## Department of Mathematics, Duke University

Upcoming Seminars:
• Friday, February 22, 2019, 12:00pm, MSRB1 #001 Seminar Room, Biostatistics & Bioinformatics Seminar Series
Quantifying Gerrymandering or A Mathematician goes to court
Jonathan Mattingly (Duke University, Mathematics)

In October 2017, I found myself testifying for hours in a Federal court. I had not been arrested. Rather I was attempting to quantify gerrymandering using a mathematical analysis which grew from asking if a surprising 2012 election was in fact, surprising. It hinged on probing the geopolitical structure of North Carolina using a Markov Chain Monte Carlo algorithm. I will start at the beginning and describe the mathematical ideas involved in our analysis. The talk will lead to an interesting question in high dimensional sampling. This work is the result of a team effort by a number of researcher’s at Duke including a number of undergraduates. The court case Rucho vs Common Cause is set to be heard by the US Supreme Court on March 26, 2019.

• Friday, February 22, 2019, 4:00pm, Gross 318, Algebraic Geometry Seminar
Bounded negativity, H-constants and combinatorics
Brian Harbourne

The Bounded Negativity Conjecture (BNC) is the old still outstanding folklore conjecture that for each smooth projective algebraic surface X there is a bound b_X such that C^2 > b_X for each effective reduced divisor C on X. The question recently has become: for which X does BNC hold and what is the bound? BNC is known to sometimes fail in positive characteristic (see Exercise V.1.10 in Hartshorne's Algebraic Geometry), but few failures are known. In particular, no failures are known for rational surfaces nor for any surfaces in characteristic 0. In an effort to better understand how negative C^2 can be, H-constants were introduced at a mini-workshop at Oberwolfach in 2010. In its simplest form, given a reduced singular plane curve C, H(C) = (d^2-\sum_p m_p(C)^2)/r where d=deg(C), the sum is over the singular points of C, m_p(C) is the multiplicity of C at p and r is the number of singular points. If H(C) is bounded below for all C, then BNC holds for rational surfaces, so the question is: how negative can H(C) be? Attempts to address this have established interesting connections to combinatorics and suggest a plausible value for b_X when X is rational.

• Monday, February 25, 2019, 3:15pm, 119 Physics, Triangle Topology Seminar
Fillability of contact surgeries and Lagrangian discs
Bulent Tosun (University of Alabama)

It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties of a contact structure are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact (r) surgery is symplectically/Stein fillable for r in (0,1]. This is joint work with James Conway and John Etnyre.

• Wednesday, February 27, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
Learning Two-Layer Neural Networks with Symmetric Inputs
Rong Ge (Duke University, Computer Science)

Deep learning has been extremely successful in practice. However, existing guarantees for learning neural networks are limited even when the network has only two layers - they require strong assumptions either on the input distribution or on the norm of the weight vectors. In this talk we give a new algorithm that is guaranteed to learn a two-layer neural network under much milder assumptions on the input distribution. Our algorithms works whenever the input distribution is symmetric - which means two inputs $x$ and $-x$ have the same probability.

Based on joint work with Rohith Kuditipudi, Zhize Li and Xiang Wang

• Wednesday, February 27, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Raphael Beuzart-Plessis (Marseille and Columbia)

• Thursday, February 28, 2019, 11:45am, Gross Hall, Ahmadieh Family Grand Hall, Room 330, Data Dialogue
Samit Sura
Samit Sura

11:45 am LUNCH Noon: Seminar

• Thursday, February 28, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
Singularity formation in geometric flows
Simon Brendle (Columbia University, Mathematics)

Geometric evolution equations like the Ricci flow and the mean curvature flow play a central role in differential geometry. The main problem is to understand singularity formation. In this talk, I will discuss recent results which give a complete picture of all the possible limit flows in 2D mean curvature flow with positive mean curvature, and in 3D Ricci flow.

• Monday, March 4, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
Classification of Nahm Pole Solutions to the KW Equations on $S^1\times\Sigma\times R^+$
Siqi He (Simons Center for Geometry and Physics)

We will discuss Witten’s gauge theory approach to Jones polynomial by counting solutions to the Kapustin-Witten (KW) equations with singular boundary conditions over 4-manifolds. We will give a classification of solutions to the KW equations over $S^1\times\Sigma\times R^+$. We prove that all solutions to the KW equations over $S^1\times\Sigma\times R^+$ are $S^1$ direction invariant and we give a classification of the KW monopole over $\Sigma\times R^+$ based on the Hermitian-Yang-Mills type structure of KW monopole equation. This is based on joint works with Rafe Mazzeo.

• Wednesday, March 6, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Haomin Zhou (Georgia Tech)

• Wednesday, March 6, 2019, 3:00pm, Gross Hall, Ahmadieh Family Grand Hall, Room 330, Machine Learning Seminar
Batch Learning from Bandit Feedback
Thorsten Joachims (Cornell CS Dept.)

Every time a system places an ad, presents a search ranking, or makes a recommendation, we can think about this as an intervention for which we can observe the user's response (e.g. click, dwell time, purchase). Such logged intervention data is one of the most plentiful types of data available, as it can be recorded from a variety of systems (e.g., search engines, recommender systems, ad placement) at little cost. However, this data provides only partial-information feedback -- aka bandit feedback'' -- limited to the particular intervention chosen by the system. We don't get to see how the user would have responded, if we had chosen a different intervention. This makes learning from logged bandit feedback substantially different from conventional supervised learning, where correct'' predictions together with a loss function provide full-information feedback. It is also different from online learning in the bandit setting, since the algorithm does not assume interactive control of the interventions. In this talk, I will explore learning methods for batch learning from logged bandit feedback (BLBF). Following the inductive principle of Counterfactual Risk Minimization for BLBF, this talk presents an approach to training linear models and deep networks from propensity-scored bandit feedback. Related to this, the talk also touches on the use of observational partial-information feedback in the context of learning-to-rank.

• Wednesday, March 6, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Asif Zaman (Stanford)

TBA

• Monday, March 18, 2019, 12:00pm, 119 Physics, Graduate/faculty Seminar
Factorization tests and algorithms arising from counting modular forms and automorphic representations
Miao Gu

• Monday, March 18, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Mark A. Stern

TBA

• Tuesday, March 19, 2019, 3:15pm, 119 Physics, Applied Math And Analysis Seminar
Gergen Lecture at Gross Hall 103

• Tuesday, March 19, 2019, 3:15pm, Gross Hall 103, Gergen Lectures Seminar
A Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets
Robert V. Kohn (New York University, Courant Institute)

The wrinkling of thin elastic sheets is very familiar: our skin wrinkles, drapes have coarsening folds, and a sheet stretched over a round surface must wrinkle or fold.

What kind of mathematics is relevant? The stable configurations of a sheet are local minima of a variational problem with a rather special structure, involving a nonconvex membrane term (which favors isometry) and a higher-order bending term (which penalizes curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns seen in thin sheets arise from energy minimization -- but not in the same way that minimal surfaces arise from area minimization. Rather, the analysis of wrinkling is an example of "energy-driven pattern formation," in which our goal is to understand the asymptotic character of the minimizers in a suitable limit (as the nondimensionalized sheet thickness tends to zero).

What kind of understanding is feasible? It has been fruitful to focus on how the minimum energy scales with sheet thickness, i.e. the "energy scaling law." This approach entails proving upper bounds and lower bounds that scale the same way. The upper bounds tend to be easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-free; in many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. A related but more ambitious goal is to identify the prefactor as well as the scaling law; Ian Tobasco's striking recent work on geometry-driven wrinkling has this character.

Lecture 1 will provide an overview of this topic (assuming no background in elasticity, thin sheets, or the calculus of variations). Lecture 2 will discuss some examples of tensile wrinkling, where identification of the energy scaling law is intimately linked to understanding the local length scale of the wrinkles. Lecture 3 will discuss our emerging undertanding of geometry-driven wrinkling, where (as Tobasco has shown) it is the prefactor not the scaling law that explains the patterns seen experimentally.

• Wednesday, March 20, 2019, 12:00pm, 119 Physics, Gergen Lectures Seminar
A Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets: What sets the local length scale of tensile wrinkling?
Robert V. Kohn (New York University, Courant Institute)

The wrinkling of thin elastic sheets is very familiar: our skin wrinkles, drapes have coarsening folds, and a sheet stretched over a round surface must wrinkle or fold.

What kind of mathematics is relevant? The stable configurations of a sheet are local minima of a variational problem with a rather special structure, involving a nonconvex membrane term (which favors isometry) and a higher-order bending term (which penalizes curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns seen in thin sheets arise from energy minimization -- but not in the same way that minimal surfaces arise from area minimization. Rather, the analysis of wrinkling is an example of "energy-driven pattern formation," in which our goal is to understand the asymptotic character of the minimizers in a suitable limit (as the nondimensionalized sheet thickness tends to zero).

What kind of understanding is feasible? It has been fruitful to focus on how the minimum energy scales with sheet thickness, i.e. the "energy scaling law." This approach entails proving upper bounds and lower bounds that scale the same way. The upper bounds tend to be easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-free; in many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. A related but more ambitious goal is to identify the prefactor as well as the scaling law; Ian Tobasco's striking recent work on geometry-driven wrinkling has this character.

Lecture 1 will provide an overview of this topic (assuming no background in elasticity, thin sheets, or the calculus of variations). Lecture 2 will discuss some examples of tensile wrinkling, where identification of the energy scaling law is intimately linked to understanding the local length scale of the wrinkles. Lecture 3 will discuss our emerging undertanding of geometry-driven wrinkling, where (as Tobasco has shown) it is the prefactor not the scaling law that explains the patterns seen experimentally.

• Wednesday, March 20, 2019, 3:15pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Ruiwen Shu (Maryland)

• Thursday, March 21, 2019, 11:45am, Gross Hall, Ahmadieh Family Grand Hall, Room 330, Data Dialogue
Libby McClure
Libby McClure (UNC Epidemiology)

Lunch: 11:45 am Seminar: NOON

• Thursday, March 21, 2019, 3:15pm, 119 Physics, Gergen Lectures Seminar
A Variational Perspective on Wrinkling Patterns in Thin Elastic Sheets: What sets the patterns seen in geometry-driven wrinkling?
Robert V. Kohn (New York University, Courant Institute)

The wrinkling of thin elastic sheets is very familiar: our skin wrinkles, drapes have coarsening folds, and a sheet stretched over a round surface must wrinkle or fold.

What kind of mathematics is relevant? The stable configurations of a sheet are local minima of a variational problem with a rather special structure, involving a nonconvex membrane term (which favors isometry) and a higher-order bending term (which penalizes curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns seen in thin sheets arise from energy minimization -- but not in the same way that minimal surfaces arise from area minimization. Rather, the analysis of wrinkling is an example of "energy-driven pattern formation," in which our goal is to understand the asymptotic character of the minimizers in a suitable limit (as the nondimensionalized sheet thickness tends to zero).

What kind of understanding is feasible? It has been fruitful to focus on how the minimum energy scales with sheet thickness, i.e. the "energy scaling law." This approach entails proving upper bounds and lower bounds that scale the same way. The upper bounds tend to be easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-free; in many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. A related but more ambitious goal is to identify the prefactor as well as the scaling law; Ian Tobasco's striking recent work on geometry-driven wrinkling has this character.

Lecture 1 will provide an overview of this topic (assuming no background in elasticity, thin sheets, or the calculus of variations). Lecture 2 will discuss some examples of tensile wrinkling, where identification of the energy scaling law is intimately linked to understanding the local length scale of the wrinkles. Lecture 3 will discuss our emerging undertanding of geometry-driven wrinkling, where (as Tobasco has shown) it is the prefactor not the scaling law that explains the patterns seen experimentally.

• Thursday, March 21, 2019, 4:30pm, Gross Hall, Ahmadieh Family Grand Hall, Room 330, MLBytes Workshop
Anna Yanchenko
Anna Yanchenko

Dinner provided!

• Monday, March 25, 2019, 3:15pm, 119 Physics, Triangle Topology Seminar
TBA
Lisa Traynor (Bryn Mawr College)

TBA

• Wednesday, March 27, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
TBA
Alina Chertock (NC State)

• Wednesday, March 27, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Michael Harris (Columbia)

TBA

• Wednesday, March 27, 2019, 3:30pm, Gross Hall, Ahmadieh Family Grand Hall, Room 330, Machine Learning Seminar
Maytal Saar- Tsechansky
Maytal Saar- Tsechansky

Reception starts at 3 p.m. and the seminar begins at 3:30 p.m.

• Thursday, March 28, 2019, 11:45am, Gross Hall, Ahmadieh Family Grand Hall, Room 330, Data Dialogue
Zach Abzug
Zach Abzug (Duke Neurobiology)

11:45 LUNCH Noon Seminar

• Thursday, March 28, 2019, 3:15pm, 119 Physics, Probability Seminar
TBA
Kavita Ramanan (Brown)

• Thursday, March 28, 2019, 4:30pm, Gross Hall, Ahmadieh Family Grand Hall, Room 330, MLBytes Workshop
Daniel McDuff, Microsoft
Daniel McDuff (Microsoft)

Dinner provided!

• Friday, March 29, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
TBA
Ravindra

TBA

• Monday, April 1, 2019, 12:00pm, 119 Physics, Graduate/faculty Seminar
TBA
Spencer Leslie

• Wednesday, April 3, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Yu-Min Chung

• Wednesday, April 3, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBD
Levent Alpoge (Princeton University)

TBD

• Thursday, April 4, 2019, 11:45am, Gross Hall, Ahmadieh Family Grand Hall, Room 330, Data Dialogue
Dennis Furlateno, Exxon Mobile
Dennis Furlateno (Exxon Mobile)

11:45 am: LUNCH Noon: Seminar

• Friday, April 5, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
TBA
Izzet Coskun (U Illinois, Chicago)

• Monday, April 8, 2019, 12:00pm, 119 Physics, Graduate/faculty Seminar
TBA
Marc Ryser

• Monday, April 8, 2019, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Nathan Dowlin (Dartmouth College, Mathematics)

• Wednesday, April 10, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Alexander Litvak (University of Alberta)

• Wednesday, April 10, 2019, 3:00pm, SAS 2102 (NCSU), Triangle Topology Seminar
TBA
Chris Scaduto (Simons Center for Geometry & Physics)

TBA

• Friday, April 12, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
TBA
Max Lieblich (University of Washington, Mathematics)

• Monday, April 15, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Benjamin Stamm (RWTH Aachen University)

• Tuesday, April 16, 2019, 3:00pm, 119 Physics, CNCS Seminar
Bifurcation theory of swarm formation

In nature, insects, fish, birds and other animals flock. A simple two-dimensional model due to Vicsek et al treats them as self-propelled particles that move with constant speed and, at each time step, tend to align their velocities to an average of those of their neighbors except for an alignment noise (conformist rule). The distribution function of these active particles satisfies a kinetic equation. Flocking appears as a bifurcation from an uniform distribution of particles whose order parameter is the average of the directions of their velocities (polarization). This bifurcation is quite unusual: it is described by a system of partial differential equations that are hyperbolic on the short time scale and parabolic on a longer scale. Uniform solutions provide the usual diagram of a pitchfork bifurcation but disturbances about them obey the Klein-Gordon equation in the hyperbolic time scale. Then there are persistent oscillations with many incommensurate frequencies about the bifurcating solution, they produce a shift in the critical noise and resonate with a periodic forcing of the alignment rule. These predictions are confirmed by direct numerical simulations of the Vicsek model. In addition, if the active particles may choose with probability p at each time step to follow the conformist Vicsek rule or to align their velocity contrary or almost contrary to the average one, the bifurcations are of either period doubling or Hopf type and we find stable time dependent solutions. Numerical simulations demonstrate striking effects of alignment noise on the polarization order parameter: maximum polarization length is achieved at an optimal nonzero noise level. When contrarian compulsions are more likely than conformist ones, non-uniform polarized phases appear as the noise surpasses threshold.

• Wednesday, April 17, 2019, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Ana Carpio (Complutense University of Madrid)

• Wednesday, April 17, 2019, 3:00pm, SAS 2102 (NCSU), Triangle Topology Seminar
TBA
Jen Hom (Georgia Tech)

• Wednesday, April 17, 2019, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Dan Yasaki

TBA

• Friday, April 19, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
TBA

TBA

• Monday, April 22, 2019, 12:00pm, 119 Physics, Graduate/faculty Seminar
Topology and Geometry for Sensor Fusion and Vehicle-Tracking
Paul Bendich

• Wednesday, April 24, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
TBA
Stefan Steinerberger (Yale University)

• Wednesday, April 24, 2019, 3:15pm, 119 Physics, Number Theory Seminar
Extremal primes for elliptic curves without complex multiplication
Ayla Gafni (University of Rochester)

Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

• Thursday, April 25, 2019, 3:30pm, 120 Physics, Gergen Lectures Seminar
TBA
TBA

• Friday, April 26, 2019, 12:00pm, 120 Physics, Gergen Lectures Seminar
TBA
TBA

• Friday, April 26, 2019, 3:15pm, 119 Physics, Algebraic Geometry Seminar
Prym varieties of genus four curves
Nils Bruin (Simon Fraser University)

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

TBA

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