|9/7/2018||Yang Zhou||Title: Counting curves in algebraic geometry
Abstract: The classical mirror symmetry predicts that counting holomorphic curves on a Calabi-Yau manifold corresponds to the variation of Hodge structure of its mirror manifold. In this talk, we will briefly talk about various techniques of counting curves, from the perspective of algebraic geometry. We will go from “through two points there is a line” to counting curves on a quintic Calabi-Yau threefold.
|9/14/2018||Yu-Wei Fan||Title: BPS data, Riemann-Hilbert problem, and curve-counting invariants
Abstract: We start with the observation that linear maps between vector spaces give rise to the simplest example of family of BPS data. Then we introduce the Riemann-Hilbert problems associated to BPS data, and sketch the relation between solutions of these problems and curve-counting invariants on Calabi-Yau threefolds.
|9/21/2018||Tsung-Ju||Title: Hypergeometric systems and relative cohomology
Abstract: The hypergeometric equations, which were studied by Euler, Gauss, Appell, Laurecilia, etc, and generalized by Gel’fand, Kapranov and Zelevinsky, are ubiquitous in mathematics. In this talk, I will briefly talk about a cohomological interpretation of the hypergeometric system. This is a joint work with Dingxin Zhang.
|9/28/2018||Jörn Boehnke||Title: How Efficient are Decentralized Auction Platforms? (joint work with A. L. Bodoh-Creed and B. R. Hickman)
Abstract: We provide a model of a decentralized, dynamic auction market platform (e.g., eBay) in which a continuum of buyers and sellers participate in simultaneous, single-unit auctions each period. Our model accounts for the endogenous entry of agents and the impact of intertemporal optimization on bids. We estimate the structural primitives of our model using Kindle sales on eBay. We find that just over one third of Kindle auctions on eBay result in an inefficient allocation with deadweight loss amounting to 14\% of total possible market surplus. We also find that partial centralization–for example, running half as many 2-unit, uniform-price auctions each day – would eliminate a large fraction of the inefficiency, but yield slightly lower seller revenues. Our results also highlight the importance of understanding platform composition effects – selection of agents into the market – in assessing the implications of market redesign. We also prove that the equilibrium of our model with a continuum of buyers and sellers is an approximate equilibrium of the analogous model with a finite number of agents.
|10/05/2018||Nishanth Gudapati||Title: Remarks on the Notion of Energy for Perturbations of Black Hole Spacetimes
Abstract: The notion of energy for perturbations of black hole spacetimes is important from both geometric and physical perspectives. In this talk, after reviewing some background work on global energy for perturbations of black holes, we shall discuss possible extensions to quasi-local energy for the perturbative theory.
|10/12/2018||Shuliang Bai||Title: Ricci-Curvature for graphs and Ricci-flat graphs
Abstract: The Ricci curvature plays a very important role on geometric analysis on Riemannian manifolds. In 2009, Ollivier gave a notion of coarse Ricci curvature of Markov chains valid on arbitrary metric spaces. His definition of coarse Ricci curvature was adapted by Lin-Lu-Yau so that it is more suitable for graphs. A graph is called Ricci-flat if Ricci curvatures varnish on all edges. In this talk, we classify connected Ricci-flat graphs with maximal degree at most 4.
|10/19/2018||Kyle Luh||Title: Embedding Large Structures in Random Graphs
Abstract: In this talk, we will survey several general techniques of random graphs in the context of some recent results on embedding large graphs. Although the results are state of the art, the emphasis will be on robust probability tools and intuition. Several open problems will be mentioned at the end.
|10/26/2018||Aghil Alaee||Title: Recent developments in geometric inequalities for black holes
Abstract: General relativity is a geometric theory of gravitation and the most fascinating prediction of general relativity is black holes. In fact, the new gravitational wave (radiation) detection of black hole mergers provides compelling evidence for this prediction. In this talk, I will review recent developments in geometric inequalities for black holes.
|11/2/2018||Jordan Keller||Title: Robinson-Trautman Spacetimes
Abstract: Spacetime dynamics are governed by Einstein’s equations, typically thought of as a second order non-linear hyperbo-elliptic system of equations. It is of great interest to produce explicit examples of spacetimes satisfying Einstein’s equations, both those which are time-independent and those which feature dynamics. The Robinson-Trautman spacetimes form an interesting example of the latter. These spacetimes are constructed by means of an ansatz on the spacetime metric, under which the Einstein equations reduce to a Calabi equation for an unknown scalar quantity related to gravitational radiation. We discuss work of Chrusciel on the existence and long-range behavior of Robinson-Trautman solutions via an analysis of gravitational radiation.
|11/9/2018||Dingxin Zhang||Title: p-adic methods.
Abstract: For decades, methods from p-adic analysis have been applied to number theory and geometry. For example, Dwork used spectral theory of p-adic Banach spaces to study zeta functions of algebraic varieties. Inspired by Dwork’s methods, Monsky–Washnitzer defined a “formal cohomology” for affine varieties using a certain “ind-p-adic-Banach algebras”. I shall recall the work of Dwork–Monsky–Washnitzer. Time permits, I shall explain my method, which defines a cohomology for an arbitrary variety, by merging Monsky–Washnitzer’s “ind-Banach algebras” approach into the classical “tubular neighborhood” approach.
|11/30/2018||Enno Kessler||Title: Super-Riemann surfaces and the superconformal action
Abstract: With the help of a toy model, I will explain how supergeometry allows to give a geometric interpretation to supersymmetry. Analogously, a supersymmetric extension of two-dimensional harmonic maps can be understood best on super-Riemann surfaces which are a generalization of Riemann surfaces in supergeometry.
|2/1/2019||Xiaojue Zhu||Title: Exploring the ultimate of turbulence
Abstract: In this talk, we will present our newest results on fully developed turbulence. We mainly focus on two systems, Rayleigh-Bénard and Taylor-Couette flows, which share many similar features. In Rayleigh-Bénard turbulence, for the first time in two-dimensional numerical simulations we find the transition to the ultimate regime, namely at critical Rayleigh number Ra*= 10^13. We reveal how the emission of thermal plumes enhances the global heat transport, leading to a steeper increase of the Nusselt number than the classical Malkus scaling. Beyond the transition, the temperature profiles are only locally logarithmic, namely within the regions where plumes are emitted, and where the local Nusselt number has an effective scaling Nu∝Ra^0.38, corresponding to the effective scaling in the ultimate regime. In Taylor-Couette turbulence, we show how wall roughness greatly enhances the overall transport properties and the corresponding scaling exponents associated with wall-bounded turbulence. We reveal that if only one of the walls is rough, the bulk velocity is slaved to the rough side, due to the much stronger coupling to that wall by the detaching flow structures. If both walls are rough, the viscosity dependence is eliminated, giving rise to asymptotic ultimate turbulence—the upper limit of transport—the existence of which was predicted more than 50 years ago. In this limit, the scaling laws can be extrapolated to arbitrarily large Reynolds numbers.
|2/8/2019||Xinqi Gong||Title: Mathematical Intelligence Applications for Bio-Macromolecular Problems
Abstract: The intersection among mathematics, information and biology has becoming more and more interesting and important. Many studies in this direction have led to developments of theories, methods and applications. But the too fast advancing of nowadays forefront information technology and biology knowledge, have triggered two obviously emerging phenomena, tremendous brand-new peaks accessible by new kinds of efforts, randomly meaningless results by in-correct intersections. Here I will show some of our recent results in developing and distinguishing efficiently intelligent approaches and applications for computational molecular biology and medical problems, such as protein structure-function-interaction prediction and pancreas cancer CT image analysis using algorithms like Fast Fourier transform, Monte Carlo, and deep learning, and some new designed physical and geometrical features.
|2/15/2019||Salem Al Mosleh||Title: Rigidity Theory and Projective Geometry.
Abstract: We will discuss the relationship between projective geometry and the rigidity of frameworks and surfaces embedded in R^d. Starting with a simple overview of rigidity theory and projective geometry separately, we then move on to explain the projective invariance of infinitesimal isometric deformations. Lastly, we will describe projective invariants built from infinitesimal isometric deformation fields of a given framework or surface and end by discussing ongoing efforts to extend this to finite deformations. This is joint work with S.-T. Yau.
|2/22/2019||Dennis Borisov||Title: Operator product expansion and factorization algebras in differential geometry
Abstract: I will start with a motivation from Physics – operator product expansion in quantum field theory – and then I will describe a joint work with K.Kremnizer (Oxford UK), where we construct factorization algebras in differential geometry out of multiplicative Deligne cohomology classes.
|3/1/2019||Guangwei Si||Title: Structures in an olfactory code
The nervous system uses a population of neurons to encode the environment. The codes are not random but appear low-dimensional structures. They could be the consequence of the invariant properties of the neurons, neuronal circuit, and the environment. Characterizing and understanding the structures in neuronal population codes are the essential questions in systems neuroscience. Here, I will share our recent progress on understanding the neuronal code for the smell.
The olfactory system uses a relatively small number of sensory neurons to encode the odor environment with a vast number of odor molecules and a broad range of odor concentration. The question is what kind of structure in the code could support the odor perception, which allows animals to distinguish odors, recognize the same odor across concentrations, and determine concentration changes? To address the question, the experiment needs to record all the olfactory sensory neurons with single-cell resolution and study a broad range of odors and concentration. We achieved that in the small animal called fruit fly larva, with the microfluidic technology. We found that odor identity and intensity are coded by orthogonal features of the population code. Each odor’s representation forms a vector in the neuronal activity space, with the distance of the vector related to the odor concentration, and the direction of the vector related to the odor’s molecular structure. To understand the mechanism underline the structures, we analyzed individual neuron’s activation property. We further found that the activity of each sensory neuron scales with the concentration of any odor via a fixed activation function with variable sensitivity. The sensitivities across odors and sensory neurons follow a power law distribution. Much of receptor sensitivity to the odor is accounted for by a single geometrical property of the odor molecular structure. These microscopic properties contribute to the structures we observed in the population olfactory code. Together, these individual and population level patterns lend structure in the neural population code to support odor perceptions.
|3/8/2019||Yu-Wei Fan||Title: Surface, Categories, and Dynamics
Abstract: We will review basic results on diffeomorphisms of Riemann surfaces. Then we will discuss ways to measure the complexity of dynamical systems formed by diffeomorphisms. Finally, we will briefly mention some analogue categorical results, which are motivated from the parallel between Teichmuller theory and the theory of stability conditions on triangulated categories.
|3/29/2018||Tianqi Wu||Title: Koebe circle domain conjecture and the Weyl problem in hyperbolic 3-space
Abstract: In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. We prove that the Koebe circle domain conjecture is equivalent to a hyperbolic Weyl type problem. This is a joint work with Prof. Feng Luo.
| 4/5/2019|| Artan Sheshmani||
Title: Embedded surfaces, dualities, and enumerative geometry on Calabi-Yau three and four folds
Abstract: I will talk about a series of results obtained over the past years, regarding the algebraic-geometric invariants of smooth projective surfaces, and their connections to geometry of higher dimensional varieties such as Calabi-Yau threefolds and 4 folds.
| 4/12/2019||Charles Doran||Title: Mirroring Towers: Fibration and Degeneration in Calabi-Yau Geometry
Abstract: Calabi-Yau manifolds play a central role in algebraic geometry. We will briefly survey known constructions, working our way up in dimension, and focus on the geometric implications of nesting one Calabi-Yau manifold in another.
Mirror symmetry — a phenomenon first suggested by physicists — links (families of) Calabi-Yau manifolds. Mirroring towers of Calabi-Yau manifolds leads us to propose a new conjecture that unifies mirror symmetry for Calabi-Yau manifolds and their Fano manifold cousins.
| 4/19/2019|| Min Zhang||
Title: A Quasi-conformal Mapping-based Data Augmentation Technique for Improving Deep Learning on Brain Tumor Segmentation
Abstract: As deep learning (DL) finds applications in almost every aspect of medical imaging, it constantly encounters the dilemma of limited data size. As a data driven approach, DL relies on the abundance of training data to achieve satisfactory performances. When data is limited due to the high cost or long time of data collection, data augmentation is often a logic choice for implementing robust DL. In this work, we designed a novel differential geometry-based quasi conformal (QC) mapping for augmenting brain MRIs to train a DL neural network in brain tumor segmentation. The QC data augmentation algorithm allows a user to specify or randomly generate a complex-valued function on the image domain via the Beltrami coefficient. Then the algorithm obtains a homeomorphic mapping by solving the Beltrami equation and then warps the input image to obtain an augmented training set. Computationally, the algorithm can generate all possible linear or non-linear image warpings, making it a highly flexible method that can be controlled by a user for desired global distortion and local deformation. We used a publicly available brain MRI database to test the data augmentation algorithm and evaluated the benefit of data augmentation using a DL method for brain tumor segmentation. Our testing results demonstrated that the QC-based data augmentation algorithm can improve the performance of DL in brain tumor segmentation.
| 4/26/2019||Jörn Boehnke ||Title: Synthetic Regression Discontinuity – Estimating Treatment Effects using Machine Learning
Abstract: In the standard regression discontinuity setting, treatment assignment is based on whether a unit’s observable score (running variable) crosses a known threshold. We propose a two-stage method to estimate the treatment effect when the score is unobservable to the econometrician while the treatment status is known to all units. We assume that a potentially large set of observable determinants of the score is available. In the first stage, we use a statistical method to predict units’ treatment status based a continuous numerical estimate. In the second stage, we apply a regression discontinuity design using the predicted synthetic score as the running variable to estimate the treatment effect. We establish conditions under which the method identifies the local treatment effect for a unit at the threshold of the unobservable score, the same parameter that a standard regression discontinuity design with known score would identify. We examine the properties of the estimator for the case of perfect and imperfect first stage prediction accuracy by means of simulations and emphasize the use of machine learning algorithms to achieve high prediction accuracy. Finally, we applied the method to measure the effect of an investment grade rating on corporate bond prices by any of the three largest credit ratings agencies. Preliminary results show an average 1% increase in the prices of corporate bonds that received an investment grade as opposed to a non-investment grade rating. (Joint work with Pietro Bonaldi.)