Osmosis and the Oracle

Fridays 3:45 - 4:45pm

Peter Hall Building, Room 162

19 May:

Zhihang Yu (University of Melbourne)

Mirror game and affine Weyl group

If you surround yourself with mirrors, how many copies of you will there be? If there's only one mirror, then the answer is obvious: only two; the real you and your reflection. But what will happen if you increase the number of mirrors? In this talk, I will discuss a "mirror game" and use it to introduce the notion of a Coxeter group; in particular the affine Weyl group. The length and reduced expression of an element in the affine Weyl group will also be introduced. Finally, I will use this "mirror game" to give a formula for the length and to find the reduced expression of an element. 

12 May:

Haris Rao (University of Melbourne)

The Jacobson-Morozov Theorem

I will motivate the proof of Jacobson-Morozov by using a simple strategy to explicitly compute sl_2 triples in various example cases. Specifically, we will work in the Lie algebras sl_n and sp_4. Next, a proof of the Jacobson-Morozov theorem will be presented which will be based on the same strategy used before to compute the sl_2 triples. Finally, we will look at some applications of the theorem. 

28 April:

Arun Ram (University of Melbourne)

Row reduction and flag varieties

All cohomology computations that I know are done by row reduction for matrices (as in first year linear algebra).  I will review row reduction for matrices and explain how to use it to make the computation of the cohomology of the flag varieties trivial.  Hopefully, I will also be able to use the same method to derive the cohomology of Hessenberg varieties. I’ll do a bit of “Ask me a question”, and a bit of “This is what I would like to ask the Oracle”.  The “Ask me a question" time will start at 3:30 and the talk proper will start at 3:48 sharp and the “This is what I would like to ask the Oracle" will be interleaved with the talk proper between 4:18 and 4:42.

21 April:

Lihexuan (Grace) Yuan (University of Melbourne)

Knot invariants built from quantum sl2

Knot theory is a fascinating field in low dimensional topology. One of the fundamental questions in knot theory is: given two knot or link diagrams, how to tell if they represent the same knot or link? The construction of knot invariants arises to distinguish knots. One of the classical knot invariants is the Jones polynomial J(L) introduced by Vaughan Jones, which will be covered in the talk. 

Quantum group is a special kind of noncommutative algebra constructed independently by Drinfeld and Jimbo. One can use the R matrix in the quantum group to construct a knot invariant: quantum invariant Q(L).

In the talk, I will show the construction of the quantum invariant and how the R matrix in the quantum \mathfrak{sl2} recovers the Jones polynomial. 

Anyone with a linear algebra background can understand the talk.

14 April: Osmosis

Apoorva Khare (Indian Institute of Science)

Groups with norms: from word games to a PolyMath project

Consider the following three properties of an arbitrary group $G$:

1. Algebra: $G$ is abelian and torsion-free.

2. Analysis: $G$ is a metric space that admits a "norm", namely, a translation-invariant metric $d(.,.)$ satisfying: $d(1,g^n) = |n| d(1,g)$ for all $g$ in $G$ and integers $n$.

3. Geometry: $G$ admits a length function with "saturated" subadditivity for equal arguments: $l(g^2) = 2 l(g)$ for all $g$ in $G$.

While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a "norm".

We will discuss connections to analysis and geometry, followed by the proof of the above equivalences. We will also see the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.

(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)

31 March: Osmosis

Riley Morris (University of Melbourne)

Russell's Paradox and Polynomials

Russell's paradox springs from the unrestricted comprehension of naive set theory. Or does it? It was proven in 1982 that in fact by changing the underlying logic Russell's paradox cannot be derived even in a naive set theory. In this talk we will examine how this works and look at an example of a naive set theory which allows for arbitrary fixed points as well as capturing the class of polynomial time computations.

24 March: Ask the Oracle

Davood Nejaty (University of Melbourne) 

Vanishing cycle perverse sheaves

We would introduce the vanishing cycle functor on the derived category of $\mathbb{C}-$schemes and give an example of it. This functor has a lift to the category of mixed Hodge modules. We would explain its properties and cohomology.

17 March: Ask the Oracle

Linfeng Wei (University of Melbourne)

Degree of coherent sheaf on \mathbb{P}^1 and and quantum difference equation

The origin of this talk is to understand a key equation used in deducing the so-called quantum difference equation, stating that for q-equivariant coherent sheaf \mathcal{F} over \mathbb{P}^1 and q-fixed points p_1,p_2 on mathbb{P}^1, we have det H^*(\mathcal{F}\otimes(O_{p_1}-O_{p_2})=q^deg(\mathcal{F}). We will try to explain the meaning of both sides and deduce the resulting power on the RHS to be c_1(\mathcal{F})=deg(\mathcal{F}) by Hirzebruch-Riemann-Roch assuming rank(\mathcal{F})=1. If time permits, we may also discuss how this result is used in deducing the quantum difference equation.

10 March: Osmosis

Changlong Zhong (State University of New York at Albany)

Generalized Schubert Calculus (Part 2)

3 March: Osmosis

Changlong Zhong (State University of New York at Albany)

Generalized Schubert Calculus (Part 1)

I will start with basic questions in Schubert calculus of Grassmannians, then introduce more general notions, including flag manifolds, Schubert polynomials, divided difference operators, and various Hecke algebras (nil-Hecke algebra, 0-Hecke algebra and formal affine Demazure algebra) and their relation with equivariant K-theory and equivariant oriented cohomology. The starting point will be more combinatorial, and then we will turn to the algebraic setting that forms the foundation of generalized Schubert calculus.