Past

Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second fundamental groups of a topological space - hoops can be composed like in π1, balloons like in π2, and hoops "act" on balloons as π1 acts on π2. We will observe that ordinary knots and tangles in 3-space map into KBH in 4-space and become amalgams of both balloons and hoops.

We give an ansatz for a tree and wheel (that is, free-Lie and cyclic word) -valued invariant ζ of KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that ζ is a complete evaluation of the BF topological quantum field theory in 4D, though we are not sure what that means. We show that a certain "reduction and repackaging" of ζ is an "ultimate Alexander invariant" that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.

For further information see http://www.math.toronto.edu/~drorbn/Talks/Oxford-130121/

Abstract: We consider the inverse problem of recovering u from a noisy, indirect observation We adopt a Bayesian approach, in which the aim is to determine the posterior distribution _y on the unknown u, given some prior information about u in the form of a prior distribution _0,together with the observation y. We are interested in the question of posterior consistency, which is the characterization of the behaviour of _y as more data become available. We work in a separable Hilbert space X, assuming a Gaussian prior _0 = N(0; _ 2C0). The theory is developed using two concrete problems: i) a family of linear inverse problems in which we want to _nd u from y where y = A

The models of Brownian motion, Poisson processes, Levy processes and martingales are frequently used as basic formulations of prices in financial market. But probability and/or distribution uncertainties cause serious problems of robustness. Nonlinear expectations (G-Expectations) and the corresponding martingales are useful tools to solve them.

DH common room at 09:45 and from 10:00 in DH12

Let K be a number field and E/K be an elliptic curve. Multiplication by n induces a map from the n^2-Selmer group of E/K to the n-Selmer group. The image of this map contains the image of E(K) in the n-Selmer group and is often smaller. Thus, computing the image of the n^2-Selmer group under multiplication by n can give a tighter bound on the rank of E/K. The Cassels-Tate pairing is a pairing on the n-Selmer group whose kernel is equal to the image of the n^2-Selmer group under multiplication by n. For n=2, Cassels gave an explicit description of the Cassels-Tate pairing as a sum of local pairings and computed the local pairing in terms of the Hilbert symbol. In this talk, I will give a formula for the local Cassels-Tate pairing for n=3 and describe an algorithm to compute it for n an odd prime. This is joint work with Tom Fisher.

The essential information contained in most large data sets is

small when compared to the size of the data set. That is, the

data can be well approximated using relatively few terms in a

suitable transformation. Compressed sensing and matrix completion

show that this simplicity in the data can be exploited to reduce the

number of measurements. For instance, if a vector of length $N$

can be represented exactly using $k$ terms of a known basis

then $2k\log(N/k)$ measurements is typically sufficient to recover

the vector exactly. This can result in dramatic time savings when

k

Algebraic geometry has become the standard language for many number theorists in recent decades. In this talk, we will define modular forms and related objects in the language of modern geometers, thereby giving a geometric motivation for their study. We will ask some naive questions from a purely geometric point of view about these objects, and try to answer them using standard geometric techniques. If time permits, we will discuss some rather deep consequences in number theory of our geometric excursion, and mention open problems in geometry whose solution would have profound consequences in number theory.

We discuss the problem of estimating a structured matrix with a large number of elements. A key motivation for this problem occurs in multi-task learning. In this case, the columns of the matrix correspond to the parameters of different regression or classification tasks, and there is structure due to relations between the tasks. We present a general method to learn the tasks' parameters as well as their structure. Our approach is based on solving a convex optimization problem, involving a data term and a penalty term. We highlight different types of penalty terms which are of practical and theoretical importance. They implement structural relations between the tasks and achieve a sparse representations of parameters. We address computational issues as well as the predictive performance of the method. Finally we discuss how these ideas can be extended to learn non-linear task functions by means of reproducing kernels.

It has been shown that the Auslander-Reiten-quiver of an indecomposable algebra contains a finite component if and only if A is representation finite. Moreover, selfinjective algebras are representation finite if and only if the tree types of the stable components are given by Dynkin Diagrams. I will present similar results for the Auslander-Reiten-quiver of a functorially finite resolving subcategory Ω. We will see that Brauer-Thrall 1 and Brauer-Thrall 1.5 can be proved for these categories with only little extra effort. Furthermore, a connection between sectional paths in A-mod and irreducible morphisms in Ω will be given. Finally, I will show how all finite Auslander-Reiten-quivers of A-mod or Ω are related to Dynkin Diagrams with a notion similar to the tree type that coincides in a finite stable component.

No Seminar this week

We first provide a brief overview of some of the key properties of the space $\textrm{BV}(\Omega;\mathbb{R}^{N})$ of functions of Bounded Variation, and the motivation for its use in the Calculus of Variations. Now consider the variational integral

\[

F(u;\Omega):=\int_{\Omega}f(Du(x))\,\textrm{d} x\,\textrm{,}

\]

where $\Omega\subset\mathbb{R}^{n}$ is open and bounded, and $f\colon\mathbb{R}^{N\times n}\rightarrow\mathbb{R}$ is a continuous function satisfying the growth condition $0\leq f(\xi)\leq L(1+|\xi|^{r})$ for some exponent $r$. When $u\in\textrm{BV}(\Omega;\mathbb{R}^{N})$, we extend the definition of $F(u;\Omega)$ by introducing the functional

\[

\mathscr{F}(u,\Omega):= \inf_{(u_{j})}\bigg\{ \liminf_{j\rightarrow\infty}\int_{\Omega}f(Du_{j})\,\textrm{d} x\, \left|

\!\!\begin{array}{r}

(u_{j})\subset W_{\textrm{loc}}^{1,r}(\Omega, \mathbb{R}^{N}) \\

u_{j} \stackrel{\ast}{\rightharpoonup} u\,\,\textrm{in }\textrm{BV}(\Omega, \mathbb{R}^{N})

\end{array} \right. \bigg\} \,\textrm{.}

\]

\noindent For $r\in [1,\frac{n}{n-1})$, we prove that $\mathscr{F}$ satisfies the lower bound

\[

\mathscr{F}(u,\Omega) \geq \int_{\Omega} f(\nabla u (x))\,\textrm{d} x + \int_{\Omega}f_{\infty} \bigg(\frac{D^{s}u}{|D^{s}u|}\bigg)\,|D^{s}u|\,\textrm{,}

\]

provided $f$ is quasiconvex, and the recession function $f_{\infty}$ ($:= \overline{\lim}_{t\rightarrow\infty}f(t\xi )/t$) is assumed to be finite in certain rank-one directions. This result is a natural extension of work by Ambrosio and Dal Maso, which deals with the case $r=1$; it involves combining work of Kristensen, Braides and Coscia with some new techniques, including a polyhedral approximation result and a blow-up argument that exploits fine properties of BV functions.

I will be looking at some conjectures and theorems closely related to the h-cobordism theorem and will try to show some connections between them and some group theoretic conjectures.

The homological dimension of a group can be computed over any coefficient ring $K$.
It has long been known that if a soluble group has finite homological dimension over $K$
then it has finite Hirsch length and the Hirsch length is an upper bound for the homological
dimension. We conjecture that equality holds: i.e. the homological dimension over $K$ is
equal to the Hirsch length whenever the former is finite. At first glance this conjecture looks
innocent enough. The conjecture is known when $K$ is taken to be the integers or the field
of rational numbers. But there is a gap in the literature regarding finite field coefficients.
We'll take a look at some of the history of this problem and then show how some new near complement
and near supplement theorems for minimax groups can be used to establish the conjecture
in special cases. I will conclude by speculating what may be required to solve the conjecture outright.

We will first review the construction of N =1

supersymmetric Yang-Mills theory in three dimensions. Then we will

construct a superloop space formulation for this super-Yang-Mills

theory in three dimensions.Thus, we will obtain expressions for loop

connection and loop curvature in this superloop space. We will also

show that curvature will vanish, unless there is a monopole in the

spacetime. We will also construct a quantity which will give the

monopole charge in this formalism. Finally, we will show how these

results hold even in case of deformed superspace.

In this talk we show how degree N maps of the form $u_{N}(z) = \frac{z^{N}}{|z|^{N-1}}$ arise naturally as stationary points of functionals like the Dirichlet energy. We go on to show that the $u_{N}$ are minimizers of related variational problems, including one whose associated Euler-Lagrange equation bears a striking resemblance to a system studied by N. Meyers in the 60s, and another where the constraint $\det \nabla u = 1$ a.e. plays a prominent role.