All Seminars
| 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. |
| Title: Some Problems at the Intersection of Algebra, Geometry, and Optimization |
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| Type: Dissertation Proposal |
| Speaker: Alex Dunbar of |
| Contact: Alex Dunbar, |
| Date: 2024-04-24 at 9:30AM |
| Venue: MSC E406 |
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| Abstract: |
| Title: Reduced Unitary Whitehead Groups over Function Fields of $p$-adic Curves |
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| Defense: Dissertation |
| Speaker: Zitong Pei of Emory Unviersity |
| Contact: Zitong Pei, zitong.pei@emory.edu |
| Date: 2024-04-22 at 11:00AM |
| Venue: MSC E406 |
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| Abstract: The study of the Whitehead group of semi-simple simply connected groups is classical with an abundance of new open questions concerning the triviality of these groups. The Kneser-Tits conjecture on the triviality of these groups was answered in the negative by Platanov for general fields. There is a relation between reduced Whitehead groups and $R$-equivalence classes in algebraic groups.\\ \\ Let $G$ be an algebraic group over a field $F$. The $R$-equivalence, defined by Manin, is the equivalence relation on $G(F)$ defined by $x\sim y$ for $x, y \in G(F)$ if there exists a $F$-rational morphism ${\mathbb A}^1_K \cdots \to G$ defined at $0$ and $1$ and sending 0 to $x$ and 1 to $y$. Let $RG(F)$ be the equivalence class of the identity element in $G(F)$. Then $RG(F)$ is a normal subgroup of $G(F)$ and the quotient $G(F)/RG(F)$ is called the group of $R$-equivalence classes of $G(F)$. It is well known that for the semi-simple simply connected isotropic group $G$ over $F$, $W(G, F)$ is isomorphic to the group of $R$-equivalence classes. Thus the group of $R$-equivalence classes can be thought as Whitehead groups for general algebraic groups. The group of $R$-equivalence classes, is very useful while studying the rationality problem for algebraic groups, the problem to determine whether the variety of an algebraic group is rational or stably rational.\\ \\ Suppose that $D_0$ is a central division $F_0$-algebra. If the group $G(F_0)$ of rational points is given by $SL_n(D)$ for some $n>1$, then $W(G, F_0) $ is the reduced Whitehead group of $D_0$. Let $F$ be a quadratic field extension of $F_0$ and $D$ be a central division $F$-algebra. Suppose that $D$ has an involution of second kind $\tau$ such that $F^{\tau}=F_0$. If the hermitian form $h_{\tau}$ induced by $\tau$ is isotropic and the group $G(F_0)$ is given by $SU(h_\tau, D)$, then $W(G, F_0)$ is isomorphic to the reduced unitary Whitehead group of $D.$\\ \\ We start from the fundamental facts on reduced unitary Whitehead groups of central simple algebras, then introduce the patching techniques. Finally, let $F/F_0$ be a quadratic field extension of the function field of a $p$-adic curve. Let $A$ be a central simple algebra over $F$. Assume that the period of $A$ is two and $A$ has a unitary $F/F_0$ involution. We provide a proof for the triviality of the reduced unitary Whitehead group of $A$. |
| Title: Quantifying the geometry of immune response and infection |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Manuchehr Aminian of Cal Poly Pomona |
| Contact: Manuela Girotti, manuela.girotti@emory.edu |
| Date: 2024-04-18 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: In improving outcomes for infection in humans and animals, it is important to understand how the body responds to an infection, whether infection has happened at all, and how this varies from individual to individual. Traditionally, this is a simple measurement -- does someone have a fever or not? With more precise, high-frequency measurements of macro-scale data (e.g. body temperature time series) and micro-scale data (e.g. protein or RNA data from biological samples, i.e. "omics"), we can develop and study the efficacy of more sophisticated algorithms and diagnostics. I will present past and ongoing work in applying ideas from geometrical data analysis and machine learning which aid us in making predictions in classification questions such as early prediction of infection, model-free learning of time series patterns and anomaly detection, and "inverse" problems such as prediction of time since infection. We will introduce algorithmic ideas to newcomers as well as our quantitative results on data coming from clinical studies with humans challenged with influenza-like illnesses, and Collaborative Cross mice studies, in work with our collaborators at Colorado State University and Texas A&M University. |