We will recall Voevodsky's definition of the presheaves of rigid motivic homotopy groups and show that they admit a canonical structure of presheaves with transfers when we consider rational coefficients and quasi-excellent base schemes.

I will first explain a pseudodifferential calculus for the canonical dynamical system associated to the noncommutative two torus which is a special case of Connes' pseudodifferential calculus for C^* dynamical systems. Then I will report on a recent joint work with M. W. Wong where we define a noncommutative residue for classical pseudodifferential operators on the noncommutative two torus, and prove that up to a constant multiple it is the unique trace on the algebra of classical pseudodifferential operators modulo infinitely smoothing operators.

There are various mathematical models and solution strategies for the sudoku puzzle. In this expository talk, I will use the language of graph theory and hyperplane arrangements to approach the sudoku enumeration problem. In particular, I will show how sudokus suggest a finite field interpretation of the characteristic polynomial of hyperplane arrangements. If time permits I will also discuss a Groebner basis algorithm for determining whether a given sudoku puzzle has a solution.

This talk will handle Drinfel'd's construction of the object of the title by means of deformations of braided monoidal categories.

This is the second part of our discussion of singularities of weighted projective spaces and their resolutions.

We continue our exposition of the Schur-Horn-Kostant theorem for Symplectomorphisms groups of toric manifolds

More than 50 years ago Michel Kervaire constructed a manifold with no differentiable structure. Constructing manifolds is the easy part - the trick is to construct an invariant, in this case the Arf-Kervaire invariant, which guarantees the existence or otherwise of the property one is after. The connection between framed manifolds and stable homotopy groups led (circa 1960) to the problem: ``Do there exist framed manifolds of odd Arf-Kervaire invariant?''. Recently Hill-Hopkins-Ravenel proved that the outcome is: ``mostly framed manifolds of Arf-Kervaire invariant one do not exist''.

Here is what Lewis Carroll has to say about non-existent things!

``I know what you're thinking about,'' said Tweedledum; ``but it isn't so, nohow.'' ``Contrariwise,'' continued Tweedledee,``if it was so, it might be; and if it were so, it would be; but as it isn't; it ain't. That's logic.''

A theorem of Beauville implies that the general smooth quartic surface in $P^3$ may be expressed as the zerolocus of a Pfaffian associated to an 8 by 8 skew-symmetric matrix of linear forms. In this talk, I will discuss how the recent work of Aprodu-Farkas on the Green conjecture for curves on K3 surfaces may be used to generalize this statement to all smooth quartic surfaces in $P^3$. This is joint work with Emre Coskun and Rajesh Kulkarni.

Let $p$ be a prime. A 1989 theorem of Ossa calculates the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$ and purports to calculate the corresponding orthogonal connective K-theory when $p=2$. Sadly the latter is wildly wrong!

Using a simple K\"{u}nneth formula short exact sequence I shall derive Ossa's unitary connective K-theory result in an elementary manner. As a corollary, I shall derive the correct version of Ossa's orthogonal theorem.

As an application of this result I shall show that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to $ {\mathbb RP}^{\infty}$ gives a stable homotopy element having Arf-Kervaire invariant one.