All Seminars
| Title: Ramsey and density results for approximate arithmetic progressions. |
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| Seminar: Combinatorics |
| Speaker: Marcelo Sales of UC Irvine |
| Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2024-02-23 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: Let AP_k={a,a+d,\ldots,a+(k-1)d} be an arithmetic progression of length k. For a given epsilon>0, we call a set AP_k(epsilon)={x_0,…,x_{k-1}} an epsilon-approximate arithmetic progression of lenght k for some a and d, if the inequality |x_i-(a+id)|<\epsilon d holds for all i in {0,1,...,k-1}. In this talk we discuss numerical aspects of Van der Waerden and Szemeredi type of results in which arithmetic progressions are replaced by their epsilon-approximation. Joint work with Vojtech Rodl. |
| Title: On the evolution of structure in triangle-free graphs |
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| Seminar: Discrete Analysis |
| Speaker: Will Perkins of Georgia Tech |
| Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2024-02-19 at 5:30PM |
| Venue: MSC W301 |
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| Abstract: Erdos-Kleitman-Rothschild proved that the number of triangle-free graphs on n vertices is asymptotic to the number of bipartite graphs; or in other words, a typical triangle-free graph is a random subgraph of a nearly balanced complete bipartite graph. Osthus-Promel-Taraz extended this result to much lower densities: when m >(\sqrt{3}/4 +eps) n^{3/2} \sqrt{\log n}, a typical triangle-free graph with m edges is a random subgraph of size m from a nearly balanced complete bipartite graph (and this no longer holds below this threshold). What do typical triangle-free graphs at sparser densities look like and how many of them are there? We consider what we call the "ordered" regime, in which typical triangle-free graphs are not bipartite but do align closely with a nearly balanced bipartition. In this regime we prove asymptotic formulas for the number of triangle-free graphs and give a precise probabilistic description of their structure. Joint work with Matthew Jenssen and Aditya Potukuchi. |
| Title: Canonical colourings in random graphs |
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| Seminar: Combinatorics |
| Speaker: Mathias Schact of University of Hamburg |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2024-02-16 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: Rodl and Rucinski established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_\ell$. We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erd\H{o}s and Rado showed that \textit{canonically coloured} copies of~$K_\ell$ can be ensured in the deterministic setting. We transfer the Erd\H os--Rado theorem to the random environment and show that for $\ell\geq 4$ both thresholds coincide. As a consequence the proof yields $K_{\ell+1}$-free graphs~$G$ for which every edge colouring yields a canonically coloured $K_\ell$. This is joint work with Nina Kamev. |
| Title: A shifted convolution problem arising from physics |
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| Seminar: Algebra |
| Speaker: Kim Klinger-Logan of Kansas State University/Rutgers University |
| Contact: Andrew Kobin, andrew.jon.kobin@emory.edu |
| Date: 2023-12-12 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Physicists Green, Russo, and Vanhove have discovered solution to differential equations involving automorphic forms appear at the coefficients to the $4$-graviton scattering amplitude in type IIB string theory. Specifically, for $\Delta$ the Laplace-Beltrami operator and $E_s(g)$ a Langlands Eisenstein series, solutions $f(g)$ of $(\Delta-\lambda) f(g) = E_a(g) E_b(g)$ for $a$ and $b$ half-integers on certain moduli spaces $G(Z)\backslash G(R)/K(R)$ of real Lie groups appear as coefficients to the analytic expansion of the scattering amplitude. We will briefly discuss different approaches to finding solutions to such equations and focus on a shifted convolution sum of divisor functions which appears as the Fourier modes associated to the homogeneous part of the solution. Initially, it was thought that, when summing over all Fourier modes, the homogeneous solution would vanish but recently we have found an exciting error term. This is joint work with Stephen D. Miller, Danylo Radchenko, and Ksenia Fedosova. |
| Title: Predicting Complex Spatiotemporal Cardiac Voltage Dynamics Using Reservoir Computing |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Elizabeth Cherry of Georgia Tech |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-12-05 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: Disruptions to the electrical behavior of the heart caused by cardiac arrhythmias can result in complex dynamics, from period-2 rhythms in single cells to spatiotemporally complex spiral and scroll waves of electrical activity, which can inhibit contraction and may be lethal if untreated. Accurate forecasts of cardiac voltage behavior could allow new opportunities for intervention and control, but predicting complex nonlinear time series is a challenging task. In this talk, we discuss our recent work using machine-learning approaches based on reservoir computing to forecast cardiac voltage dynamics. First, we show that a novel method combining an echo state network with automated feature extraction via an autoencoder can successfully and efficiently predict time series of synthetic and experimental datasets of cardiac voltage in one cell with 20-30 action potentials in advance. Building on this work, we then demonstrate a novel method for predicting the complex spatiotemporal electrical dynamics of cardiac tissue using an echo state network integrated with a convolutional autoencoder. We show that our approach can forecast complex spiral-wave behavior, including breakup several periods in advance for time series ranging from model-derived synthetic datasets to optical-mapping recordings of explanted human hearts. |
| Title: On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields |
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| Seminar: Algebra |
| Speaker: Sun Woo Park of University of Wisconsin |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-12-05 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $\mu_p$. Based on the works by Swinnerton-Dyer, Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erd\"os-Kac theorem, and the geometric ergodicity of Markov chains. |
| Title: What in the structure of data make them learnable? |
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| Seminar: Algebra |
| Speaker: Matthieu Wyart of EPFL |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-12-04 at 11:30AM |
| Venue: N215 |
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| Abstract: Deep learning algorithms have achieved remarkable successes, yet why they work is unclear. Notably, they can learn many high-dimensional tasks, a feat generically infeasible due to the so-called curse of dimensionality. What is the structure of data that makes them learnable, and how this structure is exploited by deep neural networks, is a central question of the field. In the absence of an answer, relevant quantities such as the number of training data needed to learn a given task -the sample complexity- cannot be determined. I will show how deep neural networks trained with gradient descent can beat the curse of dimensionality when the task is hierarchically compositional, by building a good representation of the data that effectively lowers the dimension of the problem. This analysis also reveals how the sample complexity is affected by the hierarchical nature of the task. If time permits, I will also discuss how the fact that regions in the data containing information on the task can be sparse affects sample complexity. |
| Title: Perfect Cuboids and Magic Squares of Squares |
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| Colloquium: Algebra |
| Speaker: Tony Várilly-Alvarado of Rice University |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-12-04 at 3:00PM |
| Venue: Atwood 240 |
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| Abstract: A perfect cuboid is a box such that the distance between any two corners is a positive integer. A magic square is a grid filled with distinct positive integers, whose rows, columns, and diagonals add up to the same number. To date, we don't know if there exists a perfect cuboid, or a 3 x 3 magic square whose entries are distinct squares. What do these problems have in common? Secretly, they are both problems about rational points on algebraic surfaces of general type with mild singularities. I believe there is no such thing as a perfect cuboid or a 3 x 3 magic square of squares, and I will try to convince you that geometry suggests this is so. |
| Title: Ascending subgraph decompositions |
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| Seminar: Combinatorics |
| Speaker: Alexey Pokrovskiy of University College London |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-11-29 at 4:00PM |
| Venue: Atwood 240 |
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| Abstract: A graph G has a decomposition into graphs H_1, ..., H_m, if the edges of G can be partitioned into edge-disjoint copies of each of H_1, ..., H_m. A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph G into copies H_1, ..., H_m are also sufficient. One such problem was posed by Alavi, Boals, Chartrand, Erd?s, and Oellerman. They conjectured that the edges of every graph with {m+1 choose 2} edges can be decomposed into subgraphs H_1, ..., H_m such that each H_i has i edges and is isomorphic to a subgraph of H_{i+1}. This talk will be about a proof of this for sufficiently large n. Joint work with Kyriakos Katsamaktsis, Shoham Letzter, and Benny Sudakov. |
| Title: Bounds on the Torsion Subgroups of Second Cohomology |
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| Seminar: Algebra |
| Speaker: Hyuk Jun Kweon of University of Georgia |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-11-28 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}\, ^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron--Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}\, ^0 X)_{\mathrm{red}}$, and the N\'eron--Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_\mathrm{et}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements in various situations. |