# Seminars and Colloquia

## Algebra Seminar

Feb 16

#### Algebra Seminar

01:20 pm

Wai Kiu Chan, Wesleyan Warings problem for integral quadratic forms Abstract : For every positive integer n , let g ( n ) be the smallest integer such that if an integral quadratic form in n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g ( n ) squares of integral linear forms. As a generalization of Lagranges Four-Square Theorem, Mordell (1930) showed that g (2) = 5 and later that year Ko (1930) showed that g ( n ) = n + 3 when n 5. More than sixty years later, M.-H. Kim and B.-K. Oh (1996) showed that g (6) = 10, and later (2005) they showed that the growth of g ( n ) is at most an exponential of n . In this talk, I will discuss a recent improvement of Kim and Oh's result showing that the growth of g ( n ) is at most an exponential of $\sqrt{n}$. . This is a joint work with Constantin Beli, Maria Icaza, and Jingbo Liu.

Nov 3

#### Algebra Seminar

01:20 pm

Jonathan Huang, Wes A Macdonald formula for zeta functions of varieties over finite fields Abstract : We provide a formula for the generating series of the zeta function Z ( X , t ) of symmetric powers Sym n X of varieties over finite fields. This realizes Z ( X , t ) as an exponentiable motivic measure whose associated Kapranov motivic zeta function takes values in W ( R ) the big Witt ring of R = W ( ). We apply our formula to compute Z (Sym n X , t ) in a number of explicit cases. Moreover, we show that all -ring motivic measures have zeta functions which are exponentiable. In this setting, the formula for Z ( X , t ) takes the form of a MacDonald formula for the zeta function.

Sep 27

#### Algebra Seminar

04:15 pm

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.

Sep 20

#### Algebra Seminar

04:15 pm

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.

Feb 5

#### Algebra Seminar

01:15 pm

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.

Dec 4

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

01:10 pm

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.

Nov 6

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

01:10 pm

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.

Oct 9

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

01:10 pm

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.

Sep 28

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

04:45 pm

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.

Sep 25

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

01:10 pm

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.

May 1

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

01:10 pm

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.

Apr 24

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

01:10 pm

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.

Apr 17

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

01:10 pm

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.

Apr 10

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

01:10 pm

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.

Apr 3

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

01:10 pm

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.

Feb 27

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

01:10 pm

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.