Monday, June 17, 2019

12:10 p.m.

BA6183

PhD Candidate: Jia Ji

Supervisor: Lisa Jeffrey

Thesis title: Volume Formula and Intersection Pairings of N-fold Reduced Products

***

Let $ G $ be a semisimple compact connected Lie group. An $ N $-fold reduced product of $ G $ is the symplectic quotient of the Hamiltonian system of the Cartesian product of $ N $ coadjoint orbits of $ G $ under diagonal coadjoint action of $ G $. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume and the intersection pairings of an $ N $-fold reduced product of $ G $. In 2008, Suzuki and Takakura gave a volume formula of $ N $-fold reduced products of $ \mathbf{SU}(3) $ via Riemann-Roch.

We compare our volume formula with theirs and prove that up to normalization constant, our volume formula completely matches theirs in the case of triple reduced products of $ \mathbf{SU}(3) $.

The draft of the thesis can be found here: ut-thesis_Ji_draft_v1_1

]]>Friday, July 5, 2019

11:10 a.m.

BA6183

PhD Candidate: Jihad Zerouali

Supervisor: Eckhard Meinrenken

Thesis title: Twisted conjugation, quasi-Hamiltonian geometry, and Duistermaat-Heckman measures

***

Let $G$ be a Lie group, and let $\kappa\in\mathrm{Aut}(G)$. Let $G\kappa$ denote the group $G$ equipped with the $\kappa$-twisted conjugation action, $\mathrm{Ad}_{g}^{\kappa}(h)=gh\kappa(g^{-1})$. A twisted quasi-Hamiltonian manifold is a triple $(M,\omega,\Phi)$, where $M$ is a $G$-space, the equivariant map $\Phi:M\to G\kappa$ is called the moment map, and $\omega$ is a certain invariant 2-form with properties generalizing those of a symplectic structure.

The first topic of this work is a detailed study of $\kappa$-twisted conjugation, for $G$ compact, connected, simply connected and simple, and for $\kappa$ induced by a Dynkin diagram automorphism of $G$. After recovering the classification of $\kappa$-twisted conjugacy classes by elementary means, we highlight several properties of the so-called \textit{twining characters} $\tilde{\chi}^{(\kappa)}:G\rightarrow\mathbb{C}$.

We show that as elements of $L^{2}(G\kappa)^{G}$, the twining characters generalize several properties of the usual characters in a natural way. We then discuss $\kappa$-twisted representation and fusion rings, in relation to recent work of J. Hong. The second topic of this work is the study of the Duistermaat-Heckman (DH) measure $\mathrm{DH}_{\Phi}\in\mathcal{D}'(G\kappa)^{G}$ of a twisted quasi-Hamiltonian manifold $(M,\omega,\Phi)$. After developing the necessary background, we prove a localization formula for the Fourier coefficients of the measure $\mathrm{DH}_{\Phi}$, and we illustrate the theory with several examples of twisted moduli spaces. These character varieties parametrize a class of local systems on bordered surfaces, for which the transition functions take values in $G\rtimes\mathrm{Aut}(G)$ instead of $G$.

A copy of the thesis can be found here:

]]>Monday, July 15, 2019

1:10 p.m.

BA6183

PhD Candidate: Val Chiche-Lapierre

Supervisor: Jacob Tsimerman

Thesis title: Length of elements in a Minkowski basis for an order in a number field

(or a ring of integers of a number field)

Exam type: One-defense

***

Abstract:

Suppose K is a number field of degree n, and R is an order in K with discriminant D. If K has r real embeddings and s pairs of complex embeddings then we can look at R as a lattice in \R^r x \C^s. We call the length of elements of R their Euclidean length in \R^r x \C^s and denote it by |.|. Let v1=1,v2,…,vn be a Minkowski basis for R. We are interested in the asymptotic lengths of these vi’s for a family or orders with arbitrarily large discriminant D. By the theory of Minkowski bases we have that 1\leq |v2| \leq … \leq |vn| and \prod |v_i| \asymp |D|^{1/2} and by \cite{J}, we also know that |v_n| << |D|^{1/n}.

We say a family of orders in number fields have Minkowski type \delta_2,…,\delta_n if the members of the family have arbitrarily large discriminant and each have a Minkowski basis of the form v1=1,v2,…,vn with |vi| \asymp |D|^{\delta_i} for each i, where D is the discriminant.

In the thesis, we are interested in possible Minkowski types. The first question is: Can we find sufficient and necessary bounds on some rational numbers \delta_2,…,\delta_n such that there is a family of orders in number fields having Minkowski type \delta_2,…,\delta_n?

We already know the following necessary conditions: \delta_2 \leq … \leq \delta_n and \delta_2+…+\delta_n=1/2 by Minkowski basis theory, and \delta_n \leq 1/n by \cite{J}. We prove that bounds of the form \delta_k << \delta_i+\delta_j for each i+j=k are sufficient bounds, and if K has no non trivial subfield, we conjecture that these bounds are actually necessary. We can prove this in some cases (of n,i,j,k). In particular, for n=3,4,5,6, we prove that all these bounds are necessary.

The second question is: For some fixed \delta_2,…\delta_n, “how many” orders in number fields have Minkowski type \delta_2,…,\delta_n. We will make sense of what we mean by “how many” using the Delone-Faddeev correspondence (n=3), and the correspondence of Bhargava (n=4,5). Using these correspondences and counting, we are also able to give a sieving argument to count those orders that are maximal (and therefore are ring of integers of number fields).

A copy of the thesis can be found in this link: val_chichelapierre_thesis

]]>Thursday, June 13, 2019

11:10 a.m.

BA6183

PhD Candidate: Francis Bischoff

Supervisor: Marco Gualtieri

Thesis title: Morita Equivalence and Generalized Kahler Geometry

***

Generalized Kahler (GK) geometry is a generalization of Kahler geometry, which arises in the study of super-symmetric sigma models in physics. In this thesis, we solve the problem of determining the underlying degrees of freedom for the class of GK structures of symplectic type. This is achieved by giving a reformulation of the geometry whereby it is represented by a pair of holomorphic Poisson structures, a holomorphic symplectic Morita equivalence relating them, and a Lagrangian brane inside of the Morita equivalence.

We apply this reformulation to solve the longstanding problem of representing the metric of a GK structure in terms of a real-valued potential function. This generalizes the situation in Kahler geometry, where the metric can be expressed in terms of the partial derivatives of a function. This result relies on the fact that the metric of a GK structure corresponds to a Lagrangian brane, which can be represented via the method of generating functions. We then apply this result to give new constructions of GK structures, including examples on toric surfaces.

Next, we study the Picard group of a holomorphic Poisson structure, and explore its relationship to GK geometry. We then apply our results to the deformation theory of GK structures, and explain how a GK metric can be deformed by flowing the Lagrangian brane along a Hamiltonian vector field. Finally, we prove a normal form result, which says that locally, a GK structure of symplectic type is determined by a holomorphic Poisson structure and a time-dependent real-valued function, via a Hamiltonian flow construction.

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

]]>Tuesday, July 16, 2019

1:10 p.m.

BA6183

PhD Candidate: Evan Miller

Supervisor: Robert McCann

Thesis title: The Navier-Stokes strain equation with applications to enstrophy growth and global regularity

***

The resulting identity allows us to prove a new family of scale-critical necessary and sufficient conditions for blow-up of the solution in finite time $T_{max}<+\infty$, which depend only on the history of the positive part of the second eigenvalue of the strain matrix. Since this matrix is trace-free,

this severely restricts the geometry of any finite-time blow-up. This regularity criterion provides analytic evidence of the numerically observed tendency of the vorticity to align with the eigenvector corresponding to the middle eigenvalue of the strain matrix.

We then consider a vorticity approach to the question of almost two-dimensional initial data, using this same identity for enstrophy growth and an isometry relating the third column of the strain matrix to the first two components of the vorticity. We prove a new global regularity result for initial data with two components of the vorticity sufficiently small. Finally, we prove the existence and stability of blowup for a toy model ODE of the strain equation.

A copy of the thesis can be found: Miller_Thesis_Draft

]]>Wednesday, May 8, 2019

11:10 a.m.

BA6183

PhD Candidate: Ozgur Esentepe

Co-Supervisors: Joel Kamnitzer, Graham Leuschke

Thesis title: Annihilation of Cohomology over Gorenstein Rings

***

One of the fundamental links between geometry and homological algebra is that smooth affine schemes have coordinate rings of finite global dimension. The roots of this link goes back to Hilbert’s syzygy theorem and later to the work of Auslander and Buchsbaum and also of Serre.

Having finite global dimension can be characterized by Ext-modules. Namely, a ring $R$ has finite global dimension if and only if there is a natural number $n$ such that $\Ext_R^n(M,N) = 0$ for every pair $M,N$ of $R$-modules. Hence, in the singular case, there are nonzero Ext-modules for arbitrarily large $n$. So, for a commutative Noetherian ring $R$, one is interested in the cohomology annihilator ideal which consists of the ring elements that annihilate all $\Ext$-modules for arbitrarily large $n$.

The main theme of this thesis is to study the cohomology annihilator ideal over Gorenstein rings. Over Gorenstein rings, the cohomology annihilator ideal can be seen as the annihilator of the stable category of maximal Cohen-Macaulay modules.

The first main result concerns the cohomology annihilator ideal of a complete local coordinate ring of a reduced algebraic plane curve singularity. We show that that the cohomology annihilator ideal coincides with the conductor ideal in this case. We use this to investigate the relation between the Jacobian ideal and the cohomology annihilator ideal.

The second main result shows that if the Krull dimension of $R$ is at most $2$, then the cohomology annihilator ideal is equal to the stable annihilator ideal of a non-singular $R$-order. We also give several generalizations of this which brings us to the second part and the closing section of this thesis. Namely, we study the dominant dimension of orders over Cohen-Macaulay rings. We provide examples and prove results on tilting modules for orders with positive dominant dimension.

A copy of the thesis can be found: ozgurthesis-1

]]>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

]]>Tuesday, June 18, 2019

2:10 p.m.

BA6183

PhD Candidate: Zhifei Zhu

Supervisor: Regina Rotman

Thesis title: Geometric inequalities on Riemannian manifolds

***

In this thesis, we will show three results which partially answer several questions in the field of quantitative geometry. We first show that there exists Riemannian metric on a 3-disk so that the diameter, volume and surface area of the boundary is bounded, but during any contraction of the boundary of the disk, there exists a surface with arbitrarily large surface area. This result answers a question of P. Papasoglu.

The second result we will prove is that on any closed 4-dimensional simply-connected Riemannian manifolds with diameter <=D, volume > v > 0 and Ricci curvature |Ric| < 3, the length of a shortest closed geodesic can be bounded by some function f(v,D) which only depends on the volume and diameter of the manifold. This result partially answers a question of M. Gromov.

As an extension of our second result, we show that on 4-dimensional Riemannian manifolds satisfying the above conditions, the first homological filling function HF_1(l) <=f_1 (v,D)l +f_2 (v,D), for some functions f_1 and f_2 which only depends on v and D. And in particular, the area of a smallest minimal surface on the manifold can be bounded by some function which only depends on v and D.

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

]]>Friday, April 26, 2019

11:10 a.m.

BA6183

PhD Candidate: Leonid Monin

Supervisor: Askold Khovanskii

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

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

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