All Seminars
| Title: Reflections and Perspectives: a look back at my teaching journey |
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| Seminar: Teaching |
| Speaker: Maryam Khaqan (she/her/hers) of Department of Mathematics, University of Toronto |
| Contact: Bree Ettinger, bree.d.ettinger@emory.edu |
| Date: 2024-11-01 at 10:00AM |
| Venue: MSC W307C |
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| Abstract: As a graduate student at Emory, I taught 6 semesters of Calculus 1 and 2 and one semester of Linear Algebra. I taught independently, as a part of a small loosely coordinated team, and as part of larger, more tightly coordinated teams of instructors with shared assessments and shared syllabi.\\ My teaching journey, just like many of yours, began with MATH590, and since then, I've taught both graduate and undergraduate courses on a small and large scale. Currently, I am a teaching postdoc at the University of Toronto, where I am co-coordinating a team of 3 instructors and 8 TAs as well as teaching my own course with ~140 students.\\ \\ Each new role has brought with itself its own unique challenges, opportunities for growth, and shifts in perspective. In this talk, I will share various stories from my teaching journey with an aim to demonstrate what is "out there" i.e., what kinds of teaching roles a MATH590 student might pursue after Emory and what skills they might learn along the way. |
| Title: Finite Field Fourier Transforms in Arithmetic Statistics |
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| Algebra and Number Theory: Algebra |
| Speaker: Frank Thorne of University of South Carolina |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-10-08 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: In many arithmetic statistics problems, it is useful to evaluate or bound certain Fourier transforms over finite fields. I will give an overview of (1) how these Fourier transforms arise, (2) some strategies that my collaborators and I (and others!) have developed to analyze them, and (3) some surprising structures one finds.\\ \\ Some of this work is older, but I will focus on forthcoming work with Anderson and Bhargava in Bhargava's averaging method, and recently finished work with Ishitsuka, Taniguchi, and Xiao on binary quartic forms. |
| Title: Strong $u$-invariant and Period-Index bound |
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| Seminar: Algebra and Number Theory |
| Speaker: Shilpi Mandal of Emory University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-10-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: For a central simple algebra $A$ over $K$, there are two major invariants, viz., \textit{period} and \textit{index}.\\ \\ For a field $K$, define the \emph{Brauer $l$-dimension of $K$} for a prime number $l$, denoted by $\mathrm{Br}_l\mathrm{dim}(K)$, as the smallest $d \in \mathbb{N} \cup \{\infty\}$ such that for every finite field extension $L/K$ and every central simple $L$-algebra $A$ of period a power of $l$, we have that $\mathrm{ind}(A)$ divides $\mathrm{per}(A)^d$.\\ \\ If $K$ is a number field or a local field (a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$, for some prime number $p$), then classical results from class field theory tell us that $\mathrm{Br}_l\mathrm{dim}(K) = 1$. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Lieblich, Harbater-Hartmann-Krashen for $K$ a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields $K$, Parimala-Suresh have given some bounds.\\ \\ Also, the $u$-invariant of $K$, denoted by $u(K)$, is the maximal dimension of anisotropic quadratic forms over $K$. For example, $u(\mathbb{C}) = 1$; for $F$ a non-real global or local field, we have $u(F) = 1, 2, 4,$ or $8$, etc.\\ \\ In this talk, I will present similar bounds for the $\mathrm{Br}_l\mathrm{dim}$ and the strong $u$-invariant of a complete non-Archimedean valued field $K$ with residue field $\kappa$. |
| Title: Average congruence class biases in the cyclicity and Koblitz conjectures |
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| Seminar: Algebra |
| Speaker: Jacob Mayle of Wake Forest University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-09-24 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Given an elliptic curve over the rationals, it is natural to ask about the distribution of primes p for which the reduction of E modulo p has certain properties. Two well-known problems of this type are the cyclicity and Koblitz problems, which ask about the primes of cyclic and prime-order reduction, respectively. In this talk, we will discuss a recent joint work with Sung Min Lee and Tian Wang in which we consider variants of these problems for primes in arithmetic progression. In particular, we will highlight a somewhat counterintuitive phenomenon: on average, primes of cyclic reduction are oppositely biased to primes of prime-order reduction over congruence classes. |
| Title: Minimal Torsion Curves in Geometric Isogeny Classes |
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| Seminar: Algebra |
| Speaker: Abbey Bourdon of Wake Forest University |
| Contact: Santiago Arango-Piñeros, santiago.arango@emory.edu |
| Date: 2024-09-17 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Let $E/\mathbb{Q}$ be a non-CM elliptic curve and let $\mathcal{E}$ denote the collection of all elliptic curves geometrically isogenous to $E$. That is, for every $E' \in \mathcal{E}$, there exists an isogeny $\varphi: E \rightarrow E'$ defined over $\overline{\mathbb{Q}}$. We will discuss the problem of identifying minimal torsion curves in $\mathcal{E}$, which are elliptic curves $E' \in \mathcal{E}$ attaining a point of prime-power order in least possible degree. Using recent classification results of Rouse, Sutherland, and Zureick-Brown, we obtain an answer to this question in many cases, including a complete characterization for points of odd degree.\\ \\ This is joint work with Nina Ryalls and Lori Watson. |
| Title: Admissible Groups Over Number Fields |
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| Seminar: Algebra |
| Speaker: Deependra Singh, PhD of Emory University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2024-09-10 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Given a field \( K \), one can ask which finite groups \( G \) are Galois groups of field extensions \( L/K \) such that \( L \) is a maximal subfield of a division algebra with center \( K \). Such a group \( G \) is called \emph{admissible} over \( K \). Like the inverse Galois problem, the question remains open in general. But unlike the inverse Galois problem, the groups that occur in this fashion are generally quite restricted. In this talk, I will discuss some results and open problems about groups that are admissible over number fields. |
| Title: Diophantine tuples over integers and finite fields |
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| Seminar: Combinatorics |
| Speaker: Kyle Yip, PhD of Georgia Institute of Technology |
| Contact: Dr. Cosmin Pohoata, apohoat@emory.edu |
| Date: 2024-09-06 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: A set $\{a_{1}, a_{2},\ldots, a_{m}\}$ of distinct positive integers is a Diophantine $m$-tuple if the product of any two distinct elements in the set is one less than a square. There is a long history and extensive literature on the study of Diophantine tuples and their generalizations in various settings. In this talk, we focus on the following generalization: for each $n \ge 1$ and $k \ge 2$, we call a set of positive integers a Diophantine tuple with property $D_{k}(n)$ if the product of any two distinct elements is $n$ less than a $k$-th power, and we denote $M_k(n)$ be the largest size of a Diophantine tuple with property $D_{k}(n)$. In this talk, I will present improved upper bounds on $M_k(n)$. I will also discuss the analogue of Diophantine tuples over finite fields, which is of independent interest. Joint work with Seoyoung Kim and Semin Yoo. |
| Title: Topics in Abelian Varieties: Canonical Rings of Stacks |
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| Defense: Dissertation |
| Speaker: Michael Cerchia of Emory University |
| Contact: Michael Cerchia, michael.cerchia@emory.edu |
| Date: 2024-06-28 at 10:00AM |
| Venue: MSC W303 |
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| Abstract: We investigate two of the problems from my thesis "Topics in Abelian Varieties". The flavor of these problems and the techniques used to solve them vary, but a common theme is the use of geometric techniques (and in particular moduli theory) to solve concrete questions from arithmetic. The two problems we will focus on today involve section rings of algebraic varieties, which are classical objects of study and play a central role in the minimal model program. In the first of these problems, we describe the section ring of elliptic curves for arbitrary divisors, and we give a complete description when the underlying divisor is supported by up to two points. In the second, we investigate canonical rings of moduli stacks of principally polarized abelian varieties, with particular focus on the $g=2$ case. These have additional arithmetic significance: the canonical ring of modular curves, when equipped with the structure of an algebraic stack, gives rise to rings of modular forms. By considering higher dimensional analogues, we can determine explicit presentations for rings of Siegel modular forms. |
| Title: Erdos-Rogers Functions |
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| Seminar: Combinatorics |
| Speaker: Jacques Verstraete of University of California San Diego |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2024-04-26 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: The Erdos-Rogers functions are generalizations of Ramsey numbers, introduced around fifty years ago. The general question given graphs $F$ and $H$ is to determine the maximum number of vertices $f(n,F,H)$ in an $F$-free induced subgraph of any $H$-free $n$-vertex graph. The case $F = K_2$ is equivalent to determining Ramsey numbers $r(H,t)$. The case $F$ and $H$ are cliques has received considerable attention. In this talk we give almost tight bounds, showing that for $s > 3$, $$ f(n,K_s,K_{s-1}) = \sqrt{n}(\log n)^{\Theta(1)} $$ where the exponent of the logarithm is between $1/2 - o(1)$ and $1 + o(1)$. We also give new bounds on Ramsey numbers $r(F,t)$. In part joint work with David Conlon, Sam Mattheus and Dhruv Mubayi. |
| Title: Bayesian Modeling and Computation for Structural and Functional Neuroimaging |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Andrew Brown of Clemson University |
| Contact: Deepanshu Verma and Julianne Chung, deepanshu.verma@emory.edu |
| Date: 2024-04-25 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: Since its advent about 30 years ago, magnetic resonance imaging (MRI) has revolutionized medical imaging due to its ability to produce high-contrast images non-invasively without the use of radiation or injection. In neuroimaging in particular, MRI has become a very popular and useful tool both in clinical settings (e.g., in vivo measurements of anatomical structures) as well as psychology (e.g., studying neuronal activations over time in response to an external stimulus). Despite the applicability and history of MR-based neuroimaging, however, considerable challenges remain in the analysis of the associated data. In this talk, I will discuss two recent projects in which collaborators and I use fully Bayesian statistical modeling to draw inference about both brain structure and brain function. The former work illustrates how prior information can be used to improve our ability to delineate the hippocampus in patients with Alzheimer’s disease. The latter work discusses an approach that makes use of the full complex-valued data produced by an MR scanner to improve our ability to not only identify task-related activation in functional MRI, but to differentiate between types of activation that might carry different biological meaning. Along the way, I will mention some computational techniques we employ to facilitate Markov chain Monte Carlo (MCMC) algorithms to approximate the posterior distributions of interest. |