2016 Daniel B. DeLury Teaching Assistant Awards winners

We are happy to announce that this year’s winners of the Daniel B. DeLury Teaching Assistant Awards for graduate students in mathematics are:

  • Tracey Balehowsky
  • Beatriz Navarro Lameda
  • Nikita Nikolaev
  • Asif Zaman

The selection committee consisted of Mary Pugh, Abe Igelfeld and Peter Crooks.  Nominations were made by faculty members, course instructors, and undergraduate students.

The awards recipients will receive a monetary award and a certificate during our awards/graduation reception scheduled for Thursday, May 26 at 3:10 p.m. in the Math lounge.

Congratulations Tracey, Beatriz, Nikita and Asif!

Ida Bulat Teaching Award for Graduate Students

The Department of Mathematics invites nominations for the inaugural:

Ida Bulat Teaching Award

This prize honors outstanding teaching by graduate students.  It is named in memory of Ida Bulat, former Graduate Administrator in the Department of Mathematics at the University of Toronto.

Nominations can be submitted online through this link, no later than Friday, May 13, 2016:

http://blog.math.toronto.edu/forms/nomination-form-2016/

Departmental PhD Thesis Exam – Trefor Bazett

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

Wednesday, May 11, 2016
11:10 a.m.
BA6183

PhD Candidate:  Trefor Bazett
Co-Supervisors:  Lisa Jeffrey/Paul Selick
Thesis title: The equivariant K-theory of commuting 2-tuples in SU(2)

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Abstract:

In this thesis, we study the space of commuting n-tuples in SU(2), $Hom(\mathbb{Z}^n, SU(2))$. We describe this space geometrically via providing an explicit G-CW complex structure, an equivariant analog of familiar CW- complexes. For the n=2 case, this geometric description allows us to compute various cohomology theories of this space, in particular the G-equivariant K-Theory $K_G^*(Hom(\mathbb{Z}^2, SU(2)))$, both as an $R(SU(2))$-module and as an $R(SU(2))$-algebra. This space is of particular interest as $\phi^{-1}(e)$ in a quasi-Hamiltonian system $M\xrightarrow{\phi} G$ consisting of the G-space $SU(2)\times SU(2)$, together with a moment map $\phi$ given by the commutator map. Finite dimensional quasi-Hamiltonian spaces have a bijective correspondence with certain infinite dimensional Hamiltonian spaces, and we additionally compute relevant components of this larger picture in addition to $\phi^{-1}(e)=Hom(\mathbb{Z}^2, SU(2))$ for this example.

A copy of the thesis can be found here: TreforBazettThesis

Departmental PhD Thesis Exam – Jennifer Vaughan

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

Monday, May 9, 2016
11:10 a.m.
BA6183

PhD Candidate:  Jennifer Vaughan
Co-Supervisors:  Yael Karshon
Thesis title:  Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization

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Abstract:

Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction.  This thesis extends certain properties of Kostant-Souriau quantization to the metaplectic-c context.  We show that the Kostant-Souriau results are replicated or improved upon with metaplectic-c quantization.

We consider two topics:  quantomorphisms and quantized energy levels.  If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle.  We present a definition for a metaplectic-c quantomorphism, and prove that the space of infinitesimal metaplectic-c quantomorphisms exhibits all of the same properties that are seen in the Kostant-Souriau case.

Next, given a metaplectic-c prequantized symplectic manifold $(M,\omega)$ and a function $H\in C^\infty(M)$, we propose a condition under which $E$, a regular value of $H$, is a quantized energy level for the system $(M,\omega,H)$.  We prove that our definition is dynamically invariant:  if two functions on $M$ share a regular level set, then the quantization condition over that level set is identical for both functions.  We calculate the quantized energy levels for the $n$-dimensional harmonic oscillator and the hydrogen atom, and obtain the quantum mechanical predictions in both cases.  Lastly, we generalize the quantization condition to a level set of a family of Poisson-commuting functions, and show that in the special case of a completely integrable system, it reduces to a Bohr-Sommerfeld condition.

The draft to the thesis can be found here: Vaughan-Draft

Departmental PhD Thesis – Iva Halacheva

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

Wednesday, June 29, 2016
2:10 p.m.
BA6183

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

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Abstract:

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

Departmental PhD Thesis Exam – Tyler Holden

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

Wednesday, May 4, 2016
2:10 p.m.
BA6183

PhD Candidate:  Tyler Holden
Supervisor:  Lisa Jeffrey
Thesis title: Convexity and Cohomology of the Based Loop Group

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Abstract:

Let $K$ be a compact, connected, simply connected Lie group and define $\Omega K$ to be the loops on $K$. Let $\Omega_\text{alg}K$ be those loops which are the restriction of algebraic maps $\Bbb C^\times \to K_\Bbb C$.  Herein we establish two distinct but related results.  In the first, we demonstrate the module structure for various generalized abelian equivariant cohomology theories as applied to equivariantly stratified spaces. This result is applied to the algebraic based loop group for the cases of equivariant singular cohomology, $K$-theory, and complex cobordism cohomology.     Subsequently, we examine the image of the based loop group under the non-abelian moment map. We show that both the Kirwan and Duistermaat convexity theorems hold in this infinite dimensional setting.

A copy of the thesis can be found here: ThesisPreview

Departmental PhD Thesis Exam – Peter Crooks

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

Wednesday, May 4, 2016
11:10 a.m.
BA1200

PhD Candidate:  Peter Crooks
Supervisor:  Lisa Jeffrey
Thesis title:  The Equivariant Geometry of Nilpotent Orbits and Associated Varieties

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Abstract:

Nilpotent orbits are highly structured algebraic varieties lying at the interface of algebraic geometry, Lie theory, symplectic geometry, and geometric representation theory. The interest in these objects has been long-standing, ranging from Kostant’s foundational work in the 1950s and 1960s to Kronheimer’s realization of nilpotent orbits as instanton moduli spaces. At the same time, nilpotent orbits are studied for the sake of understanding closely associated varieties, such as nilpotent Hessenberg varieties.

In this thesis, we study the equivariant algebraic geometry and topology of nilpotent orbits and related varieties. Our first group of results is principally concerned with presentations of $T$-equivariant cohomology rings. More specifically, we give concrete presentations of the $T$-equivariant cohomology rings of the regular and minimal nilpotent orbits, with the latter presentation providing an equivariant counterpart to existing work on the ordinary cohomology of the minimal nilpotent orbit. We also examine the family of Hessenberg varieties arising from the minimal nilpotent orbit, showing them to be GKM and obtaining presentations of their $T$-equivariant cohomology rings. In Lie type $A$, we explain how one would compute the Poincar\'{e} polynomials and irreducible components of these Hessenberg varieties.

Our second group of results includes a characterization of those semisimple real Lie algebras for which every complex nilpotent orbit contains a real one, building on Rothschild’s criterion for such a Lie algebra to be quasi-split. We also consider the role of nilpotent orbits in quaternionic K\”ahler geometry by giving a new, self-contained proof of the LeBrun-Salamon Conjecture for equivariant contact structures on partial flag varieties. This approach allows us to give an intrinsic description of the standard contact structure on the isotropic Grassmannian of $2$-planes in $\mathbb{C}^{2n}$.

We conclude on a somewhat different note by computing the generalized $T$-equivariant cohomology of a direct limit of smooth projective varieties. As a brief application, we obtain the generalized $T$-equivariant cohomology of the affine Grassmannian.

A copy of the thesis can be found here: Thesis3

 

Departmental PhD Thesis Exam – Boris Lishak

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

Monday, May 2, 1016
11:10 a.m.
BA6183

PhD Candidate:  Boris Lishak
Supervisor:  Alex Nabutovsky
Thesis title:  Balanced Presentations of the Trivial Group and 4-dimensional Geometry

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Abstract:

We construct a sequence of balanced presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial element grows faster than the tower of exponentials of any fixed height of the length of the finite presentation.

We prove that 1) There exist infinitely many non-trivial codimension one “thick” knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $\epsilon$ the set of isometry classes of Riemannian metrics with volume equal to $1$ and injectivity radius greater than $\epsilon$ is disconnected; and 3) For each closed four-dimensional $PL$-manifold $M$ and any $m$ there exist arbitrarily large values of $N$ such that some two triangulations of $M$ with $<N$ simplices cannot be connected by any sequence of $<\exp_m(N)$ bistellar transformations, where $\exp_m(N)=\exp(\exp(\ldots \exp (N)))$ ($m$ times).

 

We construct families of trivial $2$-knots $K_i$ in $\mathbb{R}^4$ such that the maximal complexity of $2$-knots in any isotopy connecting $K_i$ with the standard unknot grows faster than a tower of exponentials of any fixed height of the complexity of $K_i$. Here we can either construct $K_i$ as smooth embeddings and measure their complexity as the ropelength (a.k.a the crumpledness) or construct PL-knots $K_i$, consider isotopies through PL knots, and measure the complexity of a PL-knot as the minimal number of flat $2$-simplices in its triangulation.

For any $m$ we produce an exponential number of balanced presentations of the trivial group with four generators and four relations of length $N$ such that the minimal number of Andrews-Curtis transformations needed to connect any two of the presentations is at least $\exp_m(N)$.

A copy of the thesis can be found here: ut-thesis

Departmental PhD Thesis Exam – Parker Glynn-Adey

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

 

Monday, May 2, 2016
2:10 p.m.
BA6183

PhD Candidate:  Parker Glynn-Adey
Supervisor:  Rina Rotman
Thesis title: Width, Ricci Curvature, and Bisecting Surfaces

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Abstract:

In this thesis we studied width-volume inequalities, bisecting surfaces in three spheres, and the planar case of Larry Guth’s sponge problem.  Our main result is a width-volume inequality for conformally non-negatively Ricci curved manifolds.  We obtain several estimates on the size of minimal hypersurfaces in such manifolds.  Concerning geometric subdivision and 3-spheres, we give a positive answer to a question of Papasoglu.  Regarding the sponge problem, we show that any open bounded Jordan measurable set in the plane of small area admits an expanding embedding in to a strip of unit height.  We also prove that a generalization of the planar sponge problem is NP-complete. This thesis is partially based on joint work with Ye. Liokumovich [G-ALiokumovich2014] and Z. Zhu [G-AZhu2015]

A copy of the thesis can be found here: ut-thesis

 

Departmental PhD Thesis Title – Alex Weekes

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

Friday, May 6, 2016
11:10 a.m.
BA6183

PhD Candidate:  Alex Weekes
Supervisor:  Joel Kamnitzer
Thesis title:  Highest weights for truncated shifted Yangians

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Abstract:

Truncated shifted Yangians are a family of algebras which are conjectured to quantize slices to Schubert varieties in the affine Grassmannian. In this thesis we study the highest weight theory of these algebras, and explore connections with Nakajima quiver varieties and their cohomology. We give a conjectural parametrization of the set of highest weights in terms of product monomial crystals, which are related to Nakajima’s monomial crystal. In type A we prove this conjecture.

Our main tool in describing the set of highest weights is the B–algebra, which is a non-commutative generalization of the notion of torus fixed-point subscheme. We give a conjectural presentation for this algebra based on calculations using Yangians, and show how this presentation admits a natural geometric interpretation in terms of the equivariant cohomology of quiver varieties. We conjecture that this gives an explicit presentation for the equivariant cohomology ring of the Nakajima quiver variety of a finite ADE quiver, and show that this conjecture could be deduced from a special case. We give a proof of this conjecture in type A.

This work can be thought of in the context of symplectic duality. In our case, slices to Schubert varieties in the affine Grassmannian are expected to be symplectic dual to Nakajima quiver varieties. The relationship between B–algebras and equivariant cohomology is part of a general conjecture of Nakajima for symplectic dual varieties. These ideas represent a first approximation to expected connections between the category $\mathcal O$’s for a symplectic dual pair of varieties.

A copy of the thesis can be found here: Alex Weekes – thesis