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PhD Candidate: Selim Tawfik

Supervisor: Eckhard Meinrenken

Thesis title: Fusion Product of D/G-Valued Moment Maps

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A fusion product is defined for Hamiltonian spaces with moment maps valued in a Lie group $D$ generalizing those of Alekseev-Malkin-Meinrenken. An analogous theory for these general Hamiltonian spaces is developed and, among other results, versions of symplectic reduction, duality and the shifting trick are derived. The Hamiltonian spaces with moment maps valued in a homogeneous space $D/G$ of Alekseev-Kosmann-Schwarzbach are shown to be equivalent to certain Hamiltonian spaces with group-valued moment maps. The aforementioned theory is consequently brought to bear on that of $D/G$-valued moment maps, thereby defining a fusion product for these. This fusion product affords many new examples of $D/G$-valued moment maps, of which there was hitherto a paucity. Among said examples are moduli spaces of flat principal bundles over certain surfaces with boundary.

A copy of the thesis can be found here: main

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PhD Candidate: Arthur Mehta

Supervisor: Henry Yuen

Thesis title: Entanglement and non-locality in games and graphs

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A copy of the thesis can be found here: ArthurMehta-Thesis

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PhD Candidate: Adam Gardner

Supervisor: Michael Sigal

Thesis title: Instability of electroweak homogeneous vacua in strong magnetic fields

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We consider the classical (local) vacua of the Weinberg-Salam (WS) model of electroweak forces. These are defined as no-particle, static solutions to the WS equations minimizing the WS energy locally. In the absence of particles, the Weinberg-Salam model reduces to the Yang-Mills-Higgs (YMH) equations for the gauge group U(2).

We consider the WS system in a constant external magnetic field, b, and prove that (i) there is a magnetic field threshold b* such that for b<b*, the vacua are translationally invariant, while, for b>b*, they are not, (ii) for b>b*, there are non-translationally invariant solutions with lower energy per unit volume and with the discrete translational symmetry of a 2D lattice in the plane transversal to the external magnetic field, and (iii) the lattice minimizing the energy per unit volume approaches the hexagonal one as the magnetic field strength approaches the threshold b*.

A copy of the thesis can be found here: Adam-Gardner-Thesis

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PhD Candidate: Nathan Carruth

Supervisor: Spyros Alexakis

Thesis title: Focussed Solutions to the Einstein Vacuum Equations

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We construct solutions to the Einstein vacuum equations in polarised translational symmetry in $3 + 1$ dimensions which have $H^{^1}$ energy concentrated in an arbitrarily small region around a two-dimensional null plane and large $H^{^2}$ initial data. Specifically, there is a parameter $k$ and coordinates $s$, $x$, $v$, $y$ such that the null plane is given by $x = k^{-1/2}/2$, $v = T\sqrt{2} – k^{-1}/2$ for some $T$ independent of $k$, the $H^{^1}$ energy of the solution is concentrated on the region $[0, T’] \times [0, k^{-1/2}] \times [T\sqrt{2} – k^{-1}, T\sqrt{2}] \times \R^{^1}$, and the $H^{^2}$ norm of the initial data is bounded below by a multiple of $k^{3/4}$. The time $T’$ has a lower bound independent of $k$.

This result relies heavily on a new existence theorem for the Einstein vacuum equations with characteristic initial data which is large in $H^{^2}$. This result is proved using parabolically scaled coordinates in a null geodesic gauge.

A copy of the thesis can be found here: thesis_comm

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PhD Candidate: Justin Ko

Supervisor: Dmitry Panchenko

Thesis title: The Free Energy of Spherical Vector Spin Glasses

We study a class of vector spin models with configurations restricted to subsets of the sphere. We will prove a constrained free energy formula for these models. This formula defines a large deviations principle for the limiting distribution of the overlaps under the asymptotic Gibbs measure. The thesis builds on the mathematical results used to prove free energy formulas for the classical Sherrington–Kirkpatrick spin glass, spherical spin models, and vector spin glass models. The free energy formula proved in this thesis are true generalizations of the classical results, in the sense that these vector spin formulas restricted to one dimension coincide with the known results for classical models.

The first contribution of this thesis is a variational formula for contrained copies of classical spherical spin glasses sampled at different temperatures. The free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied by Panchenko and Talagrand. They proved an upper bound of the constrained free energy using Guerra’s interpolation. In this thesis, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman–Sims–Starr scheme and the synchronization mechanism used in the vector spin models. We derive a vector version of the Aizenman–Sims–Starr scheme for spherical spin glass and use the synchronization property of arrays obeying the overlap-matrix form of the Ghirlanda–Guerra identities to prove the matching lower bound.

The second contribution of this thesis is the simplification of this variational formula to the form originally discovered for the classical spherical spin glass model by Crisanti and Sommers. In particular, we prove the analogue of the Crisanti–Sommers variational formula for spherical spin glasses with vector spins. This formula is derived from the discrete Parisi variational formula for the limit of the free energy of constrained copies of spherical spin glasses. In vector spin models, the variations of the functional order parameters must preserve the monotonicity of matrix paths which introduces a new challenge in contrast to the derivation of the classical Crisanti–Sommers formula.

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

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PhD Candidate: Afroditi Talidou

Co-Supervisors: Michael Sigal, Almut Burchard

Thesis title: Near-pulse solutions of the FitzHugh-Nagumo equations on cylindrical surfaces

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In 1961, FitzHugh [19] suggested a model to explain the basic properties of excitability, namely the ability to respond to stimuli, as exhibited by the more complex HodgkinHuxley equations [24]. The following year Nagumo et al. [42] introduced another version based on FitzHugh’s model. This is the model we consider in the thesis. It is called the FitzHugh-Nagumo model and describes the propagation of electrical signals in nerve axons. Many features of the system have been studied in great detail in the case where an axon is modelled as a one-dimensional object. Here we consider a more realistic geometric structure: the axons are modelled as warped cylinders and pulses propagate on their surface, as it happens in nature.

The main results in this thesis are the stability of pulses for standard cylinders of small constant radius, and existence and stability of near-pulse solutions for warped cylinders whose radii are small and vary slowly along their lengths. On the standard cylinder, we write a solution near a pulse as the superposition of a modulated pulse with a fluctuation and prove that the fluctuation decreases exponentially over time as the solution converges to a nearby translation of the pulse. On warped cylinders, we write a solution near a pulse in the same way as in standard cylinders and prove bounds on the fluctuation of near-pulse solutions.

A copy of the thesis can be found here: Talidou-thesis-draft

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PhD Candidate: Jeffrey Pike

Supervisor: Eckhard Meinrenken

Thesis title: Weil Algebras for Double Lie Algebroids

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Given a double vector bundle D → M, we define a bigraded bundle of algebras W(D) → M called the ‘Weil algebra bundle’. The space W(D) of sections of this algebra bundle ‘realizes’ the algebra of functions on the supermanifold D[1, 1]. We describe in detail the relations between the Weil algebra bundles of D and those of the double vector bundles D′, D′′ obtained from D by duality operations. We show that VB-algebroid structures on D are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the third. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ‘classical’ version of Voronov’s result characterizing double Lie algebroid structures. In the case that D = T A is the tangent prolongation of a Lie algebroid, we find that W(D) is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad-Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multivector fields, and 2-term representations up to homotopy, all have natural interpretations in terms of our Weil algebras.

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

]]>Clovis obtained his bachelor’s degree from Universidad Metropolitana, Caracas, Venezuela. He is currently a second year doctoral student working in Set Theory and Model Theory under the supervision of Prof. Frank Tall.

Clovis writes: “The world-renowned Toronto Set Theory Seminar has been an ideal place for me, fostering my research and personal growth. Prof. Franklin D. Tall’s support and guidance have been essential”.

The prestigious Vanier CGS program aims to attract and retain world-class doctoral students by supporting students who demonstrate both leadership skills and a high standard of scholarly achievement in graduate studies in social sciences and humanities, natural sciences and engineering, and health.

The full list of Vanier CGS recipients is available at:

https://vanier.gc.ca/en/scholar_search-chercheur_recherche_2020.html

Congratulations Clovis! NSERC VANIER winner 2020

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PhD Candidate: Anne Dranowski

Supervisor: Joel Kamnitzer

Thesis title: Comparing two perfect bases

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We study a class of varieties which generalize the classical orbital varieties of Joseph. We show that our generalized orbital varieties are the irreducible components of a Mirkovic-Vybornov slice to a nilpotent orbit, and can be labeled by semistandard Young tableaux. Furthermore, we prove that Mirkovic-Vilonen cycles are obtained by applying the Mirkovic-Vybornov isomorphism to generalized orbital varieties and taking a projective closure, refining Mirkovic and Vybornov’s result. As a consequence, we are able to use the Lusztig datum of a Mirkovic-Vilonen cycle to determine the tableau labeling the generalized orbital variety which maps to it, and, hence, the ideal of the generalized orbital variety itself. By homogenizing we obtain equations for the cycle we started with, which is useful for computing various equivariant invariants such as equivariant multiplicity. As an application, we show that the Mirkovic-Vilonen basis differs from Lusztig’s dual semicanonical basis. This is significant because it is a first example of two perfect bases which are not the same. Our comparison relies heavily on the theory of measures developed by Baumann, Kamnitzer and Knutson (The Mirkovic-Vilonen basis and Duistermaat-Heckman measures) so we include what we need. We state a conjectural combinatorial ‘formula’ for the ideal of a generalized orbital variety in terms of its tableau.

A copy of the thesis can be found here: dranowski_anne_yyyymm_phd_thesis

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PhD Candidate: Khoa Pham

Supervisor: Joel Kamnitzer

Thesis title: Multiplication of generalized affine Grassmannian slices and comultiplication of shifted Yangians

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Given a semisimple algebraic group $G$, shifted Yangians are quantizations of certain generalized slices in $G((t^{-1}))$. In this thesis, we work with these generalized slices and the shifted Yangians in the simply-laced case.

Using a presentation of antidominantly shifted Yangians inspired by the work of Levendorskii, we show the existence of a family of comultiplication maps between shifted Yangians. We include a proof that these maps quantize natural multiplications of generalized slices.

On the commutative level, we define a Hamiltonian action on generalized slices, and show a relationship between them via Hamiltonian reduction. This relationship is established by constructing an explicit inverse to a multiplication map between slices.

Finally, we conjecture that the above relationship lifts to the Yangian level. We prove this conjecture for sufficiently dominantly shifted Yangians, and for the $\mathfrak{sl}_2$-case.

A copy of the thesis can be found here: Thesis-Khoa-final

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