# All Seminars

Title: Secants, Gorenstein ideals, and stable complexes |
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Seminar: Algebra |

Speaker: Brooke Ullery of Emory University |

Contact: David Zureick-Brown, david.zureick-brown@emory.edu |

Date: 2022-09-06 at 4:00PM |

Venue: MSC N304 |

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Abstract:As Bertram describes in his thesis, certain rank two vector bundles on curves can be parametrized by a projective space into which the curve itself embeds. The idea is that the least so-called ``stable" bundles correspond to points on the embedded curve, the next most unstable lie on the first secant variety, then on the second secant variety, and so on. In this talk, I'll describe joint work with Bertram in which we generalize this to $\mathbb{P}^2$ by replacing vector bundles with complexes of vector bundles. This leads to a surprising connection between height three Gorenstein ideals, secant varieties (and their generalizations), and stability conditions on the bounded derived category of coherent sheaves on $\mathbb{P}^2$. |

Title: Effects of elastic shear modulus on soil liquefaction modelling and effective stress analysis |
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Seminar: Algebra |

Speaker: Jimena Tempestti of Emory University |

Contact: Alessandro Veneziani, avenez2@emory.edu |

Date: 2022-09-01 at 10:00AM |

Venue: MSC W301 |

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Abstract:This research builds upon a well-established constitutive model for fully coupled effective stress analysis of liquefaction problems, the Stress Density Model (SDM). Recently, SDM has been calibrated based on semi-empirical relationships between liquefaction resistance and penetration resistance, and in this approach, SDM requires only CPT data as input. As traditionally SDM has been used with somewhat degraded initial elastic modulus, this study investigates in particular, the influence of the elastic parameter on the performance and calibration of the model. Elastic parameters have a complex role in liquefaction modelling because they simultaneously affect the dynamic response of the system and stress-strain behavior of the soil. When investigated these changes, specific attention is given to the development of parameters for a generic sand with the ability of the model to concurrently simulate the effects of density and confining stress on the liquefaction resistance and their effects on the rate of strain development during cyclic mobility. First, the initial shear modulus G0 was obtained from literature for 41 clean sands tested at different densities and confining stresses. From each test, a value of A (i.e. SDM material parameter that defines the material constant in the relationship for the elastic shear modulus) was back-calculated. The shear modulus degradation curves for clean sands were scrutinised to quantify how the initial value of A changes at small strains so that a proper link between SDM and experimental data is established. Subsequently, the average value and dispersion of the data were computed. Secondly, a separate series of undrained cyclic laboratory tests on clean sand published in the literature were compiled and investigated to quantify the strain development during cyclic mobility (an aspect indirectly related to the elastic parameter A). Effects of relative density Dr and CSR (Cyclic Stress Ratio) on the strain-rate development were evaluated. As only Dr was found to correlate to the rate of deformation, mathematical expressions were developed to describe the effect of Dr on the deformation rate, both before and after achieving the selected liquefaction triggering criterion. Incorporating the relationships resulting from the laboratory data scrutiny required a minor modification in SDM, as the identified initial value of A constraints excessively the development of shear strain during cyclic mobility, particularly for dense soils and relative low CSRs. This was the principal reason why in the original SDM, a degraded value for A was used. In this thesis, different alternatives were studied including to use A as a variable, which is justified in principle, as A is strictly speaking strain-dependent. At small strains, at the beginning of the simulation, the value of A was set as suggested by laboratory data, allowing more rigorous modelling of the elastic shear stiffness. Then, as the effective stress path approaches and enters cyclic mobility and the deformation increases, the value of A was degraded to allow for the development of larger strains. Three types of representative relationships for sand were considered to evaluate the modification introduced in SDM and its calibration: (i) effects of soil density on the liquefaction resistance; (ii) effects of overburden stress on the liquefaction resistance; and (iii) effects of soil density on strain development during liquefaction and cyclic mobility. Once the simulation results were satisfactory at the element level, the modified model was evaluated in 1D effective stress analyses using the software FLAC. Two sites from Christchurch that liquefied during the 2010 Darfield (Mw=7.1) and 2011 Christchurch (Mw=6.2) earthquakes were the subject of 1D analyses. The input motion (deconvoluted record from the 2010 Darfield earthquake) was scaled to produce two levels of liquefaction response. The results were not entirely satisfactory, as a few anomalies (high dilation peaks) were noted in the acceleration time histories of the modified version. A detailed explanation of a plausible source of these peculiarities is provided regarding the interaction of the modified model and the selected numerical platform. |

Title: Algebraic Relations Between Solutions of Order One Differential Equations |
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Seminar: Algebra |

Speaker: Taylor Dupuy of Mathematics at University of Vermont |

Contact: David, david.zureick-brown@emory.edu |

Date: 2022-08-02 at 4:00PM |

Venue: MSC W301 |

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Abstract: |

Title: Local-global principle for hermitian spaces over semi-global fields |
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Defense: Dissertation |

Speaker: Jayanth Guhan of Emory University |

Contact: Jayanth Guhan, jayanth.guhan@emory.edu |

Date: 2022-06-20 at 11:00AM |

Venue: MSC E408 |

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Abstract: |

Title: Counting rational points on some moduli stacks or if that sounds scary on some modular curves |
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Seminar: Algebra |

Speaker: Jordan Ellenberg of University of Wisconsin, Madison |

Contact: David Zureick-Brown, david.zureick-brown@emory.edu |

Date: 2022-04-29 at 2:00PM |

Venue: MSC W301 |

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Abstract:How many elliptic curves are there which have a cyclic N-isogeny? You may note that this question is not very carefully stated. I will talk about the surprisingly subtle issue of how this question should best be precisely posed, and talk about recent work in this direction and related ones due to Rob Harron and Andrew Snowden, Brendan Boggess and Soumya Sankar, Maggie Pizzo + Carl Pomerance + John Voight, Tristan Phillips, John Yin, and to some extent the speaker + Matt Satriano + David Zureick-Brown. |

Title: Relative Arbitrage Opportunities in Stochastic Games and its Numerical Scheme |
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Seminar: Computational Math |

Speaker: Nicole Yang of Morgan Stanley |

Contact: Lars Ruthotto, lruthotto@emory.edu |

Date: 2022-04-28 at 10:00AM |

Venue: Virtual (RSVP by email to lruthotto@emory.edu) |

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Abstract:The Relative arbitrage portfolio outperforms a market portfolio over a given time horizon with probability one. When an investor competes with both market and peers, does relative arbitrage opportunity exist? What is the best performance one can achieve? What is the impact on the market when a large group of investors competes in a similar way? We construct a framework of multi-agent optimization for relative arbitrage problems to answer these questions. Under a dynamical system with interacting investors, the objective is characterized by the smallest non-negative continuous solution of the Cauchy problem for the associated partial differential equation. We solve the optimal strategies by deriving the Nash equilibrium in finite player and mean field games. However, solving this numerically presents many challenges due to the non-uniqueness and the curse of dimensionality. We provide a deep learning approach to tackle minimal solutions in the high-dimensional PDEs based on the associated obstacle problem and Deep Galerkin Method. We show that the minimal deep learning based solution is a good approximation in the volatility-stabilized models when compared to the grid-based numerical solution. We lay out a few future research topics related to deep learning, mean field type problems, and inverse problems. |

Title: Half Covering, Half Coloring |
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Defense: Dissertation |

Speaker: Alexander Clifton of Emory University |

Contact: Alexander Clifton, alexander.james.clifton@emory.edu |

Date: 2022-04-22 at 10:30AM |

Venue: MSC W307C |

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Abstract:We will discuss two types of problems in extremal combinatorics. First, we discuss problems about covering sets of points using affine hyperplanes. We consider a higher multiplicity generalization of a result of Alon and F\"{u}redi about the minimum number of hyperplanes needed to cover all but one vertex of an $n$-cube. We then discuss related covering problems for triangular grids. Next, we answer a question in arithmetic Ramsey theory. For a fixed set $D$, of positive integers, let $\Delta(D,k;2)$ be the smallest $N$ such that any $2$-coloring of $\{1,2,\cdots,N\}$ contains a monochromatic sequence $a_1 |

Title: Directed Reading Program presentations II |
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Type: N/A |

Speaker: Several Undergraduates of Emory University |

Contact: Chris Keyes, christopher.keyes@emory.edu |

Date: 2022-04-21 at 2:30PM |

Venue: Psychology and Interdisciplinary Sciences Building: PAIS 280 (tentative) |

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Abstract:Join us to celebrate the conclusion of the Spring 2022 Directed Reading Program! Students will give short presentations on what they learned, covering a wide range of topics in pure and applied mathematics. \\ \textbf{Speakers and topics for part II}: \begin{enumerate} \item Michael Liu, \emph{Percolation theory} (Mentor: Alexander C.) \item Catherine Baker, \emph{Numerical analysis} (Mentor: Riti) \item Judy Hao, \emph{Partial differential equations} (Mentor: Ben) \item Cecilia Garcia and Siwei Xu, \emph{Computational algebra} (Mentor: Alex D.) \end{enumerate} |

Title: Directed Reading Program presentations I |
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Type: N/A |

Speaker: Several Undergraduates of Emory University |

Contact: Chris Keyes, christopher.keyes@emory.edu |

Date: 2022-04-19 at 4:00PM |

Venue: MSC W303 |

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Abstract:Join us to celebrate the conclusion of the Spring 2022 Directed Reading Program! Students will give short presentations on what they learned, covering a wide range of topics in pure and applied mathematics. \\ \textbf{Speakers and topics for part I}: \begin{enumerate} \item Hayden Truong, \emph{Commutative algebra} (Mentor: Ariella) \item Meg Ruder, \emph{Number theory} (Mentor: Shilpi) \item Ezra Arovas, \emph{Sports analytics} (Mentor: Chris) \item Zoe Ji, \emph{Math for computer graphics} (Mentor: Abbey) \end{enumerate} |

Title: Ranking Instagram Preferences: Get to know your friends better through experimental mathematics |
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Defense: Honors Thesis |

Speaker: Urshila Choubal of Emory University |

Contact: Urshila Choubal, urshila.vikram.choubal@emory.edu |

Date: 2022-04-07 at 10:00AM |

Venue: MSC E406 |

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Abstract:Ranking methods offer remarkable potential in creating and revamping recommendation systems. The task of suggesting relevant and attractive content to users is directly benefited by improving ranking techniques. With graph ranking as the mathematical foundation on which recommendation systems are built, vertex prestige is a critical problem to be addressed. Several models exist that rank vertices in a graph. However, we explore the following methods: HITS, Dominant Eigenvector, and PageRank. We aim to emulate a recommendation system by first gathering primary data from Instagram by tracking the activity of nine participants on the app. With the help of the three ranking methods, we intend to provide our recommendation to the participants based on having accessed their past preferences. |