concept of "reduced divisors" (which is the main ingredient needed to prove the Riemann-Roch theorem) and explain their various interpretations. We then discuss "break divisors" from multiple points of view. If time permits we discuss the classical analogues of these representatives and give some applications. No prior knowledge in the subject will be assumed. This talk is based on joint works with M. Baker, with M. Baker, G. Kuperberg, A. Yang, and with Ye Luo. Fri02/22/20134:00pm Seminar: CombinatoricsAsymptotic distribution for the birthday problem with multiple coincidencesSkip Garibaldi, Emory UniversityContact: Dwight Duffus, dwight@mathcs.meory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 44.7 kB)Show abstractThis talk is about joint work with Richard Arratia and Joe Kilian on a version of the birthday problem. We study the random variable $\mathbf{B}(c, n)$, which counts the number of balls that must be thrown into $n$ equally-sized bins in order to obtain $c$ collisions. We determine the limiting distribution for $(\mathbf{B}(c,n))^2/(2n)$ where $c$ is a function of $n$ that is $o(\sqrt{n})$, among other results. The basis for this result is a coupling. Wed02/20/20133:00pm Seminar: AlgebraThe non-Abelian Whitney theorem and the Higher Pairing on GraphsEric Katz, University of WaterlooContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 39.3 kB)Show abstractFor a connected graph G with no 1-valent vertices, the set of based reduced graphs is sufficient to recover the graph. This is non-commutative invariant of the graph. Its abelianization, the cycle space of the graph is sufficient to recover the graph up to two moves by Whitney's 2-isomorphism theorem. In this talk, we will consider a unipotent invariants that interpolates between the set of paths and its abelianization. There is a related isomorphism theorem that lets you recover the graph from the analogous unipotent invariant. In the same spirit, we will introduce a unipotent pairing between paths and ordered n-tuples of cycles which was inspired by Chen's theory of iterated integrals and which generalizes the length pairing between cycles. We conjecture a higher Picard-Lefschetz theorem relating this pairing to the asymptotics of iterated integrals on degenerating families of curves, and state a sort of Torelli theorem relating the asymptotics to the dual graph of a degeneration. Wed02/13/20133:00pm Seminar: Number TheoryPower series expansions for modular formsJohn Voight, University of VermontContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 36.2 kB)Show abstractWe exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field including the image of CM points, and equations for Shimura curves. Fri02/08/20134:00pm Seminar: CombinatoricsPhase Transitions in Ramsey-Turán TheoryJozsef Balogh, The University of Illinois at Urbana-ChampaignContact: Dwight Duffus, dwight@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 42.3 kB)Show abstractDenote by $RT(n,L,f(n))$ the maximum number of edges of an $L$-free graph with independence number at most $f(n)$. This concept was defined by Erd\H{o}s and S\'{o}s in 1970. In this talk I will survey some of the recent progress on studying $RT(n,L,f(n))$ and some related questions. The newer results are partially joint with Hu, Lenz and Simonovits. Thu02/07/20134:00pm ColloquiumA new filtration of the Magnus kernel of the Torelli group - CANCELLEDTaylor McNeill, Rice UniversityContact: Steve Batterson, sb@mathcs.emory.eduVenue: Download printable flyer (PDF, 47 kB)Show abstractFor a surface $S$, the Torelli group is the group of orientation preserving homeomorphisms of $S$ that induce the identity on homology. The Magnus representation represents the action on $F/F''$ where $F$ is the fundamental group of $S$ and $F''$ is the second term of the derived series. For many years it was unknown whether the Magnus representation of the Torelli group is faithful. In recent years there have been many developments on this front including the result of Church and Farb that the kernel of the Magnus representation, denoted $Mag(S)$, is infinitely generated. I show that, not only is $Mag(S)$ highly non-trivial but that it also has a rich structure as a group. Specifically, I define an infinite filtration of $Mag(S)$ by subgroups, called the higher order Magnus subgroups, $M_n$. I show that for each n the quotient $M_n/M_n+1$ is infinitely generated. To do this, I define a Johnson type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image. Wed02/06/20133:00pm Seminar: Number TheoryThe degrees of divisors of $x^n-1$Lola Thompson, University of GeorgiaContact: David Zureick-Brown, dzb@mathcs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 51.5 kB)Show abstractWe discuss what is known about the following questions concerning the degrees of divisors of $x^n-1 in Z[x]$, as n ranges over the natural numbers:\\ \\ 1. How often does $x^n-1$ have AT LEAST ONE divisor of every degree between 1 and n?\\ \\ 2. How often does $x^n-1$ have AT MOST ONE divisor of every degree between 1 and n?\\ \\ 3. How often does $x^n-1$ have EXACTLY ONE divisor of every degree between 1 and n?\\ \\ 4. For a given m, how often does $x^n-1$ have a divisor of degree m?\\ \\ We will also discuss what changes when Z is replaced by the finite field $F_p$. A portion of this talk is based on joint work with Paul Pollack. Wed02/06/201312:50pm Seminar: Numerical Analysis and Scientific ComputingFokker-Planck Equation Method for Predicting Viral Signal Propagation in Social NetworksXiaojing Ye, Georgia Institute of TechnologyContact: James Nagy, nagy@math.cs.emory.eduVenue: Mathematics and Science Center, Room W306Download printable flyer (PDF, 43.2 kB)Show abstractWe consider the modeling and computations of random dynamical processes of viral signals propagating over time in social networks. The viral signals of interests can be popular tweets on trendy topics in social media, or computer malware on the Internet, or infectious diseases spreading between human or animal hosts. The viral signal propagations can be modeled as diffusion processes with various dynamical properties on graphs or networks, which are essentially different from the classical diffusions carried out in continuous spaces. We address a critical computational problem in predicting influences of such signal propagations, and develop a discrete Fokker-Planck equation method to solve this problem in an efficient and effective manner. We show that the solution can be integrated to search for the optimal source node set that maximizes the influences in any prescribed time period. This is a joint work with Profs. Shui-Nee Chow (GT-MATH), Hongyuan Zha (GT-CSE), and Haomin Zhou (GT-MATH). Tue02/05/20134:00pm ColloquiumKnot Polynomials in the Melvin-Morton-Rozansky Expansion of the Colored Jones PolynomialAndrea Overbay, University of North CarolinaVenue: Mathematics and Science Center, Room W301Download printable flyer (PDF, 40 kB)Show abstractBoth the Alexander polynomial and the Jones polynomial are two well-known knot invariants. The Melvin-Morton conjecture, proved by Bar-Natan and Garoufalidis and further generalized by Rozansky, provides a relationship between these two invariants. The relationship appears when expanding the colored Jones polynomial in a certain way. Within this expansion, we get more polynomial invariants of the knot. During this talk, we will discuss some polynomial knot invariants including the Alexander polynomial and the colored Jones polynomial. Then we will describe the polynomial invariants appearing in the Melvin-Morton-Rozansky expansion for some simple knots and outline a method for computing them. Mon02/04/20134:00pm ColloquiumSuperimposed CodesZoltan Furedi, University of Illinois at Urbana-Champaign and Renyi Institute of Mathematics of the Hungarian Academy of SciencesContact: Andrzej Rucinski, andrzej@mathcs.emory.eduVenue: Mathematics and Science Center, Room W303Download printable flyer (PDF, 57.9 kB)Show abstractThere are many instances in Coding Theory when codewords must be restored from partial information, like defected data (error correcting codes), or some superposition of the strings. These lead to superimposed codes, a close relative of group testing problems. There are lots of versions and related problems, like Sidon sets, sum-free sets, union-free families, locally thin families, cover-free codes and families, etc. We discuss two cases {\it cancellative} and {\it union-free} codes. A family of sets $\mathcal F$ (and the corresponding code of 0-1 vectors) is called {\bf union-free} if $A\cup B\neq C\cup D$ and $A,B,C,D\in \mathcal F$ imply $\{ A,B\}=\{ C, D \}$. $\mathcal F$ is called $t$-{\bf cancellative} if for all distict $t+2$ members $A_1, \dots, A_t$ and $B,C\in \mathcal F$ $$A_1\cup\dots \cup A_t\cup B \neq A_1\cup \dots A_t \cup C.$$ Let $c_t(n)$ be the size of the largest $t$-cancellative code on $n$ elements. We significantly improve the previous upper bounds of K\"orner and Sinaimeri, e.g., we show $c_2(n)\leq 2^{0.322n}$ (for $n> n_0$). We introduce a method to deal with such problems, namely we first investigate the constant weight case (i.e., uniform hypergraphs).