## Departmental PhD Thesis Exam – Fabian Parsch

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

Tuesday, June 18, 2019
11:10 a.m.
BA6183

PhD Candidate:  Fabian Parsch
Supervisor:   Alex Nabutovsky
Thesis title:  Geodesic Nets with Few Boundary Points

***

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere.
We survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. A particular focus will be put on the question if the number of inner vertices (balanced vertices) in a geodesic net can be bounded by the number of boundary points (unbalanced vertices) or the total imbalance.
We prove that a geodesic net with three unbalanced vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degree of unbalanced vertices. The result seems to be new even in the Euclidean case.
We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices which can have a significantly more complicated structure. In particular, an example of a geodesic net with four unbalanced vertices and sixteen balanced vertices that is not a union of simpler geodesic nets is constructed. The previously known irreducible geodesic nets with four unbalanced vertices have at most two balanced vertices.
We provide a partial answer for a related question, namely a description of a new infinite family of geodesic nets on the Euclidean plane with 14 unbalanced vertices and arbitrarily many balanced vertices of degree three or more.

A copy of the thesis can be found here: parsch_thesis_2019-04-09

## Departmental PhD Thesis Exam – Zhifei Zhu

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

BA6183

PhD Candidate:  Zhifei Zhu
Supervisor:   Regina Rotman
Thesis title:

***

## Departmental PhD Thesis Exam – Leonid Monin

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

Friday, April 26, 2019
11:10 a.m.
BA6183

PhD Candidate:  Leonid Monin
Thesis title: Newton Polyhedra, Overdetermined system of equations, and Resultants

***

In the first part of this thesis we develop Newton polyhedra theory for overdetermined systems of equations. Let A_1…A_k be finite sets in Z^n  and let be an algebraic subvariety of (C^*)^n  defined by a system of  Laurent polynomials with supports in A_1… A_k.  Assuming that  Laurent polynomials are sufficiently generic, the Newton polyhedron theory computes discrete invariants of Y in terms of their Newton polyhedra. It may appear that the generic system with fixed supports  is inconsistent. In this case one is interested in the generic consistent system. We extend Newton polyhedra theory to this case and compute discrete invariants generic non-empty zero sets. Unlike the classical situation, not only the Newton polyhedra of Laurent polynomials, but also their supports themselves appear in the answers.

We proceed then to the study of overdetermined collections of linear series on algebraic varieties other than algebraic torus. That is let E_1…E_k be a finite dimensional subspace of the space of  regular sections of line bundles on an irreducible algebraic variety X, so that the system

s_1 = … = s_k = 0,

where s_i is a generic  element of E_i does not have any roots on X. In this case we investigate the consistency variety  (the closure of the set of all systems which have at least one common root) and study general properties of zero sets Z of a generic consistent system. Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set Z.

The second part of this thesis is devoted to the study of Delta-resultants of (n+1)-tuple of Laurent polynomials with generic enough Newton polyhedra.  We provide an algorithm for computing Delta-resultant assuming that an n-tuple f_2, …, f_{n+1} is developed. We provide a relation between the product of f_1 over roots of  f_2 = … = f_{n+1} = 0 in (C^*)^n and the product of f_2 over roots of f_1=f_3 = … = f_{n+1} = 0 in (C^*)^n assuming that the n-tuple (f_1f_2, f_3…f_{n+1} is developed. If all n-tuples contained  in (f_1…f_{n+1}) are developed we provide a signed version of Poisson formula for Delta-resultant. Interestingly, the sign of the sparse resultant is nontrivial and is defined through Parshin symbols. Our proofs are based on a topological version of the Parshin reciprocity laws.
A copy of the thesis can be found here:  ut-thesis monin

## Departmental PhD Thesis Exam – Francisco Guevara Parra

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

Friday, March 15, 2019
1:30 p.m.
Fields Institute, room 210

PhD Candidate:  Francisco Guevara Parra
Supervisor:   Stevo Todorcevic
Thesis title:  Analytic spaces and their Tukey types

***

In this Thesis we study topologies on countable sets from the perspective of Tukey reductions of their neighbourhood filters. It turns out that is closely related to the already established theory of definable (and in particular analytic) topologies on countable sets. The connection is in fact natural as the neighbourhood filters of points in such spaces are typical examples of directed sets for which Tukey theory was introduced some eighty years ago. What is interesting here is that the abstract Tukey reduction of a neighbourhood filter $\mathcal{F}_{x}$ of a point to standard directed sets like $\mathbb{N}^\mathbb{N}$ or $\ell_1$ imposes that $\mathcal{F}_{x}$ must be analytic. We develop a theory that examines the Tukey types of analytic topologies and compare it by the theory of sequential convergence in arbitrary countable topological spaces either using forcing extensions or axioms such as, for example, the Open Graph Axiom. It turns out that in certain classes of countable analytic groups we can classify all possible Tukey types of the corresponding neighbourhood filters of identities. For example we show that if $G$ is a countable analytic $k$-group then $1=\{0\},$ $\mathbb{N}$ and $\mathbb{N}^\mathbb{N}$ are the only possible Tukey types of the neighbourhood filter $\mathcal{F}_{e}^{G}$. This will give us also new metrization criteria for such groups. We also show that the study of definable topologies on countable index sets has natural analogues in the study of arbitrary topologies on countable sets in certain forcing extensions.

A copy of the thesis can be found here: Francisco_PhD_Thesis

## Departmental PhD Thesis Exam – Duncan Dauvergne

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

Monday, March 25, 2019
10:10 a.m.
BA6183

PhD Candidate:  Duncan Dauvergne
Supervisor:   Balint Virag
Thesis title:  Random sorting networks, the directed landscape, and random polynomials

***

The first part of this thesis is on random sorting networks. A sorting network is a shortest path from $12 \cdots n$ to $n \cdots 21$ in the Cayley graph of the symmetric group $S_n$ generated by adjacent transpositions. We prove that in a uniform random $n$-element sorting network $\sigma^n$, all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-$t$ permutation matrix measures of $\sigma^n$. As a corollary, we show that if $S_n$ is embedded into $\mathbb{R}^n$ via the map $\tau \mapsto (\tau(1), \tau(2), \dots \tau(n))$, then with high probability, the path $\sigma^n$ is close to a great circle on a particular $(n-2)$-dimensional sphere. These results prove conjectures of Angel, Holroyd, Romik, and Vir\’ag. To prove these results, we find the local limit of random sorting networks and prove that the local speed distribution is the arcsine distribution on $[-\pi, \pi]$.

The second part of this thesis is on last passage percolation. The conjectured limit of last passage percolation is a scale-invariant, independent, stationary increment process with respect to metric composition. We prove this for Brownian last passage percolation. We construct the Airy sheet and characterize it in terms of the Airy line ensemble. We also show that the last passage geodesics converge to random functions with H\”older-$2/3^-$ continuous paths. To prove these results, we develop a new probabilistic framework for understanding the Airy line ensemble.

The third part of this thesis is on random sums of orthonormal polynomials. Let $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d. non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$ supported on a compact set $K$. We show that the zero measure of $G_n$ converges weakly almost surely to the equilibrium measure of $K$ if and only if $\mathbb{E}\log(1 + |\xi_0|) < \infty$. We also show that the zero measure of $G_n$ converges weakly in probability to the equilibrium measure of $K$ if and only if $\mathbb{P}(|\xi_0| > e^n) = o(n^{-1})$. Our methods also work for more general sequences of asymptotically minimal polynomials in $L^p(\tau)$, where $p \in (0, \infty]$.

A copy of the thesis can be found here: MainThesisPhD

## Departmental PhD Thesis Exam – Chia-Cheng Liu

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

Wednesday, December 5, 2018
11:10 a.m.
BA1170

PhD Candidate:  Chia-Cheng Liu
Co-Supervisors:   Joel Kamnitzer/Alexander Braverman
Thesis title:  Semi-innite Cohomology, Quantum Group Cohomology, and the Kazhdan-Lusztig
Equivalence

The Kazhdan-Lusztig tensor equivalence is a monoidal functor which sends modules over ane
Lie algebras at a negative level to modules over quantum groups at a root of unity. A positive
level Kazhdan-Lusztig functor is dened using Arkhipov-Gaitsgory’s duality between ane Lie
algebras of positive and negative levels. We prove that the semi-innite cohomology functor
for positive level modules factors through the positive level Kazhdan-Lusztig functor and the
quantum group cohomology functor with respect to the positive part of Lusztig’s quantum
group. This is the main result of the thesis.

Monoidal structure of a category can be interpreted as factorization data on the associated
global category. We describe a conjectural reformulation of the Kazhdan-Lusztig tensor equivalence
in factorization terms. In this reformulation, the semi-innite cohomology functor at
positive level is naturally factorizable, and it is conjectured that the factorizable semi-innite
cohomology functor is essentially the positive level Kazhdan-Lusztig tensor functor modulo the
Riemann-Hilbert correspondence. Our main result provides an important technical tool in a
proposed approach to a proof of this conjecture.

A copy of the thesis can be found here: thesis_chiachengliu-1

## Departmental PhD Thesis Exam – Krishan Rajaratnam

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

Friday, October 19, 2018
2:10 p.m.
BA6183

PhD Candidate:  Krishan Rajaratnam
Supervisor:   Michael Sigal
Thesis title: Abrikosov lattice solutions of the ZHK Chern-Simons equations

****

In this thesis we study the ZHK Chern-Simons equations which occur in the study of the fractional quantum hall effect of condensed matter physics. After stating basic properties of these equations, we first prove the existence of Abrikosov lattice solutions of them.  Among these solutions, we find the physically interesting one whose lattice shape minimizes the average energy per lattice cell. In addition to the Abrikosov lattice solutions, we find solutions of the ZHK Chern-Simons equations on Riemann surfaces of higher genus $g$, by utilizing similar results for the Ginzburg-Landau equations.

Finally, we study the orbital stability of the Abrikosov lattice solutions under perturbations which preserve the lattice.

A copy of the thesis can be found here:

## Departmental PhD Thesis Exam – Steven Amelotte

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

Tuesday, June 26 2018
1:10 p.m.
BA6183

PhD Candidate:  Steven Amelotte
Supervisor:   Paul Selick
Thesis title: Unstable Homotopy Theory Surrounding the Fibre of the $p^\text{th}$ Power Map on Loop Spaces of Spheres

***

In this thesis, we study the fibre of the $p^\text{th}$ power map on loop spaces of spheres with a view toward obtaining homotopy decompositions. For the prime $p=2$, we give an explicit decomposition of the fibre $\Omega^3S^{17}\{2\}$ of the squaring map on the triple loop space of the $17$-sphere, building on work of Campbell, Cohen, Peterson and Selick, who showed that such decompositions are only possible for $S^5$, $S^9$ and $S^{17}$. This induces a splitting of the mod $2$ homotopy groups $\pi_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of the integral homotopy groups of the fibre of the double suspension $E^2 \colon S^{2n-1} \to \Omega^2S^{2n+1}$ and refines a result of Cohen and Selick, who gave similar decompositions for $S^5$ and $S^9$. For odd primes, we find that the decomposition problem for $\Omega S^{2n+1}\{p\}$ is equivalent to the strong odd primary Kervaire invariant problem. Using this unstable homotopy theoretic interpretation of the Kervaire invariant problem together with what is presently known about the $3$-primary stable homotopy groups of spheres, we give a new decomposition of $\Omega S^{55}\{3\}$.

We relate the $2$-primary decompositions above to various Whitehead products in the homotopy groups of mod $2$ Moore spaces and Stiefel manifolds to show that the Whitehead square $[i_{2n},i_{2n}]$ of the inclusion of the bottom cell of the Moore space $P^{2n+1}(2)$ is divisible by $2$ if and only if $2n=2,4,8$ or $16$. As an application of our $3$-primary decomposition, we prove two new cases of a longstanding conjecture which states that the fibre of the double suspension is a double loop space.

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

## Departmental PhD Thesis Exam – Vincent Gelinas

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

Monday, June 18,  2018
11:10 a.m.
BA6183

PhD Candidate:  Vincent Gelinas
Co-Supervisors:   Joel Kamnitzer, Colin Ingalls
Thesis title:  Contributions to the Stable Derived Categories of Gorenstein Rings

***

Abstract:

The stable derived category ${\rm D}_{sg}(R)$ of a Gorenstein ring $R$ is defined as the Verdier quotient of the bounded derived category ${\rm D}^b(\modsf R)$ by the thick subcategory of perfect complexes, that is, those with finite projective resolutions, and was introduced by Ragnar-Olaf Buchweitz as a homological measure of the singularities of $R$. This thesis contributes to its study, centered around representation theoretic, homological and Koszul duality aspects.

In Part I, we first complete (over $\C$) the classification of homogeneous complete intersection isolated singularities $R$ for which the graded stable derived category ${\rm D}^{\Z}_{sg}(R)$ (respectively, ${\rm D}^b(\coh X)$ for $X = \proj R$) contains a tilting object. This is done by proving the existence of a full strong exceptional collection of vector bundles on a $2n$-dimensional smooth complete intersection of two quadrics $X = V(Q_1, Q_2) \subseteq \mathbb{P}^{2n+2}$, building on work of Kuznetsov. We then use recent results of Buchweitz-Iyama-Yamaura to classify the indecomposable objects in ${\rm D}_{sg}^{\Z}(R_Y)$ and the Betti tables of their complete resolutions, over $R_Y$ the homogeneous coordinate rings of $4$ points on $\mathbb{P}^1$ and $4$ points on $\mathbb{P}^2$ in general position.

In Part II, for $R$ a Koszul Gorenstein algebra, we study a natural pair of full subcategories whose intersection $\mathcal{H}^{\mathsf{lin}}(R) \subseteq {\rm D}_{sg}^{\Z}(R)$ consists of modules with eventually linear projective resolutions. We prove that such a pair forms a bounded t-structure if and only if $R$ is absolutely Koszul in the sense of Herzog-Iyengar, in which case there is an equivalence of triangulated categories ${\rm D}^b(\mathcal{H}^{\mathsf{lin}}(R)) \cong {\rm D}_{sg}^{\Z}(R)$. We then relate the heart to modules over the Koszul dual algebra $R^!$. As first application, we extend the Bernstein-Gel’fand-Gel’fand correspondence beyond the case of exterior and symmetric algebras, or more generally complete intersections of quadrics and homogeneous Clifford algebras, to any pair of Koszul dual algebras $(R, R^!)$ with $R$ absolutely Koszul Gorenstein. In particular the correspondence holds for the coordinate ring of elliptic normal curves of degree $\geq 4$ and for the anticanonical model of del Pezzo surfaces of degree $\geq 4$. We then relate our results to conjectures of Bondal and Minamoto on the graded coherence of Artin-Schelter regular algebras and higher preprojective algebras; we characterise when these conjectures hold in a restricted setting, and give counterexamples to both in all dimension $\geq 4$.

A copy of the thesis can be found here:  thesis

## Departmental PhD Thesis Exam – Huan Vo

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

Wednesday, June 20 2018
11:10 a.m.
BA6183

PhD Candidate:  Huan Vo
Supervisor:   Dror Bar-Natan
Thesis title:  Alexander Invariants of Tangles via Expansions

****

Abstract:

In this thesis we describe a method to extend the Alexander polynomial to tangles. It is based on a
technology known as expansions, which is inspired by the Taylor expansion and the Kontsevich integral.
Our main object of study is the space of w-tangles, which contains usual tangles, but has a much simpler
expansion. To study w-tangles, we introduce an algebraic structure called meta-monoids. An expansion
of w-tangles together with a particular Lie algebra, namely the non-abelian two-dimensional Lie algebra,
gives us a meta-monoid called Γ-calculus that recovers the Alexander polynomial. Using the language
of Γ-calculus, we rederive certain important properties of the Alexander polynomial, most notably the
Fox-Milnor condition on the Alexander polynomials of ribbon knots [Lic97, FM66]. We argue that our
proof has some potential for generalization which may help tackle the slice-ribbon conjecture. In a sense
this thesis is an extension of [BNS13].

A copy of the thesis can be found here:  Thesis_HuanVo_V1