Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Wednesday, June 29, 2016
2:10 p.m.

PhD Candidate:  Iva Halacheva
Co-Supervisors:  Dror Bar-Natan, Joel Kamnitzer
Thesis title:   Alexander-type invariants of tangles, Skew Howe duality for crystals and the cactus group



In the first part of this work, we study generalizations of a classical link invariant–the multivariable Alexander polynomial–to tangles. The starting point is Archibald’s tMVA invariant for virtual tangles which lives in the setting of circuit algebras. Using the Hodge star map, we define a reduction of the tMVA to an invariant (rMVA) which is valued in matrices with entries equal to certain Laurent polynomials. When restricted to tangles without closed components, we show the rMVA has the structure of a metamonoid morphism and is further equivalent to another tangle invariant defined by Bar-Natan. This invariant also reduces to the Gassner representation on braids and has a partially defined trace operation for closing open strands of a tangle.

In the second part, we look at crystals and the cactus group. The  crystals for a finite-dimensional, complex, reductive Lie algebra $\mathfrak g$ encode the structure of its representations, yet can also reveal surprising new structure of their own. In this work, we construct a group $J_{\mathfrak g}$, the “cactus group”, using the Dynkin diagram of $\mathfrak g$ and show that it acts combinatorially on any $\mathfrak g$-crystal via the Sch\”{u}tzenberger involutions. For ${\mathfrak g} =\mathcal g l_n$, the cactus group was studied by Henriques and Kamnitzer, who construct an action of it on $n$-tensor products of $\mathfrak g$-crystals. We study the crystal corresponding to the $\mathfrak g l_n \times \mathfrak g l_m$-representation $\Lambda^N(\Bbb C^n \otimes \Bbb C^m)$, derive skew Howe duality on the crystal level and show that the two cactus group actions agree in this setting. An application of this result is discussed in studying a family maximal commutative subalgebras of the universal enveloping algebra, the shift of argument and Gaudin algebras, where an algebraically constructed monodromy action is expected to match that of the cactus group.

A copy of the thesis can be found here:  Halacheva thesis


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