## Departmental PhD Thesis Exam – Mihai Alboiu

Wednesday, March 24, 2021
10:00 a.m. (sharp)

PhD Candidate:  Mihai Alboiu
Supervisor:   George Elliott
Thesis title: The Stable Rank of Diagonal Ash Algebras

***

Building on the work of Lutley, we study a certain subclass of recursive subhomogeneous algebras, called DSH algebras, in which the pullback maps are all diagonal in a suitable sense. We examine inductive limits of DSH algebras, where each bonding map is itself diagonal in an appropriate way, and show that every simple algebra thus obtained has stable rank one. We are therefore able to show that every simple dynamical crossed product has stable rank one and that the Toms-Winter Conjecture holds for such algebras.

We also introduce the class of non-unital DSH algebras and make partial progress towards showing that inductive limits of such algebras with diagonal maps have stable rank one. Moreover, we investigate more intrinsic notions of a diagonal map and matrix unit compatibility and show that in the full matrix algebra setting they agree with their usual (global) counterparts.

A copy of the thesis can be found here:  ALBOIU_Thesis_Draft_Updated

## Departmental PhD Thesis Exam – Selim Tawfik

Monday, January 11, 2021
1:00 p.m.

PhD Candidate:  Selim Tawfik
Supervisor:   Eckhard Meinrenken
Thesis title:  Fusion Product of D/G-Valued Moment Maps

***

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

## Departmental PhD Thesis Exam – Arthur Mehta

Friday, January 22, 2021
4:00 p.m.

PhD Candidate:  Arthur Mehta
Supervisor:   Henry Yuen
Thesis title:  Entanglement and non-locality in games and graphs

***

This thesis is primarily based on two collaborative works written by the author and several coauthors. These works are presented in Chapters 4 and 5 and are on the topics of quantum graphs, and self-testing via non-local games, respectively.

Quantum graph theory, also known as non-commutative graph theory, is an operator space generalization of graph theory. The independence number, and Lova’sz theta function were generalized to this setting by Duan, Severini, and Winter and two different version of the chromatic number were introduced by Stahlke and Paulsen. In Chapter 4, we introduce two new generalizations of the chromatic number to non-commutative graphs and provide an upper bound on the parameter of Stahlke. We provide a generalization of the graph complement and show the chromatic number of the orthogonal complement of a non-commutative graph is bounded below by its theta number. We also provide a generalization of both Sabidussi’s Theorem and Hedetniemi’s conjecture to non-commutative graphs.

The study of non-local games considers scenarios in which separated players collaborate to provide satisfying responses to questions given by a referee. The condition of separating players makes non-local games an excellent setting to gain insight into quantum phenomena such as entanglement and non-locality. Non-local games can also provide protocols known as self-tests. Self-testing allows an experimenter to interact classically with a black box quantum system and certify that a specific entangled state was present, and a specific set of measurements were performed. The most studied self-test is the CHSH game which certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. In Chapter 5, we introduce an algebraic generalization of CHSH and obtain a self-test for non-Pauli operators resolving an open question posed by Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012).

The results of Chapter 5 make use of sums of squares techniques in the settings of group rings and *-algebras. In Chapter 3, we review these techniques and discuss how they relate to the study of non-local games. We also provide a weak sum of squares property for the ring of integers. We show that if a Hermitian element is positive under all unitary representations then it must be expressible as a sum of Hermitian squares.

A copy of the thesis can be found here:  Thesis_Version_3 (1)-1

## Departmental PhD Thesis Exam – Adam Gardner

Monday, November 30, 2020
5:00 p.m.

PhD Candidate:  Adam Gardner
Supervisor:   Michael Sigal
Thesis title: Instability of electroweak homogeneous vacua in strong magnetic fields

****

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

## Departmental PhD Thesis Exam – Nathan Carruth

Friday, November 13, 2020
11:00 a.m.

PhD Candidate:  Nathan Carruth
Supervisor:   Spyros Alexakis
Thesis title: Focussed Solutions to the Einstein Vacuum Equations

***

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

## Departmental PhD Thesis Exam – Justin Ko

Wednesday, August 25, 2020
11:00 a.m.

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

## Departmental PhD Thesis Exam – Afroditi Talidou

Wednesday, August 5, 2020
4:00 p.m.

PhD Candidate:  Afroditi Talidou
Co-Supervisors:   Michael Sigal, Almut Burchard
Thesis title:  Near-pulse solutions of the FitzHugh-Nagumo equations on cylindrical surfaces

****

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

## Departmental PhD Thesis Exam – Jeffrey Pike

Friday, August 14, 2020
11:00 a.m.

PhD Candidate:  Jeffrey Pike
Supervisor:   Eckhard Meinrenken
Thesis title:  Weil Algebras for Double Lie Algebroids

***

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

## Departmental PhD Thesis Exam – Anne Dranowski

Monday, July 27, 2020
2:00 p.m.

PhD Candidate:  Anne Dranowski
Supervisor:   Joel Kamnitzer
Thesis title:  Comparing two perfect bases

****

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

## Departmental PhD Thesis Exam – Khoa Pham

Monday, July 13, 2020
2:00 p.m.

PhD Candidate:  Khoa Pham
Supervisor:   Joel Kamnitzer
Thesis title:  Multiplication of generalized affine Grassmannian slices and comultiplication of shifted Yangians

***

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