## Department of Mathematics, Duke University

Upcoming Seminars:
• Monday, September 25, 2017, 3:15pm, 119 Physics, Geometry/topology Seminar
Binet-Legendre metric and applications of Riemannian results in Finsler geometry

We introduce a construction that associates a Riemannian metric $g_F$ (called the \emph{Binet-Legendre} metric) to a given Finsler metric $F$ on a smooth manifold $M$. The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or isometric transformations of the Finsler metric $F$ that makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named problems in Finsler geometry. In particular, we extend a classical result of Wang to all dimensions. We answer a question of Matsumoto about local conformal mapping between two Berwaldian spaces and use it to investigate essentially conformally Berwaldian manifolds. We describe all possible conformal self maps and all self similarities on a Finsler manifold, generalizing the famous result of Obata to Finslerian manifolds. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of Deng and Hou on locally symmetric Finsler spaces. We prove smoothness of isometries of Holder-continuous Finsler metrics. We construct new easy to calculate' conformal and metric invariants of Finsler manifolds. The results are based on the papers arXiv:1104.1647, arXiv:1409.5611, arXiv:1408.6401, arXiv:1506.08935, arXiv:1406.2924 partially joint with M. Troyanov (EPF Lausanne) and Yu. Nikolayevsky (Melbourne)

• Tuesday, September 26, 2017, 3:00pm, 119 Physics, CNCS Seminar
Assessing the Effects of Protein Load on Protein Function in Living Cells
Brenton D. Hoffman (Duke University, Biomedical Engineering)

Cells exist in a complex mechanical environment that is both a source of applied forces and a means of mechanical support. An incomplete understanding of the mechanisms cells use to detect mechanical stimuli, a process termed mechanotransduction, is currently preventing advances in tissue engineering and hindering the understanding of several mechanosensitive disease states. Mechanical stimuli are sensed at focal adhesions (FAs), complex dynamic structures comprised of several hundred types of proteins that mediate physical connections between the extracellular matrix and the cytoskeleton. Detection of mechanical cues is thought to be mediated by mechanically-induced changes in protein structure, which, in elegant in vitro single molecule experiments, have been shown to induce new biochemical functions, such as changes in binding affinity as well as the formation of distinct protein-protein interactions. However, the existence and role of these mechanically-induced changes in protein function in living cells are not well understood. To enable the visualization of protein loading, we create Forster Resonance Energy Transfer (FRET)-based tension sensors that emit different colors of light in response to applied forces. The next step in the development of this technology is the use of these sensors to study the effects of mechanical loading on protein functions in living cells. To begin this process, we have refined two commonly used and powerful approaches, Fluorescence Recovery After Photobleaching (FRAP) and fluorescence co-localization to be compatible with FRET-based tension sensors. Initial efforts have focused on the mechanical linker protein vinculin due to its established role in regulating the response of FAs to mechanical loading. These novel techniques reveal that force affects both vinculin turnover as well as its ability to form distinct protein-protein interactions. Further use of these techniques should enable a wide variety of studies in mechanobiology involving different load-bearing proteins, subcellular structures, extracellular contexts, and cellular functions.

• Wednesday, September 27, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
Efficient methods for curvature based variational imaging models
Sung Ha Kang (Georgia Institute of Technology)

Starting with an introduction to multiphase image segmentation, this talk will focus on inpainting and illusory contour using variational models with curvature terms. Recent developments of fast algorithms, based on operator splitting, augmented Lagrangian, and alternating minimization, enabled us to efficiently solve functional with higher order terms. Main ideas of the models and algorithms, some analysis and numerical results will be presented.

• Wednesday, September 27, 2017, 3:15pm, 119 Physics, Number Theory Seminar
Absolute convergence of the twisted Arthur-Selberg trace formula
Abhishek Parab (Purdue University)

We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven among other cases, when the group is split. This result extends the work of Finis-Lapid (and Muller, spectral side) in the non-twisted setting. In the end, we will give an application towards residues of Rankin-Selberg L-functions suggested by J. Getz.

• Thursday, September 28, 2017, 11:45am, Ahmadieh Family Grand Hall (Gross 330), Data Dialogue
Machine Learning in 10 Epochs
Nishant Shukla (UCLA)

Running nightly experiments is prohibitive when you're just getting started with TensorFlow or any other machine learning platform. There are some first-principles we should all internalize before tackling a data-driven research problem. The primary question I will address is, 'how do I know what to do next?,' in the context of machine learning

• Thursday, September 28, 2017, 3:15pm, 119 Physics, Probability Seminar
Geometric ergodicity of stochastic splitting schemes for (generalized) Langevin dynamics
Matthias Sachs (SAMSI/Duke)

• Monday, October 2, 2017, 12:00pm, 119 Physics, Graduate/faculty Seminar
TBA
Alexander Watson (Duke University)

• Monday, October 2, 2017, 4:00pm, at NCSU, Triangle Topology Seminar
Flow and Yamada polynomials, planar triangulations, and TQFT
Slava Krushkal (University of Virginia)

In the 1960s Tutte observed that the value of the chromatic polynomial of planar triangulations at (golden ratio +1) obeys a number of remarkable properties. In this talk I will explain how TQFT gives rise to a conceptual framework for studying planar triangulations. I will discuss several extensions of Tutte's results and applications to the structure of the chromatic and flow polynomials of graphs, and the Yamada polynomial of spatial graphs. This talk is based on joint works with Ian Agol and with Paul Fendley.

• Tuesday, October 3, 2017, 3:00pm, 119 Physics, CNCS Seminar
Shear Banding, Discontinuous Shear Thickening, and Rheological Transitions in Athermally Sheared Frictionless Disks
Stephen Teitel (University of Rochester)

• Wednesday, October 4, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Zongming Ma (University of Pennsylvania)

• Wednesday, October 4, 2017, 3:15pm, 119 Physics, Number Theory Seminar
Theta integrals and generalized error functions
Stephen Kudla (University of Toronto)

Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature (n-2,2). Their functions, which depend on two pairs of time like vectors, are obtained by completing' a non-modular holomorphic generating series by means of a non-holomorphic theta type series involving generalized error functions. We show that their completed modular series arises as integrals of the 2-form valued theta functions, defined in old joint work of the author and John Millson, over a surface S determined by the pairs of time like vectors. This gives an alternative construction of such series and a conceptual basis for their modularity. If time permits, I will discuss the generalization to the case of arbitrary signature and a curious `convexity' problem for Grassmannians that arises in this context.

• Thursday, October 5, 2017, 3:15pm, 119 Physics, Probability Seminar
Genealogies in growing sold tumors
Rick Durrett

Over the past decade, the theory of tumor evolution has largely focused on the selective sweeps model. According to this theory, tumors evolve by a succession of clonal expansions that are initiated by driver mutations. In a 2015 paper, Sottoriva et al collected genetic data of various types from 349 individual tumor glands were sampled from the opposite sides of 15 colorectal tumors and large adenomas. Based on this the authors proposed an alternative theory of tumor evolution, the so-called {\bf Big Bang model}, in which one or more driver mutations are acquired by the founder gland, and the evolutionary dynamics within the expanding population are predominantly neutral. In this talk we will describe a simple mathematical model that reproduces the observed phenomena and makes quantitative predictions.

• Tuesday, October 10, 2017, 4:30pm, at UNC, Triangle Topology Seminar
Hall algebras and the Fukaya category
Peter Samuelson (University of Edinburgh)

The Hall algebra is an invariant of an abelian (or triangulated) category C whose multiplication comes from "counting extensions in C." Recently, Burban and Schiffmann defined the "elliptic Hall algebra" using coherent sheaves over an elliptic curve, and this algebra has found applications in knot theory, mathematical physics, combinatorics, and more. In this talk we discuss some background and then give a conjectural description of the Hall algebra of the Fukaya category of a topological surface. This is partially motivated by an isomorphism between the elliptic Hall algebra and the skein algebra of the torus, which we also discuss. (Joint works with H. Morton and with B. Cooper.)

• Wednesday, October 11, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Chongchun Zeng (Georgia Institute of Technology)

• Wednesday, October 11, 2017, 3:15pm, 119 Physics, Number Theory Seminar
TBA
W. Spencer Leslie (Boston College)

• Thursday, October 12, 2017, 3:15pm, 119 Physics, Probability Seminar
Frogs!
Erin Beckman

• Monday, October 16, 2017, 12:00pm, 119 Physics, Graduate/faculty Seminar
TBA
Dmitry Vagner

• Monday, October 16, 2017, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Ziva Myer (Duke University)

TBA

• Tuesday, October 17, 2017, 3:00pm, 119 Physics, CNCS Seminar
Lagrangian irreversibility and inversions in 3 and 2 dimensional turbulence
Andrew D Bragg (Duke University, Civil and Environmental Engineering)

• Wednesday, October 18, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Bob Pego (Carnegie Mellon University)

• Wednesday, October 18, 2017, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Michael Lipnowski (IAS)

• Thursday, October 19, 2017, 3:15pm, 119 Physics, Probability Seminar
Sharp Threshold for K_4 percolation
Brett Kolesnik (NSERC postdoc at Berkeley)

• Monday, October 23, 2017, 3:15pm, 119 Physics, Triangle Topology Seminar
Complex curves through a contact lens
Kyle Hayden (Boston College)

Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)

• Wednesday, October 25, 2017, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
TBA
Changhui Tan (Rice University)

• Thursday, October 26, 2017, 3:15pm, UNC Hanes Hall 125, Probability Seminar
TBA
Natalie Stanley (UNC)

• Monday, October 30, 2017, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Daniel Scofield (North Carolina State University)

TBA

• Monday, October 30, 2017, 3:15pm, Physics 119, Algebraic Geometry Seminar
TBA
Brian Krummel (UC Berkely)

• Thursday, November 2, 2017, 3:15pm, 119 Physics, Probability Seminar
TBA
Gerandy Brito (Georgia Tech)

• Monday, November 6, 2017, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Tori Akin (Duke University)

• Tuesday, November 7, 2017, 4:30pm, at NCSU, Triangle Topology Seminar
TBA
Faramarz Vafaee (California Institute of Technology)

TBA

• Thursday, November 9, 2017, 4:15pm, Hanees Hall 125 UNC, Probability Seminar
TBA
Suman Chakraborty (UNC)

• Monday, November 13, 2017, 3:15pm, 119 Physics, Geometry/topology Seminar
TBA
Chen-Yun Lin (Duke University)

• Thursday, November 16, 2017, 3:15pm, 119 Physics, Probability Seminar
TBA
David Sivakoff

• Thursday, November 23, 2017, 3:15pm, 119 Physics, Probability Seminar
Thanksgiving
no seminar

• Wednesday, November 29, 2017, 3:15pm, 119 Physics, Number Theory Seminar
TBA
Birgit Speh (Cornell)

• Thursday, November 30, 2017, 4:15pm, UNC Hanes Hall 125, Probability Seminar
TBA
Eric Friedlander

• Tuesday, December 5, 2017, 3:30pm, TBA, CTMS Adventures In Theory Lectures Seminar
TBA