It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb C)$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak g$ is any simple Lie algebra over $\mathbb C$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak g$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak g^L$.
It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal O_1<\mathcal O_2$ are adjacent in the partial order then so are their duals $\mathcal O_1^t>\mathcal O_2^t$, and the isolated singularity attached to the pair $(\mathcal O_1,\mathcal O_2)$ is dual to the singularity attached to $(\mathcal O_2^t,\mathcal O_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.
In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal O_1<\mathcal O_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.
 

With the evolution towards the IoT and cyber-physical systems, the role that the underlying hardware plays in securing an application is becoming more prominent. Hardware can be used constructively, e.g. for accelerating computationally- intensive cryptographic algorithms. Hardware can also be used destructively, e.g., for physical attacks or transistor-level Trojans which are virtually impossible to detect. In this talk, we will present case studies for high-speed cryptography used in car2x communication and recent research on low-level hardware Trojans. 

In this talk I will discuss acoustic and electromagnetic transmission problems; i.e. problems where the wave speed jumps at an interface. I will focus on what is known mathematically about resonances and trapped waves (e.g. When do these occur? When can they be ruled out? What do we know in each case?). This is joint work with Andrea Moiola (Pavia).

The Landau-Lifshitz-Gilbert equation (LLG) is a continuum model describing the dynamics for the spin in ferromagnetic materials. In the first part of this talk we describe our work concerning the properties and dynamical behaviour of the family of self-similar solutions under the one-dimensional LLG-equation.  Motivated by the properties of this family of self-similar solutions, in the second part of this talk we consider the Cauchy problem for the LLG-equation with Gilbert damping and provide a global well-posedness result provided that the BMO norm of the initial data is small.  Several consequences of this result will be also given.

In the study of variational models for non-linear elasticity in the context of proving regularity we are led to the challenging so-called Ball-Evan's problem of approximating a Sobolev homeomorphism with diffeomorphisms in its Sobolev space. In some cases however we are not able to guarantee that the limit of a minimizing sequence is a homeomorphism and so the closure of Sobolev homeomorphisms comes into the game. For $p\geq 2$ they are exactly Sobolev monotone maps and for $1\leq p<2$ the monotone maps are intricately related to these limits. In our paper we prove that monotone maps can be approximated by diffeomorphisms in their Sobolev (or Orlicz-Sobolev) space including the case $p=1$ not proven by Iwaniec and Onninen.