Structure and Symmetry Day

Homological mirror symmetry is a conjecture due to Maxim Kontsevich, predicting an equivalence between the derived category of coherent sheaves of an algebraic variety X and the Fukaya category of the “mirror” symplectic manifold Y. One of the most notable mirror constructions are Batyrev-Borisov mirrors, where both spaces are presented as complete intersections inside toric varieties, which means that a lot of their properties can be understood in purely combinatorial terms. In this talk, I will discuss some new results concerning Lagrangian skeleta of complete intersections in (C*)^n and explain how they could be used as the first step towards proving HMS for general Batyrev-Borisov pairs, following an approach that has already been applied in several special cases. 

To order a group means to find a total ordering of its elements that is invariant under (right, or left) multiplication. Orders on groups connect to many other areas, in particular dynamics and topology.
In this talk I will explore in which groups and situations a partial order can be extended to a total order. In particular, in nilpotent groups the question about extending partial to total orders is connected to the Identity Problem: this asks if the subsemigroup generated by a given finite set of elements of a group contains the identity element of the group. I plan to explain these connections, and why the Identity Problem is decidable in nilpotent groups.
Joint work with Corentin Bodart and George Metcalfe.

Quantum toroidal algebras exist as the double affine objects within the quantum setting. In this talk we’ll work towards their definition from the ground up, briefly mentioning various motivations and connections along the way. We shall then highlight some of the major successes and difficulties in studying these algebras, and outline results by the speaker in this area. Time permitting, we’ll finish with possible future directions.