Graduate Blog

Zoom Web Conference

PhD Candidate: Gaurav Patil
Supervisor: V. Kumar Murty
Thesis title:

***

***

The draft of the thesis can be found here:

Zoom Web Conference

PhD Candidate: Eva Politou
Supervisor: Stefanos Aretakis
Thesis title:

***

***

The draft of the thesis can be found here:

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

***

***

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

****

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.

****

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

****

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

****

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

****

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.

***

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)

***

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.

***

The draft of the thesis can be found here: Janisse_Thad_date_PHD_thesis

Zoom Web Conference

PhD Candidate: Yucong Jiang
Supervisor: Marco Gualtieri
Thesis title:

****

***

The draft of the thesis can be found here:

Wednesday, June 28, 2023
10:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Surya Raghavendran
Supervisor: Kevin Costello
Thesis title:  Twisted eleven-dimensional supergravity and exceptional
simple infinite dimensional super-Lie algebras

****

***

The draft of the thesis can be found here: main

Thursday, May 18, 2023 at 10:00 a.m. (sharp)

Zoom Web Conference

PhD Candidate: Hyungseop Kim
Supervisor: Michael Groechenig
Thesis title: Descent techniques in algebraic K-theory

****

We investigate two different approaches to describing algebraic K-theory of schemes through descent techniques, one of global nature and the other of local nature.

The first half of the thesis is devoted to the study of an adelic descent statement for algebraic K-theory of Noetherian schemes, or more generally for any localizing invariants in place of algebraic K-theory. Given a Noetherian scheme $X$ of finite Krull dimension, Beilinson’s cosemisimplicial ring $A^{\bullet}_{\mathrm{red}}(X)$ of reduced adeles on $X$ provides a resolution of the structure sheaf of $X$. We prove that for any localizing invariant $E$ of small stable $\infty$-categories, e.g., nonconnective algebraic K-theory of Bass-Thomason, there is a natural equivalence $E(X)\simeq\lim_{\Delta_{s}}E(A^{\bullet}_{\mathrm{red}}(X))$. This can be viewed as a variant of the formal glueing problem for algebraic K-theory which concerns all irreducible closed subsets at once. We prove the descent statement by first converting the question to a cubical descent statement, and then constructing exact sequences of perfect module categories over adele rings.

In the second half of the thesis, we turn our attention to the study of $p$-adic K-theory of characteristic $p$ rings. Specifically, we provide an alternative proof of Kelly-Morrow’s generalization of Geisser-Levine theorem to the Cartier smooth case. Our approach puts emphasis on utilizing motivic filtration and descent spectral sequence. Using the homological smoothness of Cartier smooth rings, we first compute their prismatic cohomology and syntomic cohomology complexes. Through motivic filtration, this computation gives a description of topological cyclic homology for Cartier smooth rings. Then, we use the pro-\’etale descent spectral sequence for topological cyclic homology and rigidity properties of the cyclotomic trace and syntomic cohomology complexes to deduce the result, computing algebraic K-theory of local Cartier smooth rings in terms of their logarithmic de Rham-Witt groups. We also collect some direct consequences of our arguments to prismatic cohomology complexes of Cartier smooth rings and their $p$-torsion free liftings to mixed characteristic.

****

The draft of the thesis can be found here: main