Previous Members’ Seminars

Spring 2020:

Date Speaker Title/Abstract

1/31/2020

Bogdan Stoica

Title: From p-adic to Archimedean Physics: Renormalization Group Flow and
Berkovich Spaces

Abstract: We introduce the p-adic particle-in-a-box as a free particle with periodic boundary conditions in the p-adic spatial domain. We compute its energy spectrum, and show that the spectrum of the Archimedean particle-in-a-box can be recovered from the p-adic spectrum via an Euler product formula. This product formula arises from a flow equation in Berkovich space, which we interpret as a space of theories connected by a kind of renormalization group flow. We propose that Berkovich spaces can be used to relate p-adic and Archimedean quantities generally. Talk based on arXiv:2001.01725.

2/7/2020

Sergiy Verstyuk 

Title: Some Shallow Explorations in Deep Learning for Finance

Abstract: I will introduce the existing approaches to understanding (rather than predicting) prices on financial assets. I will then discuss some simple ways of improving upon them using modern machine learning methods.

2/14/2020

Yifan Wang 

Title: Modularity in Physics and Mathematics

Abstract: I’ll discuss several incarnations of the modular group SL(2,Z) in quantum field theories and string theories, and how they relate to different areas of mathematics. We’ll see examples where mathematical frameworks lead to nontrivial predictions for physical systems, and how physics methods lead to conjectures that call for new mathematical understanding.

2/21/2020

 Du Pei

Title: Quantization: theory and applications

Abstract: How to quantize a classical system to get a quantum system? After briefly surveying the history of this problem, I will explain how to use the topological A-model to better understand quantization. As an application, I will discuss how this approach can shed light on the representation theory of double affine Hecke algebras.

2/28/2020

Cancelled

 

3/6/2020

G02

Yuewen Chen 

Title: Introduction to WENO scheme

Abstract: In this talk, we introduce the analysis and applications of WENO scheme for hyperbolic conservation law and Hamilton-Jacobi equation. WENO scheme is a powerful numerical tool to solve partial differential equationswith shock solutions. 

3/13/2020

Michael McBreen 

This meeting will be taking place virtually on Zoom.

Title: Modular representations and Lagrangian branes
 
Abstract: I will give an elementary introduction to representations of Lie algebras in characteristic p, and explain how to study them using symplectic geometry and mirror symmetry.

3/20/2020

Postponed 

  

3/27/2020

Juven Wang

This meeting will be taking place virtually on Zoom.

Title: Quantum Matter Adventure to Fundamental Physics and Mathematics
Abstract: In 1956, T. D. Lee and C. N. Yang questioned the Parity Conservation in Weak Interactions in particle physics. In less than one year, experimentalists confirmed the weak interactions are indeed maximally parity-violating. The parity violation now is a Standard Model physics textbook statement, but it goes without any down-to-earth explanation for long. Why? We will see how the recent physics development in Quantum Matter may guide us to give an adventurous story and possibly a new elementary explanation. We will see how the topology and cobordism in mathematics may come into play of anomalies and non-perturbative interactions in fundamental physics. Perhaps some of you (geometers, string theorists, etc.) can team up with me to understand the “boundary conditions” of the Standard Model and Beyond.

Work-based on and Refs therein: arXiv:1809.11171, 1904.00994, and 1910.14668

p.s. Before the seminar, you may please read the Physics Review Landmarks—Breaking the Mirror https://physics.aps.org/story/v22/st19 so you know parity-violation https://en.wikipedia.org/wiki/Parity_(physics)#Parity_violation.

4/3/2020

4:00pm

Tianqi Wu 

This meeting will be taking place virtually on Zoom.

Title: Convergence of Discrete Conformal Change and Computation of Uniformizations

Abstract: The classical uniformization theorem indicates that any closed Riemannian surface is conformally equivalent to a surface of constant curvature 1 or 0 or -1, depending on the genus of the surface. Using a simple notion of discrete conformality for triangulated surfaces, we can introduce the notion of discrete uniformization, and prove the convergence of this discrete uniformization to the classical uniformization. The key ingredient of the proof is an L^\infty estimates for discrete harmonic functions on triangulated surfaces.

4/1/2020

Yingying Wu

This meeting will be taking place virtually on Zoom.

Title: Examples of Singularity Models for Z/2 Harmonic 1-forms and Spinors in R^3

Abstract: We use the symmetries of the tetrahedron, octahedron, and icosahedron to construct local models for a Z/2 harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. These Z/2 harmonic gadgets (1-forms and spinors) characterize in part the behavior of non- convergent sequences of solutions to certain first-order gauge theory equations. The Z/2 harmonic 1-forms characterize in part the behavior of non-convergent sequences of equivalence classes of flat Sl(2; C) connections on X [Taubes, 2015]. The Z/2 harmonic spinors characterize in part the behavior of non-convergent sequences of equivalence classes of solutions to the 2-spinor generalization of the Seiberg-Witten equations [Haydys and T. Walpuski, 2015].

4/17/2020

Yang Zhou 

This meeting will be taking place virtually on Zoom.

Title: Wall-crossing in quasimap theory and its applications

 

Abstract: Enumerative geometry uses intersection theory on moduli spaces to count geometric objects. Different compactifications of the moduli space usually give different answers to the counting problem. The moduli space of holomorphic maps from a Riemann surface into a fixed complex projective manifold has a natural compactification by stable maps, which gives rise to the Gromov–Witten invariants. For a large class of manifolds, the theory of quasimaps provides a sequence of different compactifications, parametrizing the so-called epsilon-stable quasimaps.

In this talk, I will give a brief introduction to the theory of quasimaps, and describe the wall-crossing phenomenon when varying the stability parameter epsilon. We will see how those wall-crossings will simplify the moduli space step by step and finally leads to the solution of many enumerative problems.

4/23/2020

Thursday

Aghil Alaee 

Title: Localized Penrose inequalities and Hoop conjecture

Abstract:  In 1972, Roger Penrose conjectured an inequality between the total energy of a black hole and its event horizon area. Around the same time, Kip Thorne conjectured a formation of a black hole due to an inequality between the energy of a bounded region and a measure of its boundary. In this talk, we review some recent results regarding these conjectures. This is joint work with M. Khuri, M. Lesourd, and S.-T. Yau.

4/30/2020

Thursday

Tsung-Ju Lee 

Title: D-modules and tautological systems

Abstract: The tautological system was introduced by Lian, Song and Yau to handle periods of Calabi–Yau hypersurfaces or complete intersections in a projective manifold endowed with an auxiliary Lie group action. In this talk, I will review some basic notions of algebraic D-modules, give a brief introduction to tautological systems, and describe the solution space if time permits.

5/15/2020 Enno Keßler Title: Super J-holomorphic curves

Abstract: Superstring theory has motivated to work with generalizations of Riemann surfaces that incorporate anti-commuting variables: Super Riemann surfaces encode spinors on Riemann surfaces as supergeometric dimensions. Recently, together with Artan Sheshmani and Shing-Tung Yau, we have initiated the study of super J-holomorphic curves, that is, maps from super Riemann surfaces to almost Kähler manifolds preserving the almost complex structure. In this talk I will explain how super J-holomorphic curves couple the classical J-holomorphic curves equations with spinors, the construction of the moduli space and give an outlook on stable super J-holomorphic curves.
5/15/2020

9:00am
Xiaojue Zhu Title: The future of numerical models: big simulations vs big data

Fall 2019:

Date Speaker Title/Abstract
9/6/2019 Spiro Karigiannis Title: Constructions of compact torsion-free $G_2$-manifolds

 

 

Abstract: Compact torsion-free $G_2$-manifolds are 7-dimensional analogues of Calabi-Yau threefolds, being compact Ricci-flat Riemannian manifolds with reduced holonomy that are important ingredients in theories of physics. All known constructions use an abstract existence theorem of Dominic Joyce to perturb “almost” solutions of a quasilinear elliptic PDE to honest solutions, and construct the “almost” solutions via glueing methods. I will first summarize some basic facts about $G_2$-manifolds and Joyce’s existence theorem, and then briefly mention the previous constructions by Joyce (1994), Kovalev (2003), and Corti-Haskins-Nordstrom-Pacini (2014). Then I will focus on a new construction (joint work of myself and Joyce, to appear in JDG) that is significantly more involved for several reasons, which I will elucidate. In particular one key step in our construction involves solving a linear first order elliptic PDE on a noncompact 4-manifold with prescribed asymptotics at infinity. (arXiv: 1707.09325)

9/13/2019 Wei Gu Title: Sigma models and mirror symmetry

 

 

Abstract: In this talk, I will roughly review why physicists be interested in Calabi-Yau manfiolds, and will introduce some tools we used to probe Calabi-Yaus and other spaces like Fanos which we called sigma models.  I will also briefly mention how physicists using sigma models to study mirror symmetry. This is not a technical talks, rather, I will just focus on the pictures of the connections between math and physics from sigma models.

9/20/2019 Ryohei Kobayashi Title: Fermionic phases of matter on unoriented spacetime

 

 

Abstract: We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin± case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin− invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the Z8 classification of (1+1)d topological superconductors. We also compute the indicator formula of Z16 valued time-reversal anomaly for (2+1)d pin+ TQFT based on our construction.

9/27/2019 Yun Shi  Title: On motivic Donaldson-Thomas theory on local P2

 

 

Abstract: Donaldson-Thomas (DT) theory is an enumerative theory which counts ideal sheaves of curves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory is a categorification of the DT theory. This categofication contains more refined information of the moduli space, just like the topological space or cohomology contains more information than an Euler characteristic. In this talk, I will give a brief introduction to motivic DT theory. I will also discuss some results on this theory for moduli spaces of sheaves on the local projective plane.

10/4/2019 Yoosik Kim  Title: Towards SYZ mirror symmetry of flag varieties.

 

 

Abstract: SYZ mirror symmetry has provided a geometric way of understanding mirror symmetry via T-duality. In this talk, I will discuss how to obtain SYZ mirrors of partial flag varieties using Floer theory.

10/11/2019 Rongxiao Mi Title: On the change of Gromov-Witten theory under extremal transitions.

 

 

Abstract: Extremal transitions are a topological surgery that conjecturally connects the moduli space of Calabi-Yau 3-folds (often known as “Reid’s Fantasy”). Through extremal transitions, we may be able to build new mirror pairs from old ones, provided we understand how mirror symmetry is preserved. In this talk, I will outline a conjectural framework that relates the genus zero Gromov-Witten theory under an extremal transition. I will explain how it works for a large family of extremal transitions among toric hypersurfaces.

10/18/2019 No Seminar  
10/25/2019 Ruth J Williams  Title: Reflected Diffusions and (Bio)Chemical Reaction Networks

 

 

Abstract: Continuous-time Markov chain models are often used to describe them stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. Discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations (e.g., linear noise and Langevin), do not respect the constraint that chemical concentrations are never negative.

In this talk, we propose an approximation for such Markov chains, via reflected diffusion processes, that respects the fact that concentrations of chemical species are non-negative. This fixes a difficulty with Langevin approximations that they are frequently only valid until the boundary of the positive orthant is reached. Our approximation has the added advantage that it can be written down immediately from the chemical reactions. Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.

11/1/2019 No Seminar  
11/8/2019 Zhengping Gui  Title:  Deformation quantization and Algebraic index theorem

 

 

Abstract: Deformation quantization is one approach to encapsulating the algebraic aspects of observables in a quantum mechanical system. By constructing a trace map on the algebra of quantum observables, correlation functions are defined. Using this paradigm, an algebraic analogue of the Atiyah-Singer index theorem was established by Fedosov and jointly by Nest and Tsygan.

In this talk, I will discuss how to use topological quantum mechanics to prove the algebraic index theorem when quantum algebra is twisted by vector bundles.

11/15/2019 Ryan Thorngren   Title: Introduction to Bulk-Boundary Correspondences in Condensed Matter Physics

 

Abstract: A hallmark of topological phases are featureless insulators which are metallic at their edges. The bulk-boundary correspondence relates the ground state entanglement of the bulk with the anomalous properties of the boundary. I will give a gentle introduction to these ideas in a couple of simple models relevant to graphene and superconducting nanowires, respectively. If time permits, I will describe some recent work extending these ideas to bulk “phases” (actually critical points) described by conformal field theory.

11/22/2019 Cancelled  
11/29/2019 Cancelled  
12/6/2019 Sergiy Verstyuk  

DateSpeakerTitle/abstract
9/7/2018Yang ZhouTitle: 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.
9/14/2018Yu-Wei FanTitle: 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.
9/21/2018Tsung-JuTitle: 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.
9/28/2018Jörn BoehnkeTitle: 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.
10/05/2018Nishanth GudapatiTitle: 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.
10/12/2018Shuliang BaiTitle: 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.
10/19/2018Kyle LuhTitle: 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.
10/26/2018Aghil AlaeeTitle: 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.
11/2/2018Jordan KellerTitle: 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.
11/9/2018Dingxin ZhangTitle: 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.
11/30/2018Enno KesslerTitle: 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.
2/1/2019Xiaojue ZhuTitle: 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.
2/8/2019Xinqi GongTitle: 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.
2/15/2019Salem Al MoslehTitle: 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.
2/22/2019Dennis BorisovTitle: 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.
3/1/2019Guangwei SiTitle: 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.
3/8/2019Yu-Wei FanTitle: 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.
3/15/2019Jingyu ZhaoTBA
3/29/2018Tianqi WuTitle: 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.
 4/12/2019Charles DoranTitle: 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.
 4/26/2019Jörn 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.)

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