All Seminars
| 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. |
| Title: Rigorous computations for linear response and sampling |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Nisha Chandramoorthy of Georgia Tech |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-11-17 at 11:00AM |
| Venue: MSC W301 |
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| Abstract: In this talk, we discuss new algorithms for two distinct computational problems, that of linear response and sampling. Linear response refers to the smooth change in the statistics of an observable in a dynamical system in response to a smooth parameter change in the dynamics. The computation of linear response in chaotic systems has been a challenge, despite work pioneered by Ruelle giving a rigorous formula in Anosov systems (mathematically idea chaotic systems). This is because typical linear perturbation-based methods are not applicable due to their instability in chaotic systems. Here, we give a new differentiable splitting of the parameter perturbation vector field, which leaves the resulting split Ruelle's formula amenable to efficient computation. A key ingredient of the overall algorithm, called space-split sensitivity, is a new recursive method to differentiate quantities along the unstable manifold. Of particular importance is the score -- gradient of log density -- of the conditional density of the physical measure, which we are differentiating, along the unstable manifold. This fast algorithm for the conditional scores motivates our attack of another longstanding computational challenge in high-dimensional statistics -- sampling from complex probability distributions, which we discuss in the second half of the talk. We present Score Operator Newton (SCONE) transport -- a novel approach to sample from a target probability distribution given its score. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. Our SCONE transport map is a constructive solution of an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score of a transported distribution from the target score. The Newton iteration enjoys fast convergence under smoothness assumptions and does not make a parametric ansatz on the transport map. |
| Title: The Structural Szemerédi–Trotter problem |
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| Seminar: Combinatorics |
| Speaker: Adam Sheffer of City University of New York |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-11-15 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: The Szemerédi–Trotter theorem can be considered as the fundamental theorem of geometric incidences. This combinatorial theorem has an unusually wide variety of applications, in combinatorics, theoretical computer science, harmonic analysis, number theory, model theory, and more. Surprisingly, hardly anything is known about the structural question - characterizing the cases where the theorem is tight. In this talk, we will survey the status of the structural (or inverse) Szemerédi–Trotter problem, including several recent results. This is a basic survey talk and does not require previous knowledge of the field. Joint works with Shival Dasu, and Olivine Silier. |
| Title: Intersection of Ekedahl-Oort strata with the supersingular locus in unitary Shimura varieties of $sgn(q-2,2)$ |
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| Seminar: Algebra |
| Speaker: Sandra Nair of Colorado State University |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-11-14 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Unitary Shimura varieties are moduli spaces of abelian varieties in characteristic $p$ with certain extra structures, including CM by an imaginary quadratic field, of which we consider the case of signature $(q-2,2)$. A fruitful way to understand them is by stratifying these spaces. We focus on two such stratifications: the Ekedahl-Oort (EO) stratification, defined with respect to the p-torsion group scheme structure up to isomorphism; and the Newton stratification, defined with respect to the p-divisible group structure up to isogeny. We study the intersection of the supersingular Newton stratum with various E-O strata in some low signature cases. We present the main methods used to conduct this study. This is joint work with Deewang Bhamidipati, Maria Fox, Heidi Goodson, Steven Groen and Emerald Stacy. |
| Title: Numerical simulations for quantum many-body systems |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Yao Wang of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-11-10 at 1:00PM |
| Venue: PAIS 230 |
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| Abstract: The rapidly evolving quantum science calls for profound understanding and predictive control of entangled quantum many-body systems, such as quantum materials, synthetic qubits, and correlated molecules. I will use this talk to give an introduction to the application of numerical linear algebra theory and techniques in quantum applications. I will start with an introduction to quantum many-body systems and their mathematical descriptions. A second-quantized quantum system can be mapped to variational and algebraic problems, whose popular solvers include exact diagonalization, matrix-product state, and a few other variational techniques. These techniques have been tightly embedded into high-performance computing hardware nowadays. Then, I will use three examples to show a few aspects of exotic phenomena in quantum many-body systems and their solutions through large-scale linear algebra. The first example is the entangled spin polaron states discovered in a material-relevant model called the Hubbard model; the second example is the unconventional spectral features in quantum materials, where I will also introduce typical methods for excited-state spectral simulations; the last example is the nonequilibrium dynamics and time-resolved x-ray theory for entanglement probe and control, where I will also introduce typical methods for time evolution simulations. This talk won’t cover all aspects where large-scale numerical linear algebra challenges existing intuitions and constructs new frameworks for quantum science, but is expected to stimulate discussions about possible collaborations. |
| Title: Monopoles and the Sen Conjecture |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Chris Kottke of New College of Florida |
| Contact: David Borthwick, dborthw@emory.edu |
| Date: 2023-11-10 at 2:00PM |
| Venue: MSC W301 |
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| Abstract: Sen's 1994 conjecture for the L^2 cohomology of the moduli spaces of SU(2) monopoles is a test case for geometric analysis on non-compact spaces. The charge 2 moduli space, known as the Atiyah-Hitchin manifold, is well-understood as an example of a `fibered boundary' manifold, but for higher charges the moduli spaces are more complicated and admit an increasingly wide variety of different asymptotic regimes. I will report on a combination of joint projects which lead to a systematic understanding of these spaces as examples of `quasi-fibered boundary' (QFB) manifolds, and, through careful analysis of the decay of harmonic forms on such spaces, to a proof of Sen's conjecture in the new case of charge 3. |
| Title: Packing the largest trees in the tree packing conjecture |
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| Seminar: Combinatorics |
| Speaker: Richard Montgomery of Warwick University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2023-11-08 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: The well-known tree packing conjecture of Gyárfás from 1976 says that, given any sequence of n trees in which the ith tree has i vertices, the trees can be packed edge-disjointly into the complete n-vertex graph. Packing even just the largest trees in such a sequence has proven difficult, with Bollobás drawing attention to this in 1995 by conjecturing that, for each k, if n is sufficiently large then the largest k trees in any such sequence can be packed. This has only been shown for k at most 5, by Zak, despite many partial results and much related work on the full tree packing conjecture. I will discuss a result which proves Bollobás's conjecture by showing that, moreover, a linear number of the largest trees can be packed in the tree packing conjecture. This is joint work with Barnabás Janzer. |
| Title: How do points on plane curves generate fields? Let me count the ways. |
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| Seminar: Algebra |
| Speaker: Renee Bell of CUNY Lehman College |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2023-11-07 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: In their program on diophantine stability, Mazur and Rubin suggest studying a curve $C$ over $\mathbb{Q}$ by understanding the field extensions of generated by a single point of $C$; in particular, they ask to what extent the set of such field extensions determines the curve . A natural question in arithmetic statistics along these lines concerns the size of this set: for a smooth projective curve $C$ how many field extensions of $\mathbb{Q}$ — of given degree and bounded discriminant — arise from adjoining a point of $C$? Can we further count the number of such extensions with specified Galois group? Asymptotic lower bounds for these quantities have been found for elliptic curves by Lemke Oliver and Thorne, for hyperelliptic curves by Keyes, and for superelliptic curves by Beneish and Keyes. We discuss similar asymptotic lower bounds that hold for all smooth plane curves $C$, using tools such as geometry of numbers, Hilbert irreducibility, Newton polygons, and linear optimization. |
| Title: Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Le Chen of Auburn University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2023-11-03 at 11:00AM |
| Venue: Atwood 240 |
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| Abstract: This talk consists of two parts of about the same length. In the first part, we make a general introduction of stochastic partial differential equations (SPDEs) from our own perspective. We will particularly emphasize their deep relationship with statistical physics. This part is intended to be accessible to the general audience. In the second part, we will focus on the nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the random field solution under measure-valued initial data $\nu$. We also study the two-point correlation function of the solution and obtain explicit upper and lower bounds. For $C^{1, \alpha}$-domains with Dirichlet condition, the initial data $\nu$ is not required to be a finite measure and the moment bounds can be improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to $|\nu|$. As an application, we show that the solution is fully intermittent for sufficiently high level $\lambda$ of noise under the Dirichlet condition, and for all $\lambda > 0$ under the Neumann condition. The second part of the talk is based on a recent joint-work with David Candil and Cheuk-Yin Lee to appear at Stochastic PDE: Analysis and Computations, 2023 (preprint available at arXiv:2301:06435). |
| Title: Estimating Kernel Matrix Eigenvalues |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Mikhail Lepilov of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2023-11-02 at 10:00AM |
| Venue: MSC N306 |
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| Abstract: Kernel matrices have appeared over the past few decades as intermediate structures when computing with "big data," such as during support vector machine classification or kernel ridge regression. Naive matrix algorithms quickly become too computationally intensive once such matrices reach moderate size; in fact, even explicitly forming such matrices is undesirable when the number of points is large. Hence, various low-rank approximations to such matrices become indispensable. If the underlying points come from the real world, however, it is a priori not often clear what the numerical rank of the resulting kernel matrix is for a given tolerance: existing methods like rank-revealing QR factorization or its randomized variants only apply in the case when the full matrix to be approximated has already been formed. In this work, we attempt to approximate the spectral decay of a kernel matrix that comes from a known distribution of points by that of a smaller matrix formed by sampling a few points from a related distribution. To do so, we use only information about the distribution and the analytical properties of the kernel. We explore how and when this may yield a useful approximation of the full spectrum using various sampling schemes. |