This thesis is essentially a compendium of the author's completed work as a graduate student. It includes a variety of different investigations centering around monogeneity and division fields. In the defense, we will introduce these concepts with some historical backing. The speaker hopes this portion will be friendly to a relatively general audience. After moving through some classical results, we will discuss the author's earlier work (joint with Katherine Stange and T. Alden Gassert) on the monogeneity of partial torsion fields. This segues into the author's work on the monogeneity of quartic fields generated by trinomials. We will move to more recent work on the radical extensions. Finally, we will discuss results on the non-monogeneity of division fields of certain abelian varieties, including explicit non-monogenic families in the case of dimension 1.

Everyone is invited. After the public presentation, the general audience will be asked to leave during private questioning, but a re-invite for a celebration/congratulations will be emailed out afterward, so please watch for that.

Monogeneity and Torsion

CANCELED Wed, Apr. 1

Math Phys

Andrea Carosso (CU Boulder Physics) Lattice Theory of Quantum Matter and Fields

CANCELED Wed, Apr. 1

Grad Student Seminar

Howie Jordan (CU Boulder)

X

We’ve all heard the hype: all you have to do is forget about “excluded middle” and then BOOM, your logic is constructive (in a certain sense). But like… why would you do that? And in what sense is it more constructive?

If you aren’t aware, the Law (ha!) of Excluded Middle is the assertion that, for any particular sentence, either that sentence or its negation holds in your logic. In this talk we’ll look at what kind of logic you get when you drop excluded middle and discuss various motivations and interpretations of this, from the philosophical hooey of Brouwer to the structural interpretations arising from Gentzen’s methods. We will also look at a sample of some of the differences in practice and theory that arise when you don’t allow yourself the luxury of excluded middle in your proofs.

Noo don’t exclude the middle you’re so constructive aha

Join the Zoom meeting at https://cuboulder.zoom.us/j/4348160464

The first topic is based on work in [17], we study the so-called “coin-turning” walks. Let S be the random walk obtained from “coin turning” with some sequence {pn}n?2, as introduced in [16].

We investigate the scaling limits of S in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for “not too small” sequences, the order const·n?1 (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed.

In the second topic, we study equilibrium of stopping problem for a multi-dimensional continuous strong Markov processes X. We use the framework for definition of equilibrium as in [20], especially, the discount function is of decreasing impatience, which means it is log sub-additive and may not be exponential. We provide properties of nearly Borel equilibria, and build up the equivalence be- tween an equilibrium and its finely closure.

Under certain regular conditions on X, we give results of optimality of an equilibrium and prove that there exists an optimal equilibrium among a quite general family M of equilibria with some proper regularity condition for M. Moreover, the optimal one can be achieved by taking the intersection of all equilibria.

Join the Zoom meeting at https://cuboulder.zoom.us/j/511678834

Let G be a connected, nonabelian, reductive, linear algebraic group over an algebraically closed field of characteristic 0. Let T be a maximal torus of G, and let Q be a closed, connected, solvable subgroup of G whose unipotent radical is stable under the conjugation action of T . We determine necessary and sufficient conditions for the algebraic action of G on G/Q to be at least generically 2-transitive. For semisimple groups G, we determine when this action is precisely generically 2-transitive. Through examples, we show that there exist closed, connected, solvable Q such that gtd(G, G/Q) = 2, thereby showing how the conditions of our theorem can be met. We also show, through example, that not every closed, connected, solvable Q such that gtd(G, G/Q) = 2 must have its unipotent radical stable under conjugation by T.

Quotient algorithms have been a principal tool for the computational investigation of finitely presented groups as well as for constructing groups. I will describe a method for a nonsolvable quotient algorithm, that extends a known finite quotient with a module.

This is joint work with Heiko Dietrich (Monash U., Melbourne, Australia)

Towards a Nonsolvable Quotient Algorithm

Thu, Apr. 2 3pm (MATH 350)

Topology

Katharine Adamyk (CU Boulder)

X

The mod 2 Steenrod algebra, A, appears in stable homotopy theory as the algebra of stable operations on cohomology with coefficients in Z/2. In this talk, we present results that hold in the stable category of modules over A(1), a small subalgebra of A. In this category, the invariants Q_0- and Q_1-Margolis homology together detect a large amount of information about a given A(1)-module. We show that all bounded below A(1)-modules of finite type whose Q_1-Margolis homology is trivial are stably equivalent to direct sums of suspensions of a distinguished family of A(1)-modules.

The classification theorem is then used to simplify computations of localized Ext groups and to provide necessary conditions for lifting A(1)-modules to A-modules. We give the first differential for a spectral sequence converging to an h_0-localized Ext group associated to a given bounded below A(1)-module of finite type. The classification theorem allows the computation of all differentials if the desired decomposition of a module with trivial Q_1-Margolis homology can be given explicitly. This is not always possible, but we give some examples where the higher differentials can be computed. The differentials in this spectral sequence detect obstructions to lifting an A(1)-module to an A-module.

Defense : A classification of Q_0-local A(1)-modules

Thu, Apr. 2 3pm

Thesis Defenses

Athena Sparks

X

Promise Constraint Satisfaction Problems (PCSP) are a generalization of Constraint Satisfaction Problems (CSP) that provide a common framework for many classical decision problems like graph colorability, satisfiability of Boolean formulas, solvability of linear equations, etc. Informally, a PCSP asks to distinguish between whether a given instance of a CSP has a solution or not even a specified relaxation can be satisfied. In the formulation for a fixed template of relational structures A and B, the input of PCSP(A,B) is a structure I of the same type as that of A and B. The problem is to determine if there exists a homomorphism from I into A or if there does not exist a homomorphism from I into B. The “promise” is that exactly one of these two alternatives occurs for the input I. As with CSP, the computational complexity of a PCSP is determined by the higher arity symmetries or relation preserving functions, called polymorphisms, from A to B. All polymorphisms from A to B form a clonoid, a set of finitary functions from A to a target algebra B that is closed under the usual variable operations like identifying, permuting and introducing arguments as well as closed under operations from B. We adapt the established algebraic approach for analyzing CSP to investigate the complexity of PCSP and take steps towards classifying clonoids on finite sets in general.

Clonoids and Promise Constraint Satisfaction Problems

Mon, Apr. 6 2pm

Thesis Defenses

Sebastian Bozlee

X

We use log geometric data to construct contractions of families of curves of arbitrary genus. We use log blowups and this contraction construction to resolve the rational map between the moduli space of stable curves and some alternate semistable compactifications of the moduli space of curves. In at least two cases, the target space appears to be new.

An Application of Logarithmic Geometry to Moduli of Curves in Genus Greater Than One

Tue, Apr. 7 10am

Thesis Defenses

Carly Matson (University of Colorado Boulder)

X

One of the main difficulties in studying the Galois groups of field extensions is that, in general, the set of roots of a polynomial need not have any additional algebraic structure. In 1965, Lubin and Tate showed that a certain type of power series f(x) over a p-adic field can be recognized as the "multiplication-by-pi" endomorphism of a formal group law with complex multiplication, or a larger-than-usual endomorphism ring. They used this to put a module structure on the roots of f(x) and to show that the splitting field of f(x) is Galois and abelian, culminating in a new proof of the main theorem of local class field theory. In this talk we will offer a new and more general construction of formal group laws with complex multiplication in higher dimensions and will examine how this differs from the one-dimensional case. Finally, we will once again see that the torsion points of such a formal group generate abelian extensions over p-adic fields.

This defense will be held via zoom, and all are welcome:

Everyone is welcome for the presentation, and will be asked to leave for the questioning and deliberations.

Higher dimensional formal group laws with complex multiplication

Tue, Apr. 7 11am (MATH 350)

Number Theory

TBA Number Theory thesis defenses today

Tue, Apr. 7 1pm (Zoom)

Thesis Defenses

Daniel Martin (University of Colorado Boulder)

X

We will take a geometric perspective on Euclideaneity in imaginary quadratic fields. The result is a pseudo-Euclidean algorithm with applications to continued fractions, the class group, and the special linear group in two dimensions.

This defense will be held via zoom, and all are welcome:

Due to "zoombombing" concerns, please check your Friday April 3rd seminar announcement email for the zoom id, or email Kate Stange or Daniel Martin.

Everyone is welcome for the presentation, and will be asked to leave for the questioning and deliberations. After the defense, an announcement will be made and congratulations can be sent via email.

The geometry of imaginary quadratic fields

CANCELED Tue, Apr. 7

Lie Theory

Bryan Gillespie (CSU) TBA Sponsored by the Meyer Fund