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Graduate Thesis Defenses 2013

Giorgia Fortuna

Title: The Beilinson-Bernstein Localization Theorem for the affine Grassmannian
Date: Tuesday, May 21st 2013
3:00pm, Room: 2-142
Committee: Dennis Gaitsgory, Pavel Etingof, Roman Bezrukavnikov

Abstract

Consider the 2-category whose objects are categories C acted on by the loop group G((t)). The local geometric Langlands correspondence conjectures an equiv- alence between this 2-category and the 2-category consisting of categories D over LocSysG (D), where LocSysG (D) denotes the stack of G-local systems on the punctured disc D.

In the first part of this talk we start by explaining how to construct a pair of adjoint functors inducing the above equivalence. For this, we construct a chiral algebra B over a curve X and a universal object over LocSysG (D), endowed with an action of G((t)). We then explain how the above equivalence implies an equivalence between the category Acrit-modG((t)) of G((t))-equivariant objects in Acrit-mod and the category of quasi-coherent sheaves on the space Opunr g;x of unramied opers, where Acrit denotes the chiral algebra corresponding to the Kac-Moody algebrabg crit at the critical level. This equivalence will be used to deduce an equivalence between the category of D-modules over the ane Grassmannian GrG;x = G((t))=G[[t]] and the category of B-modules at x 2 X that are supported on Opunr g;x , regarded as a sub- scheme of LocSysG (D).

In the second part of the talk, we explain how one could prove the equivalence between Acrit-modG((t)) and the category of quasi-coherent sheaves on Opunr g;x directly. For this, we construct a chiral algebra Ccrit and study its properties. In particular we describe its 0-th part C0 crit in terms of the commutative chiral algebra Zcrit dened as the center of Acrit. The chiral algebra Zcrit admits a canonical deformation into a non-commutative chiral algebra W~, and we express C0 crit using the resulting rst order deformation of Zcrit.

Jiawei Chiu

Title: Matrix probing, skeleton decompositions, sparse Fourier transform
Date: Wednesday, May 15th 2013
10:00am, Room: 2-135
Committee: Laurent Demanet (thesis advisor), Alan Edelman, Scott Sheffield

Abstract

The thesis consists of three parts. In the first part, we discuss matrix probing, a technique for preconditioning a matrix by applying it to random Gaussian vectors. We show that if a matrix can be expressed as a linear combination of p known matrices which are well-conditioned and act differently in a certain sense, then the p coefficients can be recovered in a numerically stable way when the amount of randomness is roughly proportional to p. In the second part, we discuss a randomized algorithm for computing skeleton decompositions of matrices. A skeleton decomposition of A is written as CUR where C, R consist of a few columns, rows of A respectively. We show that if A is nearly rank k and its top k generating vectors are incoherent, then the algorithm, which runs in O(k3) time ignoring log factors, can compute a skeleton decomposition with an operator norm error bound relative to the (k+1)-th singular value of A. In the third part, we discuss a randomized algorithm that computes the FFT of any vector which is known to be nearly S-sparse after being transformed, in O(S) time ignoring log factors. We believe that the algorithm is currently the fastest robust O(S) time (ignoring log factors) algorithm that is implemented. It incorporates a so-called mode collision detector using the matrix pencil method, so as to significantly reduce the number of samples of the input vector. We also demonstrate numerically that the algorithm works for compressible signals.

Bhairav Singh

Title: Some Results Related to the (quantum) Geometric Langlands Program
Date: Tuesday, May 7th 2013
2:45pm, Room: TBA
Committee: Roman Bezrukavnikov (advisor), Pavel Etingof, and David Vogan

Abstract

One of the fundamental results in geometric representation theory is the geometric Satake equivalence, between the category of spherical perverse sheaves on the affine Grassmannian of a reductive group $\mathbf G$ and the category of representations of its Langlands dual group. The category of spherical perverse sheaves sits naturally in an equivariant derived category, and this larger category was described in terms of the dual group by Bezrukavnikov-Finkelberg. Recently, Finkelberg-Lysenko proved a "twisted" version of the geometric Satake equivalence, which involves perverse sheaves associated to twisted local systems on a line bundle over the affine Grassmannian.

In this thesis we extend the Bezrukavnikov-Finkelberg description of the equivariant de- rived category to the twisted setting. Our method builds on theirs, but some additional subtleties arise. In particular, we use localization techniques in equivariant cohomology in a more detailed way. We also use show how our methods can be extended to explain an equiv- alence between Iwahori-equivariant peverse sheaves and twisted Iwahori-equivariant perverse sheaves on dual affine Grassmannians. This equivalence was observed earlier by Arkhipov- Bezrukavnikov-Ginzburg by combining several deep results, and they posed the problem of finding a more direct explanation. Finally, we explain how our results fit into the (quantum) geometric Langlands program.

Höeskulder Halldórsson

Title: Self-Similar Solutions to the Mean Curvature Flow in Euclidean and Minkowski Space
Date: Wednesday, May 1st 2013
2:00pm, Room: 4-163
Committee: Tobias Colding (advisor), William Minicozzi, Sigurdur Helgason

Abstract

In the first part of this thesis, we give a classification of all self-similar solutions to the curve shortening flow in the Euclidean plane $\mathbf R2$ and discuss basic properties of the curves. The problem of finding the curves is reduced to the study of a two-dimensional system of ODEs with two parameters that determine the type of the self-similar motion.

In the second part, we describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at their limiting behaviour as the pitch of the helicoidal motion goes to $0$ and compare it with the limiting behaviour of the classical helicoidal minimal surfaces. Finally, we give a classification of the immersed cylinders in the family of constant mean curvature helicoidal surfaces.

In the third part, we introduce the mean curvature flow of curves in the Minkowski plane $\mathbf R^{1,1}$ and give a classification of all the self-similar solutions. In addition, we demonstrate five non-self-similar exact solutions to the flow.

Rune Haugseng

Title: Weakly enriched higher categories
Date: Tuesday, April 30th 2013
3:30pm, Room: 4-257
Committee: Haynes Miller, Clark Barwick, Mark Behrens

Abstract

The goal of this thesis is to begin to lay the foundations for a theory of enriched $\infty$-categories. We introduce a definition of such objects, based on a non-symmetric version of Lurie's theory of $\infty$-operads. Our first main result is a construction of the correct homotopy theory of enriched $\infty$-categories as a localization of an ``algebraic'' homotopy theory defined using $\infty$-operads; this is joint work with David Gepner. We then prove some comparison results: When a monoidal $\infty$-category arises from a nice monoidal model category we show that the associated homotopy theory of enriched $\infty$-categories is equivalent to the homotopy theory induced by the model category of enriched categories; when the monoidal structure is the Cartesian product we also show that this is equivalent to the homotopy theory of enriched Segal categories. Moreover, we prove that the homotopy theory of $(\infty,n)$-categories enriched in spaces, obtained by iterating our enrichment procedure, is equivalent to that of $n$-fold complete Segal spaces. We also introduce notions of natural transformations and correspondences in the setting of enriched $\infty$-categories, and use these to construct $(\infty,2)$-categories of enriched $\infty$-categories, functors, and natural transformations, and double $\infty$-categories of enriched $\infty$-categories, functors, and correspondences. Finally, we briefly discuss a non-iterative definition of enriched $(\infty,n)$-categories, based on a version of $\infty$-operads over Joyal's categories $\Theta_{n}$, and define what should be the correct $\infty$-category of these.

Luís Pereira

Title: Goodwillie Calculus and Algebras over a Spectral Operad
Date: Monday, April 29th 2013
4:30pm, Room: 2-131 (the MIT Topology seminar)
Committee: Mark Behrens, Haynes Miller, Michael Ching

Abstract

The overall goal of this talk is to apply the theory of Goodwillie calculus to the category $Alg_{\mathcal{O}}$ of algebras over a spectral operad. Its first part will deal with generalizing many of the original results of Goodwillie so that they apply to a larger class of model categories and hence be applicable to $Alg_{\mathcal{O}}$. The second part will apply that generalized theory to the $Alg_{\mathcal{O}}$ categories. The main results here are: an understanding of finitary homogeneous functors between such categories; identifying the Taylor tower of the identity in those categories; showing that finitary n-excisive functors can not distinguish between $Alg_{\mathcal{O}}$ and $Alg_{\mathcal{O}_{\leq n}}$, the category of algebras over the truncated $O_{\leq n}$; and a weak form of the chain rule between such algebra categories, analogous to the one studied by Arone and Ching in the case of Spaces and Spectra.

Taedong Yun

Title: Diagrams of Affine Permutations and Their Labellings
Date: Monday, April 29th 2013
4:00pm, Room: 2-136
Committee: Richard Stanley, Alexander Postnikov, Jacob Fox

Abstract

We study affine permutation diagrams and their labellings by positive integers. A Balanced labelling of a Rothe diagram of a finite permutation was defined by Fomin-Greene-Reiner-Shimozono and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column-strict balanced labellings is the affine Stanley symmetric function which plays an important role in the geometry of the affine Grassmannian. Furthermore, we define a set-valued balanced labelling in which the labels are sets of positive integers, and we investigate the relations between set-valued balanced labellings and nilHecke words in the nilHecke algebra. A signed generating function of column-strict set-valued balanced labellings are shown to coincide with the affine stable Grothendieck polynomial which is related to the K-theory of the affine Grassmannian. Moreover, for finite permutations, we show that the usual Grothendieck polynomial of Lascoux-Schutzenberger can be obtained by flagged column-strict set-valued balanced labellings. Using the theory of balanced labellings, we give a necessary and sufficient condition for a diagram to be a permutation diagram. An affine diagram is an affine permutation diagram if and only if it is NorthWest and admits a special content map. We also characterize and enumerate the patterns of permutation diagrams.

Elette Boyle

Title: Secure Multi-party Protocols Under a Modern Lens
Date: Thursday, April 25th 2013
3:00pm, Room: 32-D507
Committee: Shafi Goldwasser (Advisor, CSAIL). Peter Shor, Jonathan Kelner, Yael Tauman Kalai (Microsoft)

Abstract

Since its inception in the 1980s, secure multi-party computation (MPC) has served as a cornerstone of modern cryptography, enabling mutually distrusting parties to collectively compute on their secret inputs while guaranteeing that malicious parties learn nothing beyond the evaluated function outputs. However, to effectively use MPC today, we need protocols which both scale up and address adversarial settings dictated by current-day computing platforms. I will discuss two lines of research toward this goal: designing protocols whose communication requirements scale reasonably to the current regime of massive data, and hardening MPC protocols to resist a new, powerful class of physical attacks. I will focus on work in the latter category. Namely, we consider the setting of a malicious adversary who may corrupt some parties and leak information about the secret state of all honest parties during the protocol. I will discuss techniques for achieving two types of security guarantees: 1) Standard (strong) MPC security, with a (necessary) one-time leak-free preprocessing phase. Our resulting protocol remains secure in the continual leakage model, where the rate of leakage is bounded but the overall amount can be unbounded over time. 2) "Gracefully degrading" security, guaranteeing that an adversary learns "no more" about the honest parties' inputs than the amount of information that was leaked. The corresponding protocol does not require any a priori bound on the rate or amount of leakage, and captures more general notions, such as "noisy" or computationally hard-to-invert leakage.

Jan Molacek

Title: Walking and Bouncing Droplets: Towards a Hydrodynamic Pilot-Wave Theory
Date: Thursday, April 25th 2013
2:00pm, Room: 36-153
Committee: John Bush (Advisor), Ruben Rosales, Gareth McKinley

Abstract

Coalescence of a liquid drop hitting a liquid bath can be prevented by vibration of the bath. In a certain parameter regime, the sustained purely vertical motion that ensues can be destabilized by the wavefield created on the bath surface by the repeated drop impacts, leading to a motion with a horizontal component called walking. Such walking drops exhibit many interesting phenomena reminiscent of quantum mechanics, yet no quantitative theoretical model had existed to effectively guide the experiments before our work. In this thesis we develop a theoretical model that describes the drop's vertical and horizontal motion in the relevant parameter range. The standard way of modeling the drop-bath interaction via linear spring has been found lacking and therefore a more refined, logarithmic spring model is derived. We first introduce this model in the context of drop impacting a rigid substrate and demonstrate its accuracy by comparison with existing numerical and experimental data. We then extend the model in the second part to the case of impact on a liquid substrate, and apply it to rationalize the regime diagrams of vertical bouncing motion. New experimental data on the coefficients of restitution and regime diagrams are presented. In the third part, we model the evolution of the standing waves created on the bath, which enables us to predict the onset of walking and its speed. A detailed examination of the walking region reveals new phenomena which are validated by our experimental results. A trajectory equation for the horizontal motion is obtained by filtering out the vertical bouncing. In the last part of the thesis, a simplified model of the emergence of long-term statistical behaviour from the trajectory equation is studied.

Yan Zhang

Title: The Combinatorics of Adinkras
Date: Tuesday, April 23rd 2013
4:30pm, Room: 2-135
Committee: Richard Stanley, Alexander Postnikov, Henry Cohn

Abstract

Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown nontrivial connections with coding theory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computational nature. In this work, we make a self-contained treatment of the mathematical foundations of adinkras that slightly generalizes the existing literature. Then, we make new connections to other areas including homological algebra, theory of polytopes, Pfaffian orientations, graph coloring, and poset theory. Selected results include the enumeration of odd dashings for all adinkraizable chromotopologies, the notion of Stiefel-Whitney classes for codes and their pleasantly nice vanishing conditions, and the enumeration of all Hamming cube adinkras up through dimension 5.

Po-Ru Loh

Title: Algorithms for Genomics and Genetics: Compression-Accelerated Search and Admixture Analysis
Date: Tuesday, April 23rd 2013
10am, Room: 32G-575 (Stata Center)
Committee: Bonnie Berger, Alan Edelman, Peter Shor

Abstract

Rapid advances in next-generation sequencing technologies are revolutionizing genomics, with data sets at the scale of thousands of human genomes fast becoming the norm. These technological leaps promise to enable corresponding advances in biology and medicine, but the deluge of raw data poses substantial mathematical, computational and statistical challenges that must first be overcome. I will discuss two research thrusts along these lines. First, I will describe the "compressive genomics" algorithmic framework that accelerates bioinformatic computations through analysis-aware compression. We have thus far demonstrated this methodology with proof-of-concept implementations of compression-accelerated search; extending the compressive framework to additional application areas is a current area of work. Second, I will discuss new computational tools for investigating population admixture, a phenomenon of importance in understanding demographic histories of human populations and facilitating association mapping of disease genes. Our recently released ALDER and MixMapper software packages provide fast, sensitive, and robust methods for detecting and analyzing signatures of admixture created by genetic drift and recombination on genome-wide, large-sample scales.

Geoffroy Horel

Title: Operads, Modules and higher Hochschild cohomology.
Date: Monday, April 22rd 2013
4:30pm, Room: 2-131
Committee: Haynes Miller (advisor), Clark Barwick, Michael Hopkins (Harvard)

Abstract

In this thesis, we describe a general theory of modules over an algebra over an operad. Specializing to the operad Ed of little d-dimensional disks, we show that each d-1 manifold gives rise to a theory of modules. we then describe a geometric construction of the homomorphisms objects in these categories of modules inspired by factorization homology (also called chiral homology). A particular case of this construction is higher Hochschild cohomology (i.e. Hochschild cohomology for Ed-algebras). This construction enlightens the relationship between Hocshchild cohomology and geometric objects like the cobordism category and the spaces of long knots.

John Ullman

Title: On the Regular Slice Spectral Sequence
Date: Monday, April 1st 2013
4:30 pm, Room: 2-131 (the MIT Topology seminar)
Committee: Mark Behrens (advisor, chair), Haynes Miller, Mike Hopkins

Abstract

The equivariant slice spectral sequence was introduced by Hill, Hopkins and Ravenel in their solution of the Kervaire invariant problem, and is rapidly becoming an important computational tool in equivariant stable homotopy theory. In this talk, I will describe new results on a variant called the regular slice spectral sequence (or RSSS). I will explain how geometric fixed point and norm functors interact with the slice filtration, giving a Leibniz formula for the latter. I will then use Brown-Comenetz duality to relate the RSSS to the homotopy orbit and homotopy fixed point spectral sequences. Next, I will use model theory to obtain Toda bracket operations in the RSSS. Finally, I will use some of these tools to obtain a formula for the slice tower of a cofree spectrum, prove real Bott periodicity and prove a special case of the Atiyah-Segal completion theorem.

Xiangdong Liang

Title: Modeling of Fluids and Waves with Analytics and Numerics
Date: Monday, January 14th 2013
10:30am, Room: 4-331
Committee: Steven Johnson (advisor), Martin Bazant, John Joannopoulos

Abstract

Capillary instability (Plateau-Rayleigh instability) has been playing an important role in experimental work, such as multimaterial fiber-drawing and multilayer particle fabrication. Motivated by complex multi-fluid geometries currently being explored in these applications, we theoretically and computationally studied capillary instabilities in concentric cylindrical flows of N fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the Stokes regime and for the full Navier-Stokes problem. The mathematical model with linear stability analysis can quickly predict the breakup lengthscale and timescale of concentric cylindrical fluids, and provides useful guidance for material selections and design parameters in fiber-drawing experiments. A three-fluid system with competing breakup processes at very different length scales is demonstrated with full Stokes flow simulation.

In the second half, we study the large-scale PDE-constrained microcavity topology optimization. Applications such as lasers and nonlinear devices require optical microcavities with long lifetimes Q and small modal volumes V. While most microcavities are basically designed by hand using some understanding of the physical principles of the confinement, we let the computer discover these kinds of structures. We formulate and solve a full 3d optimization scheme, over all possible 2d-lithography patterns in a thin dielectric film. The key to our formulation is a frequency-averaged local density of states (LDOS), where the frequency averaging corresponds to the desired bandwidth, evaluated by a novel technique: solving a single electromagnetic wave scattering problem at a complex frequency.