Mathematics & Computer Science

Seminars and Colloquia

Algebra Seminar

Tuesday, September 27, 2016

04:15 pm - 05:15 pm

Algebra Seminar

Bweong-Kweon Oh (Seoul National University): The number of representations of squares by integral quadratic forms Abstract: Let f be a positive definite integral ternary quadratic form and let r ( k , f ) be the number of representatives of an integer k by f . We say that the genus of f is indistinguishable by squares if for any integer n , r ( n 2 , f ) = r ( n , f ) for any quadratic form f in the genus of f . In this talk, we will give some examples of non trivial genera of ternary quadratic forms which are indistinguishable by squares. Also, we give some relations between indistinguishable genera by squares and a conjecture by Cooper and Lam, and we resolve their conjecture completely. This is a joint work with Kyoungmin Kim.

Exley Science Center Tower ESC 618

Tuesday, September 20, 2016

04:15 pm - 05:15 pm

Algebra Seminar

Christopher Rasmussen (Wes):A (Necessarily Incomplete) Introduction to Frobenioids Abstract: A man who knows a little is sometimes more dangerous than a man who knows nothing at all. In his approach to proving the ABC Conjecture, Mochizuki relies on the concept of a Frobenioid, which in his own words is ``a sort of a category-theoretic abstraction of the theory of divisors on [models of global fields].'' In the present talk, we will attempt to carefully introduce the notion of a Frobenioid and provide a small amount of context. Nothing will be assumed beyond a basic knowledge of category theory and some standard algebra.

Exley Science Center Tower ESC 618

Friday, February 05, 2016

01:15 pm - 02:30 pm

Algebra Seminar

Michael Wijaya, Dartmouth College: A function-field analogue of Conway's topograph Abstract: In "The Sensual (Quadratic) Form", Conway introduced a new visual method to display values of a binary quadratic form Q(x,y)=ax^2+bxy+cy^2 with integer coefficients. This topograph method, as he calls it, leads to a simple and elegant method of classifying integral binary quadratic forms and answering some basic questions about them. In this talk, I will present an analogue of Conway's topograph method for binary quadratic forms with coefficients in F_q[T], where q is an odd prime power. The constructions will take place on the Bruhat-Tits tree of SL(2), which is an analogue of the real hyperbolic plane.

Exley Science Center Tower ESC 618

Friday, December 04, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Michael Kelly (University of Michigan): " Uniform Dilations in High Dimensions"

Abstract: It is a theorem of Glasner that given an infinite subset X of the torus R/Z and an epsilon greater than 0 there exists a positive integer n such that any interval of length epsilon in R/Z contains a point of the set nX (that is, nX is epsilon-dense in R/Z). The set nX is called a dilation of X by n. Alon and Peres have shown that the dilation factor n can be chosen to be a prime or n=f(m) for some integral polynomial f with degree(f)>0 and integer m. We will discuss various developments on these sorts of topics and I'll present joint work with Le Thai Hoang where we consider this phenomenon in higher dimensions.

Exley Science Center (Tower)

Friday, November 06, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Christelle Vincent (UVM): "Compuiting equations in hyperelliptic curves whose Jacobian has CM"

Abstract: It is known that given a totally imaginary sextic field with totally real cubic subfield (a so-called CM sextic field) there exists a non-empty finite set of abelian varieties of dimension 3 that have CM by this field. Under certain conditions on the field and the CM-type, this abelian variety can be guaranteed to be principally polarized and simple. In this talk, we begin by reviewing quickly the situation for elliptic curves with complex multiplication, which is the dimension 1 case of the work we present. We then move to the dimension 3 case, and present an algorithm that takes as input such a field and CM-type, and outputs a period matrix for such an abelian variety. We then check computationally if the abelian variety is the Jacobian of a hyperelliptic curve, and compute an equation for the curve if this is the case. This is joint work with J. Balakrishnan, S. Ionica and K. Lauter.

Exley Science Center (Tower)

Friday, October 09, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Michael Chou '11 (UConn): Torsion of rational elliptic curves over quartic Galois number fields

Abstract : The classification of the torsion subgroup of elliptic curves over \mathbb{Q} was determined by Mazur. The classification over quadratic number fields was completed due to work of Kamienny, Kenku, and Momose. However, over cubic fields the classification is already incomplete. In this talk we discuss a refined version of this problem: let E be an elliptic curve defined over \mathbb{Q} and K be a number field of degree d; what groups appear as E(K)_{\text{tors}}? In particular, we will present a classification over all quartic Galois number fields K and show how the techniques used may be applied to other fields.

Exley Science Center (Tower)

Monday, September 28, 2015

04:45 pm - 06:00 pm

CT Logic Seminar, Reed Solomon (UConn): "Strong reducibilities, RT^1_3 and SRT^2_2"

Abstract : Various strong reductions between Pi^1_2 principles have been used in recent years to shed light on difficult problems in reverse mathematics. I will introduce some of these reductions and discuss their connection to reverse math. The main theorem of the talk is that RT^1_3 is not strongly computably reducible to SRT^2_2. This result is joint work with Damir Dzhafarov, Ludovic Patey and Linda Brown Westrick.

Exley Science Center (Tower)

Friday, September 25, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Liang Xiao (UConn): "Zeros of zeta functions of Artin-Scheier-Witt tower of curves"

Abstract : For a projective and smooth curve over a finite field, the zeros of its zeta function determine the number of points over finite fields. In this talk, we are interested in studying the p-adic valuations of these zeros, especially its asymptotic/periodic behavior over a Z_p Artin-Scheier-Witt tower of curves ... C_n -> ... -> C_0. It turns out that the p-adic valuations of the zeros of the zeta functions for the first few curves determine those for the rest of the curves. This is a joint work with Chris Davis and Daqing Wan.

Exley Science Center (Tower)

Friday, May 01, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Anna Haensch (Duquesne, Wes PhD '13): 'Kneser-Hecke operators for quaternary codes'

Abstract: There is a well known correspondence between lattices and codes via the classical 'construction A.' With this, the weight enumerator for codes corresponds to the theta series for lattices, where one counts the number of codewords by composition, and the other counts the number of vectors in a lattice of a certain length. In this talk, we will explore how some of the attendant machinery of theta series are born out in this correspondence. In particular, we will consider the Kneser-Hecke operator, a code theoretic analogue of the classical Hecke operator.

ESC 618

Friday, April 24, 2015

01:10 pm - 02:00 pm

Masters Thesis Defense and Algebra Seminar, John Bergan: 'The Peter-Weyl Theorem'

Abstract: For finite groups, a decomposition of the regular representation into a direct sum of irreducible subrepresentations is readily obtained with elementary representation theory. Infinite groups, however, pose a far more challenging problem. But if we restrict ourselves to compact groups and use a little functional analysis, then we can still obtain a complete decomposition of the regular representation. This is the Peter-Weyl Theorem. In this talk, I will discuss all the necessary background information and the proof the theorem.

ESC 638

Friday, April 17, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Cameron Hill (Wes): 'The Lang-Weil bounds and the geometry of pseudo-finite fields.'

Abstract: The Lang-Weil bounds are a venerable fact of algebraic geometry that provide reasonably precise estimates of the cardinalities of varieties in finite fields in terms of their dimensions (as calculated in algebraic closures). One model-theoretic corollary of this theorem is the fact that every pseudo-finite field admits a very fine-grained geometry analogous to (but distinct from) the geometry of its algebraic closure, and which also accommodates a larger family of definable sets than varieties alone. In this talk, I will discuss other routes to this geometry on a pseudo-finite field that do not use the Lang-Weil bounds as a starting point. This approach also supplies a novel proof of a relaxed version the Lang-Weil bounds themselves, and I will try to point out how this proof is really different from the classical argument.

ESC 638

Friday, April 10, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Andrew Schultz (Wellesley): 'Parameterizing solutions to Galois embedding problems via modules'

Abstract: The classifying space for elementary $p$-abelian extensions of a field $K$ has long been understood. If $K$ is a Galois extension of a field $F$, then the Galois group has a natural action on this classifying space, and --- at least when $\Gal(K/F)$ is a cyclic $p$-group --- one can develop a dictionary between a certain class of embedding problems and submodules of the corresponding Galois module. Combined with some surprising results concerning the module structure of the classifying space, this allows us to recover some interesting results on the structure of absolute Galois groups. If time permits, we will also discuss some generalizations of these modules and their potential for shedding further light on absolute Galois groups.

ESC 638

Friday, April 03, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Jeremy Rouse (Wake Forest): 'Elliptic curves over $\mathbb{Q}$ and 2-adic images of Galois'

Abstract: Given an elliptic curve $E/\mathbb{Q}$, let $E[2^k]$ denote the set of points on $E$that have order dividing $2^k$. The coordinates of these points are algebraic numbersand using them, one can build a Galois representation $\rho : G_{\mathbb{Q}} \to \GL_{2}(\mathbb{Z}_{2})$.We give a classification of all possible images of this Galois representation. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

ESC 638

Friday, February 27, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Leo Goldmakher (Williams): 'Characters and their nonresidues'

Abstract: Understanding the least quadratic nonresidue (mod p) is a classical problem, with a history stretching back to Gauss. The approach which has led to the strongest results uses character sums, objects which are ubiquitous in analytic number theory. I will discuss character sums, their connection to the least nonresidue, and some recent work of myself and Jonathan Bober (University of Bristol) on a promising new approach to the problem.

ESC 638

Friday, February 20, 2015

01:10 pm - 02:00 pm

Algebra Seminar, David Pollack (Wes): 'A lifting theorem and computations towards an eigencurve for SL_3 (part II)'

Abstract: Fix a prime p. We generalize our early method of computing overconvergent lifts of ordinary p-adic homology classes for SL_3 to allow computations of lifts of non-ordinary, but numerically non-critical classes. Then we will discuss an ongoing calculation to find to any desired degree of accuracy the germ of the projection to weight space of the eigencurve Z around the point z corresponding to the system of Hecke eigenvalues of this over convergent lift. We do this under the conjecturally mild hypothesis that Z is smooth at z. This is joint work with Avner Ash.

ESC 638

Friday, February 20, 2015

01:10 pm - 02:00 pm

Algebra Seminar, David Pollack (Wes): 'A lifting theorem and computations towards an eigencurve for SL_3 (part II)'

Abstract: Fix a prime p. We generalize our early method of computing overconvergent lifts of ordinary p-adic homology classes for SL_3 to allow computations of lifts of non-ordinary, but numerically non-critical classes. Then we will discuss an ongoing calculation to find to any desired degree of accuracy the germ of the projection to weight space of the eigencurve Z around the point z corresponding to the system of Hecke eigenvalues of this over convergent lift. We do this under the conjecturally mild hypothesis that Z is smooth at z. This is joint work with Avner Ash.

ESC 638

Friday, February 06, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Beth Malmskog (Villanova): 'Picard curves with good reduction away from 3, x+y=1, p=u2+v2, and other problems solved by the LLL algorithm

Abstract: A lattice L is the set of integer linear combinations of an independent set of vectors. The set of generating vectors is called a basis for L. A particular lattice can have many bases some good (of shorter vectors which are closer to orthogonal), some bad (of longer vectors which are closer to parallel). In 1982, Henrik Lenstra, Arlen Lenstra, and Lovasz developed a fairly simple polynomial-time algorithm which, given any basis B for a lattice L, will produce a relatively good basis for L. This algorithm has found extensive application in surprising areas. This talk will discuss applications of LLL to recent work with Chris Rasmussen on Picard curves with certain reduction properties, number theoretic problems such as finding minimal polynomials for algebraic numbers, attacking lattice-based cryptosystems, and more.

ESC 638

Friday, January 30, 2015

01:10 pm - 02:00 pm

Algebra Seminar, Rachel Davis (Purdue): 'Nonabelian etale covers of an elliptic curve minus a point'

Abstract: Let E be an elliptic curve over 'Q' with identity. We can study the absolute Galois group of 'Q', 'GQ', by studying the representations associated to E. The Tate representations recover the action of 'GQ' on the abelianization of the tale fundamental group of E\{}. Grothendieck defined a much more general representation. To understand it more concretely, we study some nonabelian tale covers and the resulting Galois representations.

ESC 638

Friday, December 05, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Lilit Martirosyan (UC Berkeley): 'The representation theory of the exceptional Lie superalgebras F(4) and G(3). '

Abstract: In this talk I will give answers to three related problems for the so-called exceptional Lie superalgebras F(4) and G(3). I will describe the character and superdimension formulae for the simple modules, classify all indecomposable representations, and construct the superanalogue of Borel-Weil-Bott theorem.

ESC 618

Friday, November 21, 2014

01:10 pm - 02:00 pm

Algebra Seminar, David Pollack (Wes): 'A lifting theorem and computations towards an eigencurve for SL_3.'

A lifting theorem and computations towards an eige curve f

ESC 618

Friday, November 14, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Nathan Kaplan (Yale): 'Counting Orders in Number Fields'

Abstract: We focus on several related counting problems in number theory. How many sublattices of Z^n have index k? How many of these sublattices are actually subrings? These questions fit nicely within the theory of zeta functions of groups and rings. We will then transition from Z^n to counting in algebraic number fields, finite extensions of the rational numbers. We study orders in these fields, subrings that have size measured by discriminant. We approach these counting questions by analyzing the zeta functions that count subrings of Z^n and orders in a number field. We derive asymptotic formulas by analyzing the poles of these functions. We use a combination of methods from analytic and algebraic number theory, algebraic geometry, representation theory of finite groups, and combinatorics. This work is joint with Ramin Takloo-Bighash and Jake Marcinek.

ESC 618

Friday, October 31, 2014

01:10 pm - 02:00 pm

Algebra Seminar, David Pollack (Wes): 'A lifting theorem and computations towards an eigencurve for SL_3.'

Abstract: Fix a prime p. We generalize our early method of computing overconvergent lifts of ordinary $p$-adic homology classes for SL_3 to allow computations of lifts of non-ordinary, but numerically non-critical classes. Then we will discuss an ongoing calculation to find to any desired degree of accuracy the germ of the projection to weight space of the eigencurve Z around the point z corresponding to the system of Hecke eigenvalues of this over convergent lift. We do this under the conjecturally mild hypothesis that Z is smooth at z.

ESC 618

Friday, October 10, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Christopher Rasmussen (Wes): 'Constrained torsion over Q. The case g=p=2.'

Abstract: Fix a prime p. Related to a long-standing question of Ihara on the nature of the canonical outer pro-p Galois representation, it is natural to ask which abelian varieties A/K have p-power torsion which generate a pro-p extension after adjoining p-th roots of unity. In some cases, the reduction type of the variety is enough to determine the structure of the prime power torsion extension. We resolve a new case, namely dimension 2 Jacobians over Q with p = 2, and discuss to what extent we can hope for a generalization.

ESC 618

Friday, October 03, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Patricio Quiroz (Wes): 'Spinor genera theory in the skew-hermitian case'

Abstract: Spinor genera theory started as the abelian component of the classification problem (modulo isometry) for integral quadratic forms (or equivalently, quadratic lattices). This theory can be generalized for the study of at least two more integral structures: skew-hermitian lattices and orders in central simple algebras. In this talk, we will introduce spinor genera theory, which depends on the knowledge of the so called 'spinor norm' and then focus on spinor norm computations in the skew-hermitian case.

ESC 618

Friday, April 04, 2014

01:10 pm - 02:00 pm

Algebra Seminar and PhD Defense, Bonita Graham (Wesleyan): 'A construction of rigid analytic cohomology classes for split reductive linear algebraic groups.

Abstract: Ash and Stevens showed that it is possible to lift ordinary classical Hecke Eigensymbols on connected reductive groups to overconvergent Hecke Eigensymbols. Pollack and Pollack showed explicitly how to compute these lifts in the case of GL_3. I extend this constructive proof to any split connected reductive algebraic group G. The key step is constructing a suitable filtration on a vector space related to G. An explicit formula for the filtration is given, allowing the computation of approximations of the overconvergent eigenclasses.

ESC 618

Friday, March 28, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Keith conrad (UConn): 'The $p$-adic generalized arithmetic-geometric mean.'

Abstract: The arithmetic-geometric mean $M(x,y)$ of two positive real numbers $x$ and $y$ is the common limit of the two sequences $\{x_n\}$ and $\{y_n\}$ where $x_1 = x$, $y_1 = y$, $x_{n+1} = (x_n + y_n)/2$ and $y_{n+1} = \sqrt{x_ny_n}$. Convergence is fast. In 1989, Henniart and Mestre showed in $p$-adic fields that essentially the same construction leads to a common limit, where positivity is replaced by $y_1/x_1$ being sufficiently close to 1. The construction of the limit $M(x,y)$ can be generalized from two positive numbers to $n$ positive numbers, using elementary symmetric polynomials in place of $x+y$ and $xy$. We will discuss a $p$-adic analogue.

ESC 618

Friday, February 21, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Han Li (Yale): 'Indefinite Integral Quadratic Forms Beyond Reduction Theory'

Abstract: The classical reduction theory of integral quadratic forms was developed by Hermite, Minkowski, Siegel and many others. It is known that a non-degenerate integral quadratic form in n-variables is integrally equivalent to a form whose height (the maximum value of the coefficients) is less than its determinant (up to a multiple constant), and whose value at (1, 0,...0) is less than the n-th root of its determinant. However, for indefinite forms in at least 3 variables it turns out that neither of the estimates is optimal. In this talk we will discuss some classical results and recent effort in improving these estimates. This is a joint work with Prof. Margulis.

ESC 618

Friday, February 14, 2014

01:10 pm - 02:00 pm

Algebra Seminar, Matthew Gelvin (Wes): 'Fusion systems, orbit counting, and broken chains'

Abstract: If we're interested in studying a finite group $G$, we will often consider the action of $G$ on finite sets. There are two ways of understanding a $G$-set $X$: Either break it up into minimal pieces, each of which is itself a $G$-set, or record the number of $H$-fixed points of $X$ for all subgroups $H\leq G$. Both approaches contain the same data, with the \emph{table of marks} serving as the dictionary connecting the two. In the first part of this talk we will describe how to understand this table of marks in terms of the poset of subgroups of $G$ and a modified M\'obius inversion process. Now let $S$ be a finite $p$-group and $\mathcal{F}$ a fusion system on $S$; for our purposes, $\mathcal{F}$ is simply the data of which subgroups of $S$ are \emph{fused} together, generalizing the notion of conjugate subgroups of a finite group. The analogue of a $G$-set is an $S$-set that respects this new fusion data. Earlier work with Reeh has shown that the additive monoid of these $\mathcal{F}$-sets has a basis of minimal elements. Unfortunately, the construction went via an inductive algorithm, so it is in general difficult to describe what exactly these basis elements should look like. The last part of this talk will detail joint work with Reeh and Yal\cc in, where we give a new description of these basis elements in terms of the combinatorics of the fusion system and the group theory of $S$.

ESC 618

Friday, December 06, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Justin Lynd (Rutgers): 'Fusion systems and punctured groups'

Abstract: Transporter systems, linking systems, and fusion systems, which are categories having objects some collection of subgroups of a fixed finite p-group S, give progressively coarser encodings of the p-local structure (i.e. roughly the amalgam of normalizers of p-subgroups) of a hypothetical finite group having S as Sylow p-subgroup. Chermak recently showed that to every fusion system F there is a unique corresponding linking system L. I'll give an an introduction to these objects and Chermak's technique of 'descent' for constructing new transporter systems from old. Then I'll present some preliminary work (joint with Ellen Henke) examining in some small special cases a question arising out of Chermak's work: need there exist a 'punctured group' associated to an (exotic) fusion system F over S, that is, a transporter system over the collection of nonidentity subgroups of S? For an exotic fusion system (i.e. one not arising from a finite group), such an associated punctured group could be viewed as the most group-like object approximating the nonexistent finite group at the given prime.

ESC 618

Friday, November 22, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Kate Thompson (UGA): 'Towards Analytic Proofs of Universality of Quadratic Forms'

Abstract: The 290-Theorem of Bhargava and Hanke says that a positive-definite quadratic form in four variables over the integers is universal iff the form represents 1 through 290 and classifies up to equivalence all such forms. in this talk I will describe some of the issues arising when one tries to generalize this result to rings of integers in other number fields. The methods used to prove universality for a fixed form Q are very analytic in nature, drawing from the theory of (classical) modular forms and Siegel local densities. These tools generalize and can be used to provide universality theorems for positive-definite quaternary quadratic forms over O_K (for K a totally real number field). I will give an overview of how these analytic tools work, pinpointing what changes significantly when generalizing to a number field, and highlighting theoretical versus computational difficulties.

ESC 618

Friday, November 08, 2013

01:10 pm - 02:00 pm

Algebra Seminar, David Pollack (Wes): 'Explicit eigencurves for GL_3'

Abstract: A theorem of Hida tells us that ordinary modular forms on $\operatorname{GL}_2$ deform in families of varying p-adic weight. These families are defined over a p-adic neighborhood in weight space. Certain homology classes on $\GL_3$ are also known to deform in p-adic families. In this case the weight space (up to twists) is two-dimensional, and the family may only be defined over (an open subset of) a one-dimensional curve in weight space. In this talk we will give an introduction to these p-adic families and their relation to deformations of Galois representations, and report on ongoing work with Avner Ash to compute approximations to the curves over which these deformations exist.

ESC 618

Friday, November 01, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Holly Krieger (MIT): 'The arithmetic of dynamical sequences'

Abstract: Understanding the arithmetic of sequences that are dynamically defined, such as the Fibonacci and Mersenne numbers, is of classical interest but generally results are difficult; for example, the infinitude of primes in either of these sequences is still open. However, results on the existence of primitive prime divisors and perfect powers in these sequences have been achieved, including the notable t eorem of Bugeaud, ignotte, and Siksek listing the Fibonacci powers. These questions and methods generalize to sequences which are forward orbits under iteration of certain dynamical systems, and I will discuss results on the arithmetic of such sequences, which rely on techniques from Diophantine approximation, arithmetic dynamics, and complex dynamics.

ESC 618

Friday, October 25, 2013

01:10 pm - 02:00 pm

Algebra Seminar: Three Wesleyan Graduate Student Talks by Abbey Bourdon, Bonita Graham, and James Ricci

Abstract: This seminar will consist of three separate talks by graduate students on the material in their theses: Abbey Bourdon will present 'A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves.' Bonita Graham will present 'A Construction of Rigid Analytic Cohomology Classes for Split Reductive Linear Algebraic Groups.'James Ricci will present 'Finiteness results for regular ternary quadratic polynomials.'

ESC 618

Friday, October 11, 2013

01:10 pm - 02:00 pm

Algebra Seminar, Asher Auel (Yale): A game on the exceptional group G_2 and a conjecture concerning flag varieties

Abstract: I will introduce a simple combinatorial game that can be played on any lattice in euclidean space. When played on the weight lattice of a linear algebraic group, this game is related to a longstanding conjecture on the derived category of flag varieties. Without any use of flag varieties, I will describe the solution to this game for all weight lattices in two dimensions. The most interesting case is the exceptional group G_2, which is related to octonion algebras. At the end, I will explain the conjecture via the case of quadrics, where the classical Clifford algebras play a role. This is joint work with Alexey Ananyevskiy, Skip Garibaldi, and Kirill Zainoulline.

ESC 618