Graduate Blog

Wednesday, August 9, 2023
1:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Sina Zabanfahm
Co-Supervisors: Michael Groechenig/Lisa Jeffrey
Thesis title: Cluster pictures for Hitchin fibers of rank two Higgs bundles

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Let φ: X → Y be a degree two Galois cover of smooth curves over a local field F, where F has odd residual characteristic. Assuming that Y has good reduction, we describe a semi-stability criterion for the curve X, using the data of the branch locus of the covering φ. In the case that X has semi-stable reduction, we describe the dual graph of the minimal regular model of X over F. We do this by adopting the notion of the cluster picture defined for hyperelliptic curves for the case where Y is not necessarily a rational curve. Using these results, we describe the variation of the p-adic volume of Hitchin fibers over the semi-stable locus of the moduli space
of rank 2 twisted Higgs bundles.

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The draft of the thesis can be found here: SZabanfahm-Thesis

Wednesday, August 23, 2023
10:00 a.m. (sharp)

BA6183/ Zoom Web Conference

PhD Candidate: Peter Angelinos
Supervisor: Michael Groechenig
Thesis title: p-adic Integration for Derived Equivalent Abstract Hitchin Systems

PhD Defense – Angelinos

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Let ˆM denote the moduli space of SLn-Higgs bundles and let ˇM denote the moduli space of PGLn-Higgs bundles. Under the appropriate conditions and equipped with some additional data, ˆM and ˇM form a dual pair of abstract Hitchin systems in the sense of [GWZ20b]. The authors make the superfluous assumption in loc. cit. that the stacks defining the abstract Hitchin systems are quotient stacks. This is sufficient for their purposes as their intended application is resolving the topological mirror symmetry conjecture of Hausel-Thaddeus in [HT03]. In particular, they prove that the Hodge numbers of ˆM, which is a smooth variety, are equal to the stringy Hodge numbers of ˇM, which is a smooth orbifold.

In this thesis, the notion of abstract Hitchin systems is extended to certain tame Deligne-Mumford stacks, subject to some conditions. This allows us to remove the assumption that the abstract Hitchin systems are Zariski-locally quotient stacks. This is achieved via passing to an étale cover on which the stack is a quotient stack, using an original lemma that says we can choose étale neighbourhoods of points such that the residue field remains unchanged. This allows us to apply the methods of p-adic integration as in [GWZ20b] and [GWZ20a].

In [HT03], it is shown that the spaces ˆM and ˇM are generically fibred over the Hitchin base in Lagrangian tori. Furthermore, each space is equipped with a gerbe, and are dual in the sense that the fibres are torsors over an abelian variety and are isomorphic as torsors to the equivalence classes of trivializations of the gerbe on the dual fibre. A similar algebro-geometric version of this duality is formulated for abstract Hitchin systems in [GWZ20b] and it is shown that topological mirror symmetry follows from an arithmetic version proven via p-adic integration. We reformulate this duality condition as a twisted derived equivalence of abstract Hitchin systems equipped with certain gerbes and recover arithmetic mirror symmetry via p-adic integration in this context.

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The draft of the thesis can be found here: angelinos-thesis-draft

Wednesday, June 14, 2023
3:00 p.m. (sharp)

Zoom Web Conference/BA6183

PhD Candidate: Gaurav Patil
Supervisor: V. Kumar Murty
Thesis title: Rings of finite rank over integers

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The draft of the thesis can be found here: Thesis_GauravPatil

Wednesday, July 5, 2023
10:00 am. (sharp)

Zoom Web Conference

PhD Candidate: Eva Politou
Supervisor: Stefanos Aretakis
Thesis title: A Geometric Framework for Conservation Laws Along Null
Hypersurfaces and their Relation to Huygens’ Principle

PhD Defense – Politou

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In the present thesis we examine two main topics. In the first part, we use the general theory of local conservation laws for arbitrary partial differential equations to provide a geometric framework for conservation laws on characteristic null hypersurfaces. The operator of interest is the wave operator on general four-dimensional Lorentzian manifolds restricted on a null hypersurface. In the second part of the thesis, we investigate relations between the geometric conditions that lead to the validity of Huygens’ principle and those that give rise to conservation laws along null hypersurfaces for the wave operator. We apply our results in spacetimes such as the Minkowski, Schwarzschild, and Reissner-Nordström, as well as in general spherically symmetric spacetimes.

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The draft of the thesis can be found here: Eva’s Thesis

Tuesday, June 27, 2023
11:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Pavel Shlykov
Supervisor: Alexander Braverman
Thesis title: Certain cases of Hikita-Nakajima conjecture

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Let M0 be an affine Nakajima quiver variety, and M is the corresponding BFN Coulomb branch. Assume that M0 can be resolved by the (smooth) Nakajima quiver variety M. The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras H∗S (M, C) ≃ C[MC×s], where S ↷ M0 is a torus acting on M0 preserving the Poisson structure, Ms is the (Poisson) deformation of M over s = Lie S, C× is a generic one-dimensional torus acting on M, and C[MC×s] is the algebra of schematic C×-fixed points of Ms. In this thesis we prove the Hikita-Nakajima conjecture for M = C^2/Γ (Kleinian singularities) and M = M(n,r) Gieseker variety (ADHM space). In the latter case we produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of Ms as the spectrum of the center of rational Cherednik algebra corresponding to Sn ⋉ (Z/rZ) n and identify all the algebras that appear in the isomorphism with the center of degenerate cyclotomic Hecke algebra.

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The draft of the thesis can be found here: thesis_template_shlykov

Wednesday, June 28, 2023
3:30 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Faisal Al-Faisal
Supervisor: Steve Kudla
Thesis title: An arithmetic-geometric reciprocity between theta functions
attached to real and imaginary quadratic fields

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We use the theta correspondence to construct classical holomorphic modular forms associated to ideal classes in quadratic number fields. These modular forms are theta functions that were originally introduced by Hecke in the 1920s and have been investigated by several authors since. Our framework allows us to prove old and new results concerning the periods of these modular forms over certain geometric cycles defined by arithmetic data. In particular, we establish a reciprocity relationship between the periods of theta functions attached to ideal classes in real and imaginary quadratic fields. This provides an analogue of (and context for) Hecke’s discovery that certain periods of his imaginary quadratic theta functions are special values of classical Eisenstein series at CM points.

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The draft of the thesis can be found here: alfaisal_thesis

Wednesday, May 31, 2023
2:00 p.m. (sharp)

Zoom Web Conference/BA6183

PhD Candidate: Samprit Ghosh
Supervisor: V. Kumar Murty
Thesis title: Higher Euler-Kronecker constants

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The coefficients that appear in the Laurent series of Dedekind zeta functions and their logarithmic derivatives are mysterious and seem to contain a lot of arithmetic information. Although the residue and the constant term have been widely studied, not much is known about the higher coefficients. In this thesis, we study these coefficients $\gamma_{K,n}$ that appear in the Laurent series expansion of $\frac{\zeta_K'(s)}{\zeta_K (s)}$ about $s=1$, where $K$ is a global field. For example, when $K$ is a number field, we prove, under GRH, $$\gamma_{K,n} \ll (\log (\log(|d_K|))^{n+1}$$

$d_K$ being the absolute discriminant  of $K$.

Analogous bounds for the function field case are also shown. We prove (unconditionally) interesting arithmetic formulas satisfied by these constants.

We also study the distribution of values of higher derivatives of $\mathcal{L}(s,\chi)= L'(s, \chi)/L(s, \chi)$ at $s=1$ and $\chi$ ranges  over all non-trivial Dirichlet characters with a given large prime conductor $m$. In particular, we compute moments, i.e. the average of $P^{(a,b)}(\mathcal{L}^{(n)}(1, \chi))$, where $P^{(a,b)}(z) = z^a \overline{z}^b$ and study their asymptotic behaviour as $m \rightarrow \infty$. We then construct a density function $M_{\sigma}(z)$,  for $\sigma= $ Re$(s)$ and show that for Re$(s) > 1$

$$\text{Avg}_{\chi} \Phi(\mathcal{L}'(s, \chi)) = \int_{C} M_{\sigma}(z) \Phi(z) |dz| $$ holds for any continuous function $\Phi$ on $C$.

Thesis_poster_Samprit

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The draft of the thesis can be found here:  Thesis_Samprit_Ghosh

Wednesday, June 7, 2023
10:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Heejong Lee
Supervisor: Florian Herzig
Thesis title: Emerton–Gee stacks, Serre weights, and Breuil–Mézard conjectures for GSp4

PhD Defense – Lee

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We construct a moduli stack of rank 4 symplectic projective etale ´ (ϕ, Γ)-modules and prove its geometric
properties for any prime p > 2 and finite extension K/Qp. When K/Qp is unramified, we adapt the theory of local models recently developed by Le–Le Hung–Levin–Morra to study the geometry of potentially crystalline substacks in this stack. In particular, we prove the unibranch property at torus fixed points of local models and deduce that tamely potentially crystalline deformation rings are domain under genericity conditions. As applications, we prove, under appropriate genericity conditions, an GSp4-analogue of the Breuil–Mezard conjecture for tamely potentially crystalline deformation rings, the weight part of Serre’s conjecture formulated by Gee–Herzig–Savitt for global Galois representations valued in GSp4 satisfying Taylor–Wiles conditions, and a modularity lifting result for tamely potentially crystalline representations.

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The draft of the thesis can be found here: Heejong_Lee_thesis

Monday, May 1, 2023 at 1:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Thaddeus Janisse
Supervisor: Joe Repka
Thesis title: The Real Subalgebras of so_4(C) and G_2(2)

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Classifying the subalgebras of a simple Lie algebra is a pursuit that stretches back to the work of Cartan on representations of simple Lie algebras. Mal’cev, in classifying orthogonal and symplectic representations of simple Lie algebras also found the semisimple subalgebras of $B_n, C_n,$ and $D_n$. Following that, Dynkin and Minchenko classified the semisimple subalgebras of the complex exceptional Lie algebras.

We investigate the real subalgebras of a number of rank 2 Lie algebras: $\mathfrak{so}_4(\mathbb{C})$, its real forms, and the split real form of $G_2$, $G_{2(2)}$. In this thesis, we classify the real subalgebras of these Lie algebras up to inner automorphism (i.e., up to the adjoint action of the corresponding Lie group). For the matrix algebras above, we largely proceed with the help of copious amounts of linear algebra. For $G_{2(2)}$, we take advantage of the Cartan decomposition $G_{2(2)} = \mathfrak{k} \oplus \mathfrak{p}$, where $\mathfrak{k}$ is a compact subalgebra, to identify the semisimple and Levi-decomposable subalgebras of $G_{2(2)}$. To find the solvable subalgebras, we use the classifications of semisimple and nilpotent elements of $G_{2(2)}$, as well as our own classification of Jordan elements, to build nilpotent and solvable subalgebras.

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The draft of the thesis can be found here: Janisse_Thad_date_PHD_thesis

Friday, July 14, 2023
2:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Yucong Jiang
Supervisor: Marco Gualtieri
Thesis title: Integration of generalized Kähler structures

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Generalized Kähler (GK) geometry was first discovered in 1984 by Gates, Hull, and Roček in their study of N = (2,2) supersymmetric σ-models, which extended Zumino’s work on the relationship between Kähler geometry and N = (2, 2) supersymmetry. In this thesis we develop a new approach to the field of generalized Kähler (GK) geometry by addressing the integration problem of GK structures.

To tackle this problem, we first rephrase the definition of GK structures in terms of holomorphic Manin triples. In this way, we have discovered an intimate connection between GK geometry and double structures invented by Ehresmann and further developed by Mackenzie in the fields of Poisson geometry and Lie theory. We introduce and develop the concept of holomorphic Morita equivalence of symplectic double groupoids. This notion allows us to access the underlying holomorphic structures associated with GK structures. Additionally, we propose the concept of multiplicative Lagrangian branes as a mean to access the underlying smooth data, such as GK metrics.

We then employ techniques from Poisson Geometry, such as gauge transformations, and IM 2-forms to solve the integration problem and obtain a reconstruction theorem. As applications, we provide a definition of generalized Kähler classes and present a Hamiltonian flow construction of GK metrics.

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The draft of the thesis can be found here: thesis