## 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

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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

## 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

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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

## 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

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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

## 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

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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

## 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

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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

## Departmental PhD Thesis Exam – Xiao Ming

Thursday, June 25, 2020
2:00 p.m.

PhD Candidate:  Xiao Ming
Supervisor:   Stevo Todorcevic
Thesis title:  Borel Chain Conditions

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The subject matter of this Thesis is an instance of the Chain Condition Method of coarse classification of Boolean algebras and partially ordered sets. This method has played an important role in the measure theory, the theory of forcing, and the theory of Martin type axioms.

We focus on the posets that are Borel definable in Polish spaces and investigate the connections between the chain condition method and the chromatic numbers, a classification scheme for graphs. We then introduce Borel version of some classical chain condition and show that the Borel poset $T(\pi\mathbb{Q})$, the Borel example Todorcevic used to distinguish $\sigma$-finite chain condition and $\sigma$-bounded chain condition, cannot be decomposed into countably many Borel pieces witnessing the $\sigma$-finite chain condition, despite the fact that the non-Borel such partition exists. Starting from there, we use the variations on the $G_0$-dichotomy analyzed to construct a number of examples of Borel posets of the form $\mathbb{D}(G)$ that the new hierarchy of Borel chain conditions is proper.

A copy of the thesis can be found here:  Thesis_revised_202006210421

## Departmental PhD Thesis Exam – Abhishek Oswal

Wednesday, June 3, 2020
2:00 p.m.

PhD Candidate:  Abhishek Oswal
Supervisor:   Jacob Tsimerman
Thesis title: A non-archimedean definable Chow theorem

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O-minimality has had some striking applications to number theory. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this `tame’ property is the following surprising generalization of Chow’s theorem proved by Peterzil and Starchenko – A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this thesis, we explore a non-archimedean analogue of an o-minimal structure and prove a version of the definable Chow theorem in this context.

A copy of the thesis can be found here: thesis-draft-v4

## Departmental PhD Thesis Exam – Ren Zhu

Wednesday, May 27, 2020
2:00 p.m.

PhD Candidate:  Ren Zhu
Supervisor:   Kumar Murty
Thesis title:  The least prime whose Frobenius is an $n$-cycle

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Let $L/K$ be a Galois extension of number fields.  We consider the problem of bounding the least prime ideal of $K$ whose Frobenius lies in a fixed conjugacy class $C$. Under the assumption of Artin’s conjecture we work with Artin $L$-functions directly to obtain an upper bound in terms of irreducible characters which are nonvanishing at $C$.   As a consequence we obtain stronger upper bounds for the least prime in $C$ when many irreducible characters vanish at $C$.  We also prove a Deuring-Heilbronn phenomenon for Artin $L$-functions with nonnegative Dirichlet series coefficients as a key step.
We apply our results to the case when $\Gal(L/K)$ is the symmetric group $S_n$.  Using classical results on the representation theory of $S_n$ we give an upper bound for the least prime whose Frobenius is an $n$-cycle which is stronger than known bounds when the characters which are nonvanishing at $n$-cycles are unramified, as well a similar result for $(n-1)$-cycles.
We also give stronger bounds in the case of $S_n$-extensions over $\mathbb{Q}$ which are unramified over a quadratic field.  We also consider other groups and conjugacy classes where unconditional improvements are obtained.

A copy of the thesis can be found here: Ren Zhu PhD Thesis

## Departmental PhD Thesis Exam – Mykola Matviichuk

Wednesday, May 20, 2020
1:00 p.m.

PhD Candidate:  Mykola Matviichuk
Supervisor:   Marco Gualtieri
Thesis title:  Quadratic Poisson brackets and co-Higgs fields

This thesis is devoted to studying the geometry of holomorphic Poisson brackets on complex manifolds. We concentrate on the case when the underlying manifold admits a structure of a vector bundle, and the Poisson bracket is invariant under the dilation action of the multiplicative group of the field of complex numbers. We call such a Poisson bracket quadratic, and associate to it a Higgs type tensor, which we call a co-Higgs field. We study the interplay between these two geometric structures. A parallel theory is developed for Poisson brackets on projective bundles. Using the classical tool of the spectral correspondence available for co-Higgs fields, we construct many new examples of Poisson brackets, and provide new classification results in low dimensional cases.

A copy of the thesis can be found here: Quadratic_Poisson_brackets_and_co_Higgs_fields