Friday, October 10, 2008
01:00 pm
Arithmetic of Fundamental Groups
Speaker: Chris Rasmussen, Wesleyan University<br>Abstract: In this talk, we introduce the arithmetic fundamental group attached to an algebraic curve, and describe the Galois representations attached to its automorphism group. These representations are quite useful in answering questions about the arithmetic of algebraic varieties, and we will try to make this explicit with lots of examples and open problems.
Exley Science Center - ESC 618
Friday, October 17, 2008
01:00 pm
Representations of positive ternary quadratic forms
Speaker: Byeong-Kweon Oh, Seijong University<br>Abstract: A positive definite quadratic form f is said to be regular if it globally represents all integers that are represented by the genus of f. In 1997, Jagy, Kaplansky and Schiemann provided a list of 913 candidates of primitive positive definite regular ternary quadratic forms, and all but 22 of them are verified to be regular. In this talk we show that 8 of these 22 candidates are, in fact, regular. We also show some finiteness result on ternary forms that represent every eligible integer in some arithmetic progression.
Exley Science Center - ESC 618
Friday, October 24, 2008
01:00 pm
Arithmetic Topology
Speaker: Cam McLeman<br>Abstract: One might measure the strength of a mathematical analogy by seeing how few independent identifications are needed before two seemingly disparate theories begin to look like one and the same. In this talk, well explain a blossoming field of mathematics called arithmetic topology, stemming from the remarkably strong analogy obtained only by identifying primes in number fields with knots embedded in 3-manifolds. In addition to being a rather quirky and amusing analogy, the analogy has furthered both fields of study by adopting the others techniques and by taking cues from the others results. This talk will motivate the original analogy between knots and primes and give some examples of current research being fueled by this analogy.
Exley Science Center - ESC 618
Friday, October 31, 2008
01:00 pm
Injectivity and surjectivity of maps between curves
Speaker: Mike Zieve, IAS<br>Abstract: Let f be a morphism of curves over a finite field k. If k is sufficiently large compared to the degree of f and genera of the curves, then injectivity of f on k-rational points is equivalent to surjectivity. Surprisingly, the bounds on the size of k for these two implications seem to have wildly different orders of magnitude. I will explain this and give examples and properties of the bijective maps -- for instance, if f is bijective on k-rational points for a sufficiently large field k, then f is bijective on r-rational points for infinitely many finite extensions r of k. I will also discuss recent progress towards classifying all maps with the latter property in case the curves have genus zero, based on connections with curves having large automorphism group and instances of a positive characteristic analogue of Riemann's existence theorem.
Exley Science Center - ESC 618
Friday, December 05, 2008
01:30 pm
Class number indivisibility for function fields
Speaker: Siman Wong, UMass Amherst<br>Abstract: We will review facts about class numbers of quadratic function fields, and we will discuss new results by way of quadratic forms over function fields.*Note the unusual time.
Exley Science Center - ESC 618
Friday, February 06, 2009
01:00 pm
Deformations of Galois Representations
Speaker: Michael Broshi, UMASS<br>Abstract: In this talk we will introduce the basic notions of the deformation theory of Galois representations. This study became popularized after its use in the proof of Fermat's Last Theorem. We will discuss some applications and, time permitting, some recent developments.This will be an introductory talk and no familiarity with the subject is assumed.
Exley Science Center - ESC 618
Friday, February 20, 2009
01:00 pm
Pure generation by pure-projective modules
Speaker: Philipp Rothmaler, CUNY<br>Abstract: The behavior of, for instance, flat modules (resp. torsion-free abelian groups) is determined by that of the finitely generated projectives (resp. free abelian groups of finite rank)--whose direct limits they are--within the context of finitely presented modules (resp. finitely generated abelian groups). As an example, direct products of flat modules are flat if and only if the ring (a cyclic projective module!) is coherent, a property that can be expressed in terms of finitely presented modules. (The ring of integers is coherent, and products of torsion-free abelian groups are torsion-free.) It will be shown where this leads when the context is broadened to arbitrary pure-projective modules (in case of the integers, to arbitrary direct sums of finitely generated abelian groups). I will try and keep the talk accessible to a general algebraic audience.
Exley Science Center - ESC 618
Friday, March 27, 2009
01:00 pm
Eisenstein series on loop groups
Speaker: Kyu-Hwan Lee, UCONN<br>Abstract: After reviewing affine Lie algebras and loop groups, we will define Eisenstein series on loop groups and calculate their constant terms. It will be shown that the constant terms satisfy functional equations given by affine Weyl group symmetries.
Exley Science Center - ESC 618
Friday, April 03, 2009
01:00 pm
Schur-sigma groups and unramified p-extensions
Speaker: Michael Bush, Smith College<br>Abstract The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways which will be explained in the first part of the talk along with an overview of some earlier results on an explicit family of finite groups of this type. In the second part I will report on work in progress to catalog all such finite groups of small order when p = 3. In the process, some new infinite families have been discovered.
Exley Science Center - ESC 618
Friday, April 10, 2009
01:00 pm
Field descent method and its applications
Speaker: Ka Hin Leung, National University of Singapore<br>Abstract In the study of some combinatorial objects with a suitable group G of symmetries, it often involves certain equations in cyclotomic integers. These equations arise from applying complex representations to group ring equations characterizing the combinatoric objects in question. The three almost exclusively used methods in this direction are multiplier theorems, the self-conjugacy approach and the field descent method. Field descent method deals with equation of the form |X|2 = n where n is an integer and X is in Z[fm]. The main objective is to find a computable integer F(m,n), a divisor of m such that X f m j is in Z[fF(m,n)].In this talk, we will highlight the main idea of the method and demonstrate how it can be applied in the study of difference sets, circulant Hadamard matrix and Barker conjectures.
Exley Science Center - ESC 618
Friday, April 17, 2009
01:00 pm
Ranks of elliptic curves in towers of function field extensions
Speaker: Lisa Berger<br>Abstract: Given an elliptic curve E over a global field K, it is a theorem of Mordell and Weil that the K-rational points on E form a finitely generated abelian group. For K=\mathbb(Q) it is conjectured that there exist elliptic curves with Mordell-Weil groups E(K) of arbitrarily large rank. We'll discuss some current work on ranks of elliptic curves and higher dimensional abelian varieties in towers of function field extensions.
Exley Science Center - ESC 618
Friday, September 10, 2010
01:00 pm
- 05:00 pm
On a Certain PSL(2, Z) 1-Cocycle
Speaker: Sheldon JoynerAbstract: Classically, if some manifold M is equipped with an action of a subgroup G of PSL(2, Z) under which a certain space F of functions on M transforms via a 1-cocycle, the latter is referred to as an automorphy factor, and the functions F are said to be G-modular. In this talk, I will produce an injective 1-cocycle of PSL(2, Z) into a certain group of formal power series which extends the well-known identification of the fundamental group of P^1\{0,1,infty} with associated Chen series. This cocycle may be regarded as a quasi-automorphy factor for sections of the universal prounipotent bundle with connection on PSL(2, Z) - in particular for the polylogarithm generating series Li(z). I will go on to show that the quasi-modularity of Li(z) may be used to give a family of proofs of the analytic continuation and functional equation for the Riemann zeta function.Moreover, under this cocycle, the involutive generator of PSL(2, Z) maps to the Drinfeld associator, while the infinite cyclic generator maps to an R-matrix, in Drinfel'd's formal model of the quasi-triangular quasi-Hopf algebras, thereby producing a representation of PSL(2, Z) into tensor products of certain qtqH algebras.Underlying the whole story is a path space realization of PSL(2, Z) using Deligne's idea of tangential basepoint.
ESC 618
Friday, September 17, 2010
01:00 pm
- 02:30 pm
The Restriction Map in the Galois Cohomology of Linear Algebraic Groups
Speaker: Jodi Black, Emory UniversityAbstract: Let k be a field and let G be a connected linear algebraic group over k. We denote the first Galois cohomology set of k with values in G by H^1(k, G). Let L_1, ....L_m be a set of finite field extensions of k of coprime degree. The following is an open question of Jean-Pierre Serre: "Does the product of the restriction maps from H^1(k,G) to H^1(L_1, G) x ...xH^1(L_m, G) have trivial kernel?"We discuss the significance of a positive answer to this question for important open problems in the area and show that it has positive answer for certain families of groups G.
ESC 618
Friday, September 24, 2010
01:00 pm
- 02:30 pm
Embedding Orders in Central Simple Algebras
Speaker: Ben Linowitz, Dartmouth CollegeAbstract: Let $K$ be a number field and $B$ be a central simple algebra defined over $K$. In 1932, Albert, Brauer, Hasse and Noether explicitly characterized the finite extensions $L$ of $K$ which may be embedded into $B$. An integral refinement immediately suggestsitself: given an order $\Omega\subset \mathcal{O}_L$ and a maximal order $\mathcal M \subset B$, when does there exist an embedding of $\Omega$ into $\mathcal M$? This question turns out to be more subtle. In the case that $B$ is a quaternion algebra satisfying the Eichler condition, Chinburg and Friedman have shown that the proportion of isomorphism classes of maximal orders of $B$ admitting an embedding of $\Omega$ is either $0$, $\frac{1}{2}$ or $1$. Arenas-Carmona considered a broad class of higher rank central simple algebras and determined the maximal orders admitting an embedding of $\mathcal{O}_L$. We consider central simple algebras of dimension $p^2$ for $p$ an odd prime and show that arbitrary commutative orders $\Omega$ in a degree $p$ extension of $K$ embed into none, all or exactly one out of every $p$ isomorphism classes of maximal orders. The maximal orders admitting an embedding of $\Omega$ are explicitly characterized using the Bruhat-Tits building for $SL_p$.
ESC 618
Friday, October 01, 2010
01:00 pm
- 02:30 pm
Special Values of Goss L-Functions in Positive Characteristics
Speaker: Matt Papanikolas, Texas A&MAbstract: Values of Dirichlet L-functions at positive integers are expressible in terms of powers of pi and values of polylogarithms at algebraic numbers. In this talk we will focus on finding analogies of these results over function fields of positive characteristic. In particular, we will consider special values of Goss L-functions for Dirichlet characters, which take values in the completion of the rational function field in one variable over a finite field. Building on work of Anderson for the case of L(1,chi), we deduce various power series identities on tensor powers of the Carlitz module that are "log-algebraic" and in turn use these formulas to determine exact values of L(n,chi) for arbitrary n > 0. Moreover, we relate these L-series values to powers of the Carlitz period and values of Carlitz polylogarithms at algebraic points.
ESC 618
Friday, October 08, 2010
01:00 pm
- 02:30 pm
Almost Universal Ternary Sums of Triangular Numbers
Speaker: Anna Haensch, Wesleyan UniversityAbstract: A fundamental question in number theory is to study the representations of positive integers as sums of polygonal numbers. This problem has a rich history, beginning with Gauss who showed in 1796 that all positive integers can be expressed as a sum of three triangular numbers. This is equivalent to representing every positive integer of the form 8n+3 by a sum of three squares. Extending this idea, in 1862 Liouville determined all triples (a,b,c) of positive integers for which the ternary sums aT_x+bT_y+cT_z are universal, that is, representing all positive integers. Here T_x = x(x+1)/2 is the polynomial representing all triangular numbers. The next logical question is to determine for which triples the ternary sums would be almost universal, that is, representing all but finitely many positive integers. Recently in 2008 Kane and Sun compile a list of sufficient conditions, and formulate a conjecture for necessary conditions. Later in 2009 Chan and Oh resolve this conjecture by providing a complete classification of almost universal ternary sums of triangular numbers. In this talk I will explain the results of this paper.
ESC 618
Friday, October 15, 2010
01:00 pm
- 02:30 pm
The Dwork Family and Hypergeometric Functions
Speaker: Adriana SalernoAbstract: In his work studying the Zeta functions of families ofhypersurfaces, Dwork came upon a one-parameter family ofhypersurfaces in $\mathbb{P}^{n-1}$ (now known as the Dworkfamily), defined by:$$X_{\lambda}:x_1^n+\cdots +x_n^n-n\lambdax_1\cdots x_n=0.$$ These examples were not only useful to Dwork inhis study of his deformation theory for computing Zeta functions offamilies, but they have also proven to be extremely useful tophysicists working in mirror symmetry. A startling result is thatthese families are very closely linked to hypergeometric functions.This phenomenon was carefully studied by Dwork in the cases where$n=3,4$ and for $n=5$ by Candelas, de la Ossa, andRodr\'{i}guez-Villegas. Dwork, Candelas, et.al. observed that, forthese families, the differential equation associated to theGauss-Manin connection is in fact hypergeometric. We have developeda computer algorithm, implemented in Pari-GP, which can check thisresult for larger values of $n$ by computing the Gauss-Maninconnection and the parameters of the hypergeometric differentialequation.
ESC 618
Friday, November 05, 2010
01:00 pm
- 02:30 pm
Descent on the Congruent Number Elliptic Curves
Speaker: Nick Rogers, University of RochesterAbstract: A classical Diophantine problem asks which positive integers n occur as areas of right triangles with rational sides. Such "congruent numbers"correspond to elliptic curves y^2 = x^3 - n^2 x with positive rank. In this talk I'll describe how to compute certain Selmer groups associated to these elliptic curves. These descent calculations lead to results on non-congruent numbers and Tate-Shafarevich groups, and shed some light on the behavior of ranks in families of quadratic twists.
ESC 618
Friday, November 12, 2010
01:00 pm
- 02:30 pm
A Compactification of the Space of Algebraic Maps from P^1 to a Grassmannian
Speaker: Yijun ShaoAbstract: The moduli space of algebraic maps (i.e., morphisms) of degree d from P^1 to a Grassmannian is a nonsingular, noncompact, quasi-projective variety. In this talk, I will describe an explicit construction of a compactification for this space satisfying the following properties: 1. the compactification is a nonsingular projective variety; 2. the boundary is a divisor with normal crossings (i.e., a union of nonsingular codimension 1 subvarieties which intersect transversally). The construction is based on a compactification given by a Quot scheme. This Quot scheme is a nonsingular projective variety, but the boundary is in general singular and of codimension>1. The main tool of the construction is blowup. We first define a sequence of nested closed subschemes of the boundary, and then blowup the Quot scheme successively along these closed subschemes. The final outcome of the blowups is a compactification with the desired properties.
ESC 618
Friday, December 03, 2010
01:00 pm
- 02:30 pm
Reducible Galois representations and Hecke eigenclasses
Speaker: Avner Ash, Boston CollegeAbstract: Serre's conjecture (now a theorem due to Khare-Wintenberger and Kisin) gives a tight relationship between odd 2-dimensional mod p irreducible representations of G_Q (the absolute Galois group of Q) and mod p modular forms. In 2000, W. Sinnott and I published a generalization of this conjecture in which we connect mod p n-dimensional representations rho of G_Q and mod p cohomology of arithmetic subgroups of SL(n,Z). An interesting feature of this conjecture is the complicated parity condition on rho when rho is reducible. I have recently proved this conjecture in many cases when rho is a direct sum of 1-dimensional representations. This talk will explain these things from scratch, more or less.
ESC 618
Friday, January 28, 2011
03:00 pm
- 04:00 pm
Computing Isogeny Volcanoes
Speaker: Dustin Moody, NISTAbstract: Isogenies are maps between elliptic curves. Isogeny volcanoes are an interesting structure that have had several recent applications in cryptography. An isogeny volcano is a connected component of a larger graph. We further explore properties of and how to compute volcanoes given that we have already computed one of a different degree. This allows us to compute volcanoes of composite degree more efficiently than a direct construction using modular polynomials.
ESC 618
Friday, February 11, 2011
03:00 pm
- 04:00 pm
2 Nilpotent Real Section Conjecture
Speaker: Kirsten Wickelgren, HarvardAbstract: Grothendieck's anabelian conjectures say that hyperbolic curves over certain fields should be K(pi,1)'s in algebraic geometry. It follows that points on such a curve are conjecturally the sections of etale pi_1 of the structure map. These conjectures are analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. This talk will start with an introduction to Grothendieck's anabelian conjectures, and then present a2 nilpotent real section conjecture: for a smooth curve X over R with negative Euler characteristic, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that the set of real points equipped with a real tangent direction of the smooth compactification of X is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R) action, showing a 2-nilpotent birational real section conjecture.
ESC 618
Friday, February 18, 2011
03:00 pm
- 04:00 pm
Computing Hilbert Modular forms, or Nonsolvable number fields ramified at one prime
Speaker: John VoightAbstract: We use explicit methods for computing with quaternion algebras and the cohomology of Shimura curves to compute spaces of Hilbert modular forms. As a consequence, we construct nonsolvable finite extensions of the rational numbers ramified only at 3 and 5, respectively. This is joint work with Matthew Greenberg and Lassina Dembele.
ESC 618
Friday, February 25, 2011
03:00 pm
- 04:00 pm
Cyclic Extensions and the Local Lifting Problem
Speaker: Andrew Obus, ColumbiaAbstract: The lifting problem we will consider roughly asks: given a smooth projective curve X over an algebraically closed field of characteristic p and a finite group G of automorphisms of X, does there exist a smooth, projective curve X' in characteristic zero and a finite group of automorphisms G' of X' such that (X', G') lifts (X, G)? It turns out that solving this lifting problem reduces to solving a local lifting problem in a formal neighborhood of each point of X where G acts with non-trivial inertia. The Oort conjecture states that this local lifting problem should be solvable whenever the inertia group is cyclic. A new result of Stefan Wewers and the speaker shows that the local lifting problem is solvable whenever the inertia group is cyclic of order not divisible by p^4, and in many cases even when the inertia group is cyclic and arbitrarily large. We will discuss this result, after giving a good amount of background on the local lifting problem in general.
ESC 618
Friday, April 22, 2011
03:00 pm
- 04:00 pm
Automorphisms of Generalized GK Curves
Speaker: Beth Malmskog, Colorado State UniversityAbstract: Curves with as many points as possible over a finite field$\mathbb{F}_q$ under the Hasse-Weil bound are called maximal curves.Besides being interesting as extremal objects, maximal curves haveapplications in coding theory. Maximal curves may also have agreat deal of symmetry, i.e. have an automorphism group which islarge compared to the curve's genus. I will discuss two families ofmaximal curves and find a large subgroup of each curve's automorphismgroup. We also give an upper bound for the size of the automorphismgroup.
ESC 618
Friday, September 16, 2011
04:00 pm
- 05:00 pm
Combinatorics of the Gindikin-Karpelevich formula
Speaker: Ben Salisbury, UCONNAbstract: A crystal is a combinatorial framework to be used to discuss certain properties of Lie algebras and their representations. In particular, the crystal of the negative part of the universal enveloping algebra of a Lie algebra may be used to expand important products arising from integrals over p-adic groups as sums. In this talk, we explain how the realization of this crystal in terms of Young tableaux yields a statistic defining a coefficient making this expansion possible.
ESC 618
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