## Department of Mathematics, Duke University

Upcoming Seminars:
• Friday, September 20, 2019, 11:45am, Gross Hall 103, Data Dialogue
Arjun Devrarjan, Visnu Menon (Toucan AI)
Arjun Devrarjan, Visnu Menon (Toucan AI)

Friday Sept 20 Data Dialogue Arjun Devarajan, Vishnu Menon (Toucan AI) GH 103 11:45 - 1:00 pm

• Friday, September 20, 2019, 2:00pm, Physics 047, Number Theory Seminar
Moments of half integral weight modular L–functions, bilinear forms and applications
Alexander Dunn (University of Illinois Urbana-Champaign)

Given a half-integral weight holomorphic newform f, we prove an asymptotic formula for the second moment of the twisted L-function over all primitive characters modulo a prime. In particular, we obtain a power saving error term and our result is unconditional; it does not rely on the Ramanujan—Petersson conjecture for the form f. This gives a very sharp Lindelof on average result for L-series attached to Hecke eigenforms without an Euler product. The Lindelof hypothesis for such series was originally conjectured by Hoffstein. In the course of the proof, one must treat a bilinear form in Salie sums. It turns out that such a bilinear form also has several arithmetic applications to equidistribution. These are a series of joint works with Zaharescu and Shparlinski—Zaharescu.

• Friday, September 20, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
The Moduli Space of Matroids
Matt Baker (Georgia Tech, Mathematics)

I will begin with an introduction to hyperfields (originally introduced by Krasner for number-theoretic reasons), and then discuss a far-reaching generalization, Oliver Lorscheid’s theory of ordered blueprints. Two key examples are the sign hyperfield S and the tropical hyperfield T. I will discuss a common generalization, in this language, of Descartes' Rule of Signs (which involves polynomials over S) and the theory of Newton Polygons (which involves polynomials over T). I will then give a quick introduction to matroids and explain how the theory of ordered blueprints and ordered blue schemes allow us to construct a "moduli space of matroids", which can be viewed as an enhancement of the usual Grassmannian variety in algebraic geometry. This is joint work with Oliver Lorscheid.

• Monday, September 23, 2019, 12:00pm, Physics 119, Graduate/faculty Seminar
Static/quasi-static/dynamic models of dislocations: wellposedness and exponential convergence to equilibrium
Yuan Gao (Duke University)

Materials defects such as dislocations are important line defects in crystalline materials and they play essential roles in understanding materials properties like plastic deformation. In this talk, I will first talk about the mathematical validation of the static PN models for both straight and curved dislocation line by establishing the relationship between the PN model in the full space and the reduced problem on the slip plane in terms of both governing equations and energy minimizers. Then we study the relaxation process of dynamic Peierls-Nabarro dislocation model, which is a gradient flow with infinite nonlocal energy and double well potential describing how the materials relax to its equilibrium with the presence of a dislocation. We will show the dynamic solution will converge exponentially to a shifted steady profile which is uniquely determined.

• Monday, September 23, 2019, 3:15pm, 119 Physics, Geometry Seminar
A refinement of the Lefschetz decomposition for hyperkahler manifolds
Colleen Robles (Duke U)

The cohomology (with complex coefficients) of a compact kahler manifold M admits an action of the algebra sl(2,C), and this action plays an essential role in the analysis of the cohomology. In the case that M is a hyperkahler manifold Verbitsky and Looijenga—Lunts showed there is a family of such sl(2,C)’s generating an algebra isomorphic to so(4,b_2-2), and this algebra similarly can tell us quite a bit about the cohomology of the hyperkahler. I will describe some results of this nature for both the Hodge numbers and Nagai’s conjecture on the nilpotent logarithm of monodromy arising from a degeneration. This is joint work with Mark Green, Radu Laza and Yoonjoo Kim.

• Wednesday, September 25, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Tingran Gao (University of Chicago)

• Thursday, September 26, 2019, 3:15pm, 119 Physics, Probability Seminar
Branching diffusion processes in periodic media
Pratima Hebbar (Duke)

We investigate the asymptotic behavior of solutions to parabolic partial differential equations (PDEs) in R^d with space-periodic diffusion matrix, drift, and potential. Using this asymptotics, we describe the behavior of branching diffusion processes in periodic media. In particular, for a super-critical branching process in periodic media, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the k−th moment dominates the k−th power of the first moment for some k), while, at distances that grow sub-linearly in time, we show that all the moments converge.

• Thursday, September 26, 2019, 4:30pm, Gross Hall, Ahmadieh Family Grand Hall, Room 330, MLBytes Workshop
John Bralich, Infinia ML
John Bralich (Infinia ML)

9/26/2019 MLBytes Ahmadieh Grand Hall (GH 330) John Bralich, Infinia ML

• Friday, September 27, 2019, 11:45am, Gross Hall 103, Data Dialogue
Kelsey Sumner, UNC Epidemiology
Kelsey Sumner (UNC Epidemiology)

Fri Sep 27, 2019 11:45 am - 1pm Data Dialogues GH 103 Kelsey Sumner, UNC Epidemiology

• Friday, September 27, 2019, 12:00pm, Physics 119, Mathematics Colloquium Seminar
Model theory and complexity
Maryanthe Malliaris (University of Chicago)

The ultraproduct construction in model theory gives a way of averaging an infinite sequence of mathematical structures, such as fields, graphs, or linear orders. The talk will be about the strength of such a construction.

• Friday, September 27, 2019, 1:30pm, Physics 227, Mathematical Biology Seminar
Amy Goldberg (Duke University, Evolutionary Anthropology)

One of the major insights from the modern genomic era is the ubiquity of migration and admixture throughout human history and the animal kingdom. Admixed populations are formed through the exchange of individuals from two or more previously isolated populations. These processes shape modern human genetic and phenotypic variation, and lead to differences in disease risk between populations. Intricate sociocultural practices such as marriage customs, colonization events, and phenotypic preferences direct how parental populations interact to form admixed human populations. Despite this complexity, previous methods often considered admixed populations as simple linear combinations of their parental populations. Instead, we describe a series of mechanistic models, considering the admixture processes such as sex-specific contributions and nonrandom mating. We demonstrate that previous work misestimates population history parameters, such as the timing and intensity of migration, and may infer admixture histories that are qualitatively different. Under a related framework, we apply the model to human population genetic data to infer human migrations and interactions through time.

• Friday, September 27, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
Moduli of symmetric cubic fourfolds and nodal sextic curves
Chenglong Yu (U Penn)

Period map is a powerful tool to study geometric objects related to K3 surfaces and cubic 4-folds. In this talk, we focus on moduli of cubic 4-folds and sextic curves with specified symmetries and singularities. We identify the geometric (GIT) compactifications with the Hodge theoretic (Looijenga, mostly Baily-Borel) compactifications of locally symmetric varieties. As a corollary, the algebra of GIT invariants is identified with the algebra of automorphic forms on the corresponding period domains. One of the key inputs is the functorial property of semi-toric compactifications of locally symmetric varieties. Our work generalizes results of Matsumoto-Sasaki-Yoshida, Allcock-Carlson-Toledo, Looijenga-Swierstra and Laza-Pearlstein-Zhang. This is joint work with Zhiwei Zheng.

• Monday, September 30, 2019, 3:15pm, 119 Physics, Triangle Topology Seminar
DGA Representations, Ruling Polynomials, and the Colored HOMFLY-PT Polynomial
Caitlin Leverson (Georgia Tech)

Given a pattern braid $\beta\in J^1(S^1)$, to any Legendrian knot $\Lambda$ in $\mathbb{R}^3$ with the standard contact structure, we can associate the Legendrian satellite knot $S(\Lambda,\beta)$. We will discuss the relationship between counts of augmentations of the Chekanov­Eliashberg differential graded algebra of $S(\Lambda,\beta)$ and counts of certain representations of the algebra of $\Lambda$. We will then define an $m$­graded $n$­colored ruling polynomial from the $m$­graded ruling polynomial, analogously to how the $n$­colored HOMFLY­PT polynomial is defined from the HOMFLY­PT polynomial, and extend results of the second author, to show that the $2$­graded $n$­colored ruling polynomial appears as a specialization of the $n$­colored HOMFLY­PT polynomial. This is joint work with Dan Rutherford.

• Wednesday, October 2, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Xiaochuan Tian (UT Austin)

• Friday, October 4, 2019, 11:45am, Gross Hall 103, Data Dialogue
Tim Sell (K-Lab)
Tim Sell (K-Lab)

Fri Oct 4, 2019 11:45am - 1pm Data Dialogue GH 103 Tim Sell (K-Lab)

• Friday, October 4, 2019, 12:00pm, 119 Physics, Mathematics Colloquium Seminar
Intrinsic complexity: from approximation of random vectors and random fields to solutions of PDEs
Hongkai Zhao (University of California, Irvine)

We characterize the intrinsic complexity of a set in a metric space by the least dimension of a linear space that can approximate the set to a given tolerance. This is dual to the characterization of the set using Kolmogorov n-width, the distance from the set to the best n-dimensional linear space. In this talk I will start with the intrinsic complexity of a set of random vectors (via principal component analysis) and random fields (via Karhunen–Loève expansion) and then characterize solutions to partial differential equations of various type. Our study provides a mathematical understanding of the complexity/richness and its mechanism of the underlying problem independent of representation basis. In practice, our study is directly related to the question of whether there is a low rank approximation to the associated (discretized) linear system, which is essential for dimension reduction and developing fast algorithms.

• Friday, October 4, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
Rigidity of Schubert varieties in rational homogeneous manifolds of Picard number one
Jaehyun Hong (Korea Institute for Advanced Study)

Given a rational homogeneous manifold S=G/P of Picard number one and a Schubert variety S_0 of S, the pair (S,S_0) is said to be homologically rigid if any subvariety of S having the same homology class as S_0 must be a translate of S_0 by G. The pair (S,S_0) is said to be Schur rigid if any subvariety of S with homology class equal to a multiple of the homology class of S_0 must be a sum of translates of S_0. In this talk we use the theory of minimal rational curves to get homological rigidity and apply a refined form of transversality to reduce Schur rigidity to homological rigidity, proving that (S,S_0) exhibits Schur rigidity whenever S_0 is a non-linear smooth Schubert variety. This is joint work with N. Mok.

• Monday, October 7, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
FALL BREAK
N/A

• Wednesday, October 9, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Joan Bruna (NYU)

• Friday, October 11, 2019, 11:45am, Gross Hall 103, Data Dialogue
Tommy Lin (pondr)
Tommy Lin (pondr)

Friday October 11, 2019 Data Dialogue Tommy Lin (pondr) GH 103 11:45-1:00 pm

• Friday, October 11, 2019, 12:00pm, Physics 119, Frontiers in Mathematics Seminar
Frontiers in Mathematics Seminar
Jessica Fintzen

TBA

• Friday, October 11, 2019, 1:30pm, Physics 227, Mathematical Biology Seminar
TBD

• Friday, October 11, 2019, 3:15pm, 119 Physics, Number Theory Seminar
Local Multiplicity for Spherical Varieties
Chen Wan (MIT)

In this talk, I will discuss the local multiplicity problem for spherical varieties. I will start with the conjecture of Sakellaridis and Venkatesh. Then I will discuss the behavior of the multiplicity over the local L-packet. Finally I will explain how to use the trace formula method to study the multiplicity.

• Monday, October 14, 2019, 12:00pm, 119 Physics, Graduate/faculty Seminar
Hodge Representation
Xiayimei Han (Duke University)

• Monday, October 14, 2019, 3:15pm, Physics 119, Geometry/topology Seminar
TBA
Tom Bachmann (Massachusetts Institute of Technology, Mathematics)

TBA

• Wednesday, October 16, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Non-Convex Harmonic Parameterization
Shahar Kovalsky (Duke University)

Surface parameterization is fundamental to computational geometry and plays an important role in various applications ranging from texturing in computer graphics to the analysis and comparison of anatomical shapes in evolutionary biology. In many cases, it is desired, or even essential, that such a parameterization is invertible. The focus of this talk is invertibility, which while desirable, often presents a challenge in the computation of parameterizations. Remarkably, in the special case of harmonic parameterization onto a convex subset, invertibility is guaranteed by Tutte’s graph embedding and its continuous analog, the Rado-Kneser-Choquet Theorem. I will discuss our work on generalizing these results to the non-convex case and present simple geometric conditions for invertibility. Joint work with Stefan Steinerberger, Noam Aigerman, Misha Kazhdan, Jianfeng Lu and Ingrid Daubechies.

• Thursday, October 17, 2019, 3:15pm, 119 Physics, Probability Seminar
TBA
Mackenzie Simper (Stanford statistics)

• Friday, October 18, 2019, 11:45am, Gross Hall 103, Data Dialogue
Peter Baumgartner (RTI)
Peter Baumgartner (RTI)

Friday October 25 Data Dialogue GH 103 11:45 - 1:00 pm Peter Baumgartner (RTI)

• Friday, October 18, 2019, 3:15pm, 119 Physics, Frontiers in Mathematics Seminar
Frontiers in Mathematics Seminar
Jessica Fintzen (University of Michigan / University of Cambridge)

• Monday, October 21, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Casey Lynn Kelleher (Princeton)

TBA

• Wednesday, October 23, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
TBA
Boris Khesin (U of Toronto)

• Wednesday, October 23, 2019, 3:15pm, Physics 119, Frontiers In Mathematics Seminar
Frontiers in Mathematics Seminar
Lisa Piccirillo

• Thursday, October 24, 2019, 3:15pm, 119 Physics, Probability Seminar
Constructing extremal stationary distributions for the Voter Model in $d\geq 3$ as factors of IID
Lingfu Zhang (Princeton)

The Voters Model in $\mathbb{Z}^d$ lattice is a well studied interacting particle system. For $d \geq 3$, it has a one parameter family of extremal stationary distributions. Steif and Tykesson asked if these stationary distributions are factors of IID, or equivalently, isomorphic to Bernoulli shifts. We give an affirmative answer to this question. Our result also gives the first natural example of the so-called divide and color models, such that each cluster of the partition is infinite, while the coloring process is a factor of IID. It is a joint work with Allan Sly.

• Thursday, October 24, 2019, 4:30pm, Gross Hall, Ahmadieh Family Grand Hall, Room 330, MLBytes Workshop
Alex Sigman & Kabir Seth, Dow Jones
Alex Sigman & Kabir Seth (Dow Jones)

Thursday October 24 MLBytes Ahmadieh Grand Hall (GH 330) Alex Sigman and Kabir Seth, Dow Jones

• Thursday, October 24, 2019, 11:45pm, Gross Hall 318,
Sensing, Signals, and Communication Seminar with Jean-Francois Chamberland, Texas A&M
Jean-Francois Chamberland (Texas A&M)

Coding and Compressed Sensing for Unsourced Multiple Access Currently deployed wireless access systems based on sustained connectivity, channel estimates, and scheduling policies are ill-equipped to deal with the sporadic traffic generated by legions of unattended wireless devices. This impending technological challenge has fueled several recent research initiatives whose shared goal is to ready wireless infrastructures for the demands of tomorrow. Pertinent recent advances in this area include the introduction of unsourced, uncoordinated multiple-access models attuned to machine-driven communications and the assessment of their fundamental limits for messages with small payloads. This presentation will review recent contributions on this topic and focus on a novel communication scheme, termed coded compressed sensing, for unsourced multiple-access communication. The proposed divide-and-conquer approach leverages recent progress in compressed sensing and forward error correction to produce a novel uncoordinated access paradigm, along with a computationally efficient decoding algorithm. Within this framework, every active device partitions its data into several sub-blocks and, subsequently, adds redundancy using a systematic linear block code. Compressed sensing techniques are then employed to recover sub-blocks up to a permutation of their order, and the original messages are obtained by stitching fragments together using a tree-based algorithm.

• Friday, October 25, 2019, 9:00am, Gross Hall 103, Data Dialogue
Jascha Swisher, Laboratory of Analytic Sciences
Jascha Swisher (Laboratory of Analytic Sciences)

October 25, 2019 Data Dialogue Jascha Swisher, Laboratory of Analytic Sciences GH 103 11:45-1:00 pm

• Friday, October 25, 2019, 12:00pm, Physics 119, Frontiers In Mathematics Seminar
Frontiers in Mathematics Seminar
Lisa Piccirillo

• Friday, October 25, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Yunqing Tang (Princeton University)

• Monday, October 28, 2019, 3:15pm, 119 Physics, Triangle Topology Seminar
TBA
Tom Hockenhull (University of Glasgow, Mathematics)

• Wednesday, October 30, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Yu-Min Chung (UNC)

• Wednesday, October 30, 2019, 3:15pm, Physics 119, Mathematics Colloquium Seminar
TBA
Sarah Koch (University of Michigan)

TBA
Jacob Lurie (Institute for Advanced Study)

TBA

• Thursday, October 31, 2019, 3:15pm, 119 Physics, Probability Seminar
Jacob Lurie

• Friday, November 1, 2019, 11:45am, Gross Hall 103, Data Dialogue
Kevin Branin, Rodrigo Silva (IBM)
Kevin Branin, Rodrigo Silva (IBM)

Friday November 1 Data Dialogue Kevin Branin, Rodrigo Silva (IBM) GH 103 11:45 - 1:00 pm

• Friday, November 1, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
TBA
Jacob Lurie

• Monday, November 4, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Daniel Stern (University of Toronto)

TBA

• Wednesday, November 6, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Jeff Calder (University of Minnesota)

• Thursday, November 7, 2019, 4:15pm, at UNC, 120 Hanes Hall, Probability Seminar
TBA
Souvik Dhara (MIT)

• Friday, November 8, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
TBA
Sarah J Frei (U Oregon)

TBA

• Monday, November 11, 2019, 3:15pm, 119 Physics, Triangle Topology Seminar
TBA
Arunima Ray (Max Planck Institute for Mathematics)

• Wednesday, November 13, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
TBA
Theodore Drivas (Princeton University)

• Thursday, November 14, 2019, 3:15pm, at UNC, 130 Hanes Hall, Probability Seminar
TBA
Sayan Banerjee

• Friday, November 15, 2019, 3:15pm, Physics 119, Algebraic Geometry Seminar
TBA
Yun Shi

• Monday, November 18, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
TBD
Aleksander Doan (Columbia/Cambridge)

TBA

• Wednesday, November 20, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
TBA
Katie Newhall (UNC Chapel Hill)

• Friday, November 22, 2019, 3:15pm, Physics 119, Algebraic Geometry Seminar
TBA
Max Lieblich

• Monday, November 25, 2019, 12:00pm, 119 Physics, Graduate/faculty Seminar
TBD
Marc Ryser (Duke University)

• Monday, November 25, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Shubham Dwivedi (University of Waterloo)

TBA

• Monday, December 2, 2019, 3:15pm, 119 Physics, Triangle Topology Seminar
TBA
Maggie Miller (Princeton University, Mathematics)

• Friday, December 6, 2019, 3:00pm, 119 Physics, Number Theory Seminar
TBA
Eyal Goren (McGill University)

• Wednesday, April 1, 2020, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
Prediction of random and chaotic dynamics in nonlinear optics
Amir Sagiv (Columbia)

The prediction of interactions between nonlinear laser beams is a longstanding open problem. A traditional assumption is that these interactions are deterministic. We have shown, however, that in the nonlinear Schrodinger equation (NLS) model of laser propagation, beams lose their initial phase information in the presence of input noise. Thus, the interactions between beams become unpredictable as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model. Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) that result from differential equations with random input. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.

• Wednesday, April 8, 2020, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Jongchon Kim (UBC)

TBA

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