Past Seminars

Abstract:

In joint work with Andrea Nahmod and Haitian Yue, we prove local well-posedness for the derivative nonlinear Schrodinger's equation in Fourier-Lebesgue space which has the same scaling as H^s for any s>0. This closes the gap left open by the work of Grunrock-Herr where s>1/4. Here there is no trilinear estimate in any standard function space, instead we will construct the solution in a nonlinear submanifold (of a function space) by exploiting its structure. This is somehow inspired by the theory of para-controlled distributions that Gubinelli et al. developed for stochastic PDEs, but our arguments are purely deterministic.

Abstract:

U.S. election forecasting involves polling likely voters, making assumptions about voter turnout, and accounting for various features such as state demographics and voting history. While political elections in the United States are decided at the state level, errors in forecasting are correlated between states. With the goal of shedding light on the forecasting process and exploring how states influence each other, we develop a framework for forecasting elections in the U.S. from the perspective of dynamical systems. Through a simple approach that borrows ideas from epidemiology, we show how to combine a compartmental model with public polling data from HuffPost and RealClearPolitics to forecast gubernatorial, senatorial, and presidential elections at the state level. Our results for the 2012 and 2016 U.S. races are largely in agreement with those of popular pollsters, and we use our new model to explore how subjective choices about uncertainty impact results. We conclude by comparing our forecasts for the senatorial and gubernatorial races in the U.S. midterm elections of 6 November 2018 with those of popular pollsters. This is joint work with Daniel Linder (Augusta Univ.), Mason Porter (UCLA), and Grzegorz Rempala (Ohio State Univ.)

Abstract:

Recent advances in technology have led to a monumental increase in large-scale data across many platforms. One mathematical model that has gained a lot of recent attention is the use of sparsity. Sparsity captures the idea that high dimensional signals often contain a very small amount of intrinsic information. Using this notion, one may design efficient low-dimensional representations of large-scale data as well as robust reconstruction methods for those representations. Binary, or one-bit, representations of data for example, arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this talk, we provide a brief background to sparsity and 1-bit measurements, and present new results on the problem of data classification with low computation and resource costs. We illustrate the utility of the proposed approach on recently acquired data about Lyme disease.

Abstract:

The machinery of the Weil conjectures often allows us to relate the singular cohomology of the complex points of a scheme to the cardinality of its set of points over a finite field. When we apply these methods to a moduli scheme, we obtain an enumeration of the objects the moduli parameterizes. It's rare that we can actually fully compute the cohomology of these moduli spaces, but homological stability results often give a first order approximation to the homology.

In this talk, we'll explain how to obtain second order homological computations for a class of Hurwitz moduli spaces of branched covers; these yield second order terms in enumerating the moduli over finite fields. We may interpret these as second order terms in a function field analogue of the function which counts number fields, ordered by discriminant. Our second order terms match those of Taniguchi-Thorne/Bhargava-Shankar-Tsimerman in the cubic case, and give new predictions in other Galois settings.

This is joint (and ongoing) work with Berglund, Michel, and Tran.

Abstract:

We consider the solution to a tight-binding, periodic Schrödinger equation with a random potential evolving stochastically in time. If the potential evolves according to a stationary Markov process we obtain a positive, finite diffusion constant for the evolution of the solution. More generally, we show that the square amplitude of the wave packet, after diffusive rescaling, converges to a solution of the heat equation. (Joint work with J. Schenker and Z. Tilocco at the Michigan State University).

Abstract:

We will show the sharp estimate on the behavior of the smallest singular value of random matrices under very general assumptions. One of the key steps in the proof is a result about the efficient discretization of the unit sphere in an n-dimensional euclidean space. This result allows us to work with very general random matrices. The proof of the result will be outlined. Partially based on the joint work with Tikhomirov and Vershynin.

Abstract:

We consider the problem of reconstructing a hidden assignment of random variables (x_1,…, x_n) sitting on the nodes of a graph, given noisy observations of pairs (x_u, x_v) for every edge (u,v) of the graph. Such problems have been extensively studied when the underlying graph is ‘mean-field’ (e.g., the complete graph, Erdos-Renyi, random regular,…), in which case the existence of a ``possible-but-hard’’ phase where reconstruction is possible but computationally difficult is ubiquitous.

In contrast, the picture is dramatically different when the graph is amenable. I will represent a generic result about the optimality of local algorithms for computing marginals under a somewhat strong model of side information. Next, I will focus on Z_2 synchronization (aka planted Ising model) on the Euclidean lattice with weaker side information, and discuss a renormalization-based procedure for reconstructing the assignment.

Joint work with Andrea Montanari.

Abstract:

Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call \textit{Elser numbers} \text{els}_k(G), where G is a connected graph and k a nonnegative integer. Elser had proven that \text{els}_1(G)=0 for all G. By interpreting the Elser numbers as Euler characteristics of appropriate simplicial complexes called \emph{nucleus complexes}, we prove that for all graphs G, they are nonpositive when k=0 and nonnegative for k\geq2. The last result confirms a conjecture of Elser. At the end, we will present an open problem naturally arising from our proof of Elser's conjecture.

Abstract:

Data accessibility is important--publicly available data-sets support vital social science research, social programs and data-informed governance. In recent years, an increasing amount of data has been curated and made generally available through sites like data.gov, IPUMS, and other resources, fueling the progress of research in Big Data. However, data with the most potential value for public good can also be the most privacy sensitive--such as data on abuse, STDS, extreme poverty, or mental health. These datasets exist, but may be redacted or entirely withheld from public view due to legal restrictions and the very real danger that anonymized individuals may be reidentified.

Privacy-preserving synthetic data provides a pathway to publicly release datasets that contain very sensitive information. The basic process consists of three parts: A generative model is built which captures the distribution of the original sensitive data, perturbation steps are applied to the model to improve its privacy properties (either formal or heuristic-based), and then the model is used to synthesize a new data set of synthetic individuals. The synthetic dataset preserves the significant properties of the original data, but because it contains no real people, it can be safely released to the public. When the distributional difference between the real and synthetic data mimics the difference between two subsamples of the original data, i.e. when privacy error mimics sampling error, we can think of the synthetic data as survey results from a parallel dimension: The same pattern of information as the original data, with no real people.

In this talk, I'll cover approaches to creating synthetic data, the difference between formal and heuristic-based privacy, and, importantly, quality metrics used to verify that the synthetic data is a good substitute for the original data (a challenging problem itself in a high dimensional feature space). High quality synthetic data is a rapidly progressing research area, with both promising success stories and an exciting frontier of open problems.

Abstract:

The phase retrieval problem is the problem of reconstructing an unknown signal from its Fourier intensity function. This problem has a long history in physics and engineering and occurs in contexts such as X-ray crystallography, speech processing and computational biology. As stated, the phase retrieval problem is ill-posed as there may be up to $2^N$ non-equivalent signals (called ambiguities) with the same Fourier intensity function. To enforce uniqueness additional constraints must be imposed.

In this talk we discuss the geometry of the space of ambiguities obtained by varying the signal. By understanding this geometry we prove a general result characterizing constraints that enforce a unique solution to the phase retrieval problem. This result was applied in work with Tamir Bendory and Yonina Eldar on blind phaseless short-time Fourier transform recovery.

Abstract:

Since first defined by Deligne and Lusztig, a Deligne-Lusztig variety has become unavoidable when studying the representation theory of finite groups of Lie type. This is a certain subvariety of the flag variety of the corresponding reductive group, and its cohomology groups are naturally endowed with the action such finite groups, which in turn gives a decomposition of irreducible representations called Lusztig series. In this talk, I will briefly discuss the background of Deligne-Lusztig theory and provide a formula to calculate the homology class of Deligne-Lusztig varieties in the Chow group of the flag variety. If time permits, I will also discuss their analogues.

Abstract:

Indexed Universal Life (IUL) Insurance was developed to harness thepower of equity market returns with downside protection. However IUL iscurrently illustrated using a static credited rate which masks market returnvolatility inherent in its structure. As a result, what policyholders see as expectedperformance maybe far from reality in many cases. In our research, we modeledthe pricing algorithms of major IUL products and applied scenario testing usingMonte Carlo simulation of indices used in IUL products. The statistical variance ofindices leads to vastly different results than what is currently demonstrated inmany cases, and this variance may cause the failure of the policy. Our researchindicates a better method for demonstrating policy performance would be basedon an outcome analysis rather than the static method currently in use.Bios: Songyu Yan: https://www.linkedin.com/in/songyu-yan-826bb2126/          Ian Luo: https://www.linkedin.com/in/yifei-luo-9051b1170/

Abstract:

The Langlands Program, connecting algebra, analysis and geometry in diverse ways, is foundational to modern number theory. I will introduce this program and indicate some recent progress. As we shall see, a great deal still remains to be done.

Abstract:

We analyse compactifications of 11 dimensional or type II supergravity down to 4 dimensional Minkowski space for generic flux and generic internal Killing spinors. We note the failure of conventional differential geometry to capture the generic features of the theory and show that the correct formalism comes in the form of a closed form Leibniz algebroid - or as we call it in the physics community, generalised geometry. Our structure is similar to the generalised geometry of Hitchin, but now the structure group is the non-compact exceptional group E_{7(7)}x R^{+}. It turns out that having N=1 supersymmetry in the effective theory on Minkowski space is equivalent to an integrable SU(7) structure on the generalised tangent bundle. We provide the tensors that define the SU(7) structure and give the integrability conditions. Finally we provide an expression for the Kahler potential on the space of structures, the superpotential of the lower dimensional theory, and we explore the moduli of these structures giving explicit answers in certain cases.

Abstract:

For functions acquired through point evaluations, is there an optimal way to estimate a quantity of interest or even to approximate the functions in full? We give an affirmative answer to this question under the novel assumption that the functions belong to a model set defined by approximation capabilities. In fact, we produce implementable linear algorithms that are optimal in the worst-case setting. We present applications of the abstract theory in atmospheric science and in system identification.

Dr. Simon Foucart earned a Masters of Engineering from the Ecole Centrale Paris and a Masters of Mathematics from the University of Cambridge in 2001. In 2006, he received his Ph.D. in Mathematics at the University of Cambridge, specializing in Approximation Theory. After two postdoctoral positions at Vanderbilt University and Université Paris 6, he joined Drexel University in 2010 before moving to the University of Georgia in 2013. Since 2015, he has been with Texas A&M University, where is now professor. His recent work focuses on the field of Compressive Sensing, whose theory is exposed in the book ‘A Mathematical Introduction to Compressive Sensing’ he coauthored with Holger Rauhut. Dr. Foucart’s research was recognized by the Journal of Complexity, from which he received the 2010 Best Paper Award. Dr. Foucart’s current interests also include the mathematical aspects of Metagenomics, Optimization, Deep Learning and Data Science at Large

Abstract:

Most applied problems we encounter can be naturally written as nonconvex optimization, for which
obtaining even a local minimizer is computationally hard in theory, never mind the global minimizer. In practice, however, simple numerical methods often work surprisingly well in finding high-quality solutions, e.g., training deep neural networks.

In this talk, I will describe our recent effort in bridging the mysterious theory-practice gap for nonconvex optimization, in the context of solving practical problems in signal processing, machine learning, and scientific imaging. 1) I will highlight a family of smooth nonconvex problems that can be solved to global optimality using simple numerical methods, independent of initialization. 2) The discovery, however, does not cover nonsmooth functions, which are frequently used to encode structural objects (e.g., sparsity) or achieve robustness. I will introduce tools from nonsmooth analysis, and demonstrate how nonsmooth, nonconvex problems can also be analyzed and solved in a provable manner. 3) Toward the end, I will provide examples to show how innovative problem formulation and physical design can help to tame nonconvexity.

Ju Sun is an assistant professor at the computer science & engineering department, University of Minnesota at Twin Cities. Prior to this, he was a postdoctoral scholar at Stanford University,
working with Professor Emmanuel Cand?s. He received his Ph.D. degree from Electrical Engineering of Columbia University in 2016 (2011--2016) and B.Eng. degree in Computer Engineering (with a minor in Mathematics) from the National University of Singapore in 2008 (2004--2008). His research interests span computer vision, machine learning, numerical optimization, signal/image processing, and high-dimensional data analysis. Recently, he is particularly fascinated by why simple numerical methods often solve nonconvex problems surprisingly well (on which he maintains a bibliographic webpage: http://sunju.org/research/nonconvex/ ) and the implication on representation learning. He won the best student paper award from SPARS'15 and honorable mention of doctoral thesis for the New World Mathematics Awards (NWMA) 2017.

Abstract:

Collective dynamics is driven by alignment that tend to self-organize the crowd and by different external forces that keep the crowd together. Different emerging equilibria are self-organized into clusters, flocks, tissues, parties, etc.

I will overview recent results on the hydrodynamics of large-time, large-crowd collective behavior, driven by different “rules of engagement”. In particular, I address the question how short-range interactions lead, over time, to the emergence of long-range patterns, comparing geometric vs. topological interactions.

Abstract:

Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction- diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxis systems extremely delicate and challenging task.

In this talk, I will present a family of high-order numerical methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to to multi-scale and coupled chemotaxis–fluid system and will also be discussed.

Abstract:

Recently Corteel and Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. In this talk, we enumerate bounded lecture hall tableaux. We show that their enumeration is closely related to standard and semistandard Young tableaux. We also show that the number of bounded lecture hall tableaux is the coefficient of the Schur expansion of s?(m+y1,...,m+yn). To prove this result, we use two main tools: non-intersecting lattice paths and bijections. In particular we use ideas developed by Krattenthaler to prove bijectively the hook content formula. This is joint work with Sylvie Corteel.

Abstract:

A numerical semigroup is a subset of N that is closed under addition, contains 0 and has finite complement in N. There are several fundamental invariants of a numerical semigroup S among which are the Frobenius number and genus of S, denoted F(S) and g(S), respectively. The quotient of a numerical semigroup S by a positive integer d is the set S/d={x|dx?S} which is also a numerical semigroup. In this talk, I will present some recent results showing the relation between the genus of S/d and the genus of S. If time permits, we will also talk about identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions. This is joint work with S. Butler, C. Defant, Y. Gao, P.E. Harris, C. Hettle, Q. Liang, H. Nam, and A. Volk.

Abstract:

Calogero-Sutherland models are a class of completely integrable (quantum) many-body systems. In the most general case, particles interact pairwise through an elliptic potential. We discuss the second quantization of several elliptic Calogero-Sutherland models. In degenerate (trigonometric, rational) cases, there is a well-studied correspondence between Calogero-Sutherland systems and Benjamin-Ono equations, a nonlinear, integrable integro-differential equation. We discuss the correspondence in the elliptic regime, where second quantization of a particular system leads to a new two-component elliptic Benjamin-Ono equation. We construct a Lax pair for this equation via a Riemann-Hilbert problem on the torus. This is joint work with Edwin Langmann and Jonatan Lenells.

Abstract:

There are many papers concerning 3rd and higher order superintegrable systems on 2D Euclidean space, particularly by Pavel Winternitz, his students and collaborators. However, we are not aware of similar studies on nonflat spaces. We decided to study the 2-sphere (with separation in polar coordinates) and the 2-hyperboloid (with separation in horospherical coordinates). The computations were very complex for these nonflat systems and the results less rich than for flat space, which is to be expected. In each case we have a Hamiltonian operator H a 2nd order symmetry operator A and compute a 3rd order symmetry operator B.
We classified the potentials which yielded

a nonzero B. They fall into 4 classes: 1) Systems that are 2nd order superintegrable, 2) Systems that are 3rd order superintegrable but do not generate a cubic algebra, 3) Systems with algebraically dependent generators A and B, 4) Systems that are 3rd order superintegrable with a cubic algebra.

We pay special attention to the class 3 cases which also occur for Euclidian space with little change. We call these functionally dependent superintegrable systems, and show that they always permit a computation of the A eigenfunctions by simple quadrature. For the hyperboloid we show that the TTW method applied to a 2nd order
superintegrable system in horospherical coordinates yields a 3rd order superintegrable system.

Abstract:

We discuss the facial weak order, a poset structure that extends the poset of regions on a central hyperplane arrangement to the set of all faces of the arrangement which was first introduced on the braid arrangements by Krob, Latapy, Novelli, Phan and Schwer. We provide various characterizations of this poset including a global one, a local one, one using covectors and a geometric one using the associated zonotope. We then show that the facial weak order is in fact a lattice for simplicial hyperplane arrangements, generalizing a result by Björner, Edelman and Zieglar showing the poset of regions is a lattice for simplicial arrangements. We end by stating some properties on the facial weak order.

Abstract:

We study the problem of minimizing a strongly convex and smooth function when we have noisy estimates of its gradient. We propose a novel multistage accelerated algorithm that is universally optimal in the sense that it achieves the optimal rate both in the deterministic and stochastic case and operates without knowledge of noise characteristics. The algorithm consists of stages that use a stochastic version of Nesterov's accelerated algorithm with a specific restart and parameters selected to achieve the fastest reduction in the bias-variance terms in the convergence rate bounds.

Mert Gürbüzbalaban is an assistant professor at Rutgers University. Previously, he was a postdoctoral associate at the Laboratory for Information and Decision Systems (LIDS) at MIT. He is broadly interested in optimization and computational science driven by applications in large-scale information and decision systems. He received his B.Sc. degrees in Electrical Engineering and Mathematics as a valedictorian from Bo?aziçi University, Istanbul, Turkey, the “Diplôme d’ingénieur” degree from École Polytechnique, France, and the M.S. and Ph.D. degrees in (Applied) Mathematics from the Courant Institute of Mathematical Sciences, New York University.

Dr. Gürbüzbalaban received the Kurt Friedrichs Prize (given by the Courant Institute of New York University for an outstanding thesis) in 2013, Bronze Medal in the École Polytechnique Scientific Project Competition in 2006, the Nadir Orhan Bengisu Award (given by the electrical-electronics engineering department of Bo?aziçi University to the best graduating undergraduate student) in 2005 and the Bülent Kerim Altay Award from the Electrical-Electronics Engineering Department of Middle East Technical University in 2001. He received funding from a variety of sources including multiple programs at the U.S. National Science Foundation.

Abstract:

The (affine cone over the) Grassmannian is a prototypical example of a variety with "cluster structure"; that is, its coordinate ring is a cluster algebra. Scott (2006) gave a combinatorial description of this cluster algebra in terms of Postnikov's plabic graphs. It has been conjectured essentially since Scott's result that Schubert varieties also have a cluster structure with a description in terms of plabic graphs. I will discuss recent work with K. Serhiyenko and L. Williams proving this conjecture. The proof uses a result of Leclerc, who shows that many Richardson varieties in the full flag variety have cluster structure using cluster-category methods, and a construction of Karpman to build plabic graphs for each Schubert variety. Time permitting, I will also discuss our results on cluster structures on a larger class of positroid varieties, which involve the combinatorics of "generalized" plabic graphs.

Abstract:

This is joint with Maury Bramson, who introduced our work last week and then spoke more particularly about the existence of uncountably many mutually singular equilibria in the majority vote process on a tree of vertex degree 5 and higher, whenever the noise rate is sufficiently small. Each equilibrium is associated with a different fixed point of the majority vote operation.

Natural question: If the system starts at (or near) one of these fixed points and the noise rate is small, does it converge to the corresponding equilibrium, and if so, how fast? Answering this question can tell us much about the properties of these equilibria, but standard techniques do not apply. I will describe a new method, based on the "graphical construction" of these systems, which shows exponentially quick convergence to equilibrium when the noise rate is sufficiently small. These methods apply to other models on trees, such as the stochastic Ising model, for which the convergence result was previously unknown.

This lecture is self-contained; essential features from the previous week will be included (with some new pictures).

Abstract:

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is a long-standing open question in mathematical fluid dynamics. Recent computations have provided strong numerical evidence that the 3D Euler equations develop a finite time singularity from smooth initial data. I will report some recent progress in providing a rigorous justification of the singularity formation in the 3D Euler equations and related models.

Abstract:

We will give less precise but hopefully very motivational introduction to Langlands philosophy with many examples. A serious background in anything is not essential.

Abstract:

We show that certain sets of representations of Dynkin quivers are determined by their Jordan form data. The description is phrased in terms of the bounded derived category of representations of the quiver. This continues the study of the paper of Garver, Patrias and Thomas: Minuscule reverse plane partitions via quiver representations.

Abstract:

In many practical applications, we often need to provide solutions to quantities of interest to a large-scale problem but with only subsampled data and partial information of the physical model. Traditional PDE solvers cannot be used directly for this purpose. On the other hand, many powerful techniques have been developed in data science to represent and compress data for useful information. A crucial factor for the success of these methods is to exploit some low rank or sparsity structures in these high-dimensional data. In this talk, we will describe our recent effort in developing effective numerical methods to solve multiscale problems using subsampled data. The PDE analysis will help reveal certain important solution structures so that we can use techniques from data science to give accurate approximations for these quantities of interest.

Thomas Yizhao Hou is the Charles Lee Powell professor of applied and computational mathematics at Caltech. His research interests are centered around developing mathematical analysis and effective computational methods for vortex dynamics, interfacial flows, multiscale problems and data analysis. He received his Ph.D. in mathematics from UCLA in 1987, and joined the Courant Institute as a junior faculty member in 1989. He moved to Caltech in 1993 and was named the Charles Lee Powell Professor in 2004. He was the founding Editor-in-Chief of the SIAM Journal on Multiscale Modeling and Simulation from 2002 to 2007 and served on the IMA Board of Governors from 2010 to 2014. Dr. Hou is a Fellow of American Academy of Arts and Sciences, a SIAM Fellow and an AMS Fellow.

Abstract:

This talk focuses on generalized Bott-Samelson resolutions of Schubert varieties. These resolutions are iterated G/P-bundles (for different parabolic subgroups P of the algebraic group G). Special cases have appeared in the work of Zelevinsky, Wolper, Ryan and several other authors. After introducing these resolutions and some of their properties, we discuss a possible best resolution for a given Schubert variety X(w), based on a combinatorial condition on the inversion set of the Weyl group element w. If time permits, we will discuss a computer program that calculates local intersection cohomology (i.e., Kazhdan-Lusztig polynomials) from these resolutions. This is joint work with Jennifer Koonz.

Abstract:

The majority vote process was one of the first interacting particle systems to be investigated and can be described as follows. There are two possible opinions at each site and that opinion switches randomly to the majority opinion of the neighboring sites. Also, at a different rate epsilon, the opinion at each site randomly changes due to noise.

Despite its simple dynamics, the majority vote process is difficult to analyze. In particular, on Z^d with d>1 and epsilon chosen small, it is not known whether there exists more than one equilibrium. This is surprising due to the close analogy between the majority vote process and the Ising model.

Here, we discuss work with Larry Gray on the majority vote process on the infinite tree with vertex degree d, where it is shown that, for small noise, there are uncountably many mutually singular equilibria, and that convergence to equilibrium occurs exponentially quickly from nearby initial states. Our methods rely on graphical constructions; they are quite flexible and can be used to obtain analogous results for other models, such as the stochastic Ising model on a tree.

This is the first part of a two-lecture presentation, and will concentrate on background and on existence of equilibria. Larry Gray will give the second part the following week, which will concentrate on convergence.

Abstract:

A celebrated result of Waldspurger from the eighties express the central value of certain base-change L-functions for $GL_2$ as toric periods of modular forms or generalizations thereof. In the mid 2000s Gan, Gross and Prasad have formulated conjectural generalizations to higher rank classical groups relating the non-vanishing of central values of certain automorphic L-functions to the non-vanishing of certain explicit integrals of automorphic forms that are called 'automorphic periods'. These predictions have been subsequently refined by Ichino-Ikeda and N.Harris into precise identities relating the two invariants. These conjectures also have local counterparts which concern certain branching laws in the representation theory of real or p-adic groups. Most of these conjectures have now been established for unitary groups. This talk aims to give an introduction to this circle of ideas and to review recent results on the subject.

Abstract:

Come hear recent graduates and current grad students discuss their experiences navigating the job market. This panel is hosted by the Student Number Theory seminar, but all graduate students are welcome to attend.

Abstract:

We continue reviewing minuscule weights, their weight posets, and minuscule heaps, with the goal of eventually understanding Section 4 of the paper by Garver, Patrias and Thomas

Abstract:

The Jacquet-Rallis fundamental lemma is a local identity between (relative) orbital integrals which originates from the relative trace formula approach to the Gan-Gross-Prasad conjecture for unitary groups and is a crucial ingredient in the recent results of W. Zhang on this conjecture. It was established soon after its formulation by Z. Yun in positive characteristic using the same geometric ideas as in Ngô's proof of the endoscopic fundamental lemma and transferred to characteristic 0 by J. Gordon by model-theoretic techniques. In this talk, I will present an alternative proof of this fundamental lemma in characteristic zero which is purely local and based on harmonic analytic tools. We note that a third proof of this fundamental lemma has been recently proposed by W. Zhang through a similar although global argument.

Abstract:

We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary strength supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest as a model for the coherent vortex filament structures observed in 3d fluids, these results are the first to give well-posedness (in a certain sense) in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions which are locally approximately self-similar. Joint work with Pierre Germain and Benjamin Harrop-Griffiths.

Abstract:

We recall the self-homeomorphisms of a closed oriented reducible 3-manifold. Using this description, we discuss various problems in low-dimensional topology and dynamics, such as the existence of Anosov tori in 3-manifolds (joint work with Shicheng Wang), the simplicial volume of mapping tori of 3-manifolds (joint work Michelle Bucher) and the virtual Betti numbers of mapping tori of 3-manifolds.

Abstract:

Adrienne Sands of U of MN will continue on "Classical properties of Epstein zeta functions

Abstract:

Neural networks solve complex computer vision problems with human-like accuracy. However, it has recently been observed that neural nets are easily fooled and manipulated by "adversarial examples," in which an attacker manipulates the network by making tiny changes to its inputs. In this talk, I give a high-level overview of adversarial examples, and then discuss a newer type of attack called "data poisoning," in which a network is manipulated at train time rather than test time. Then, I explore adversarial examples from a theoretical viewpoint and try to answer a fundamental question: "Are adversarial examples inevitable?"

Tom is an Assistant Professor at University of Maryland. His research lies at the intersection of optimization and distributed computing, and targets applications in machine learning and image processing. He designs optimization methods for a wide range of platforms. This includes powerful cluster/cloud computing environments for machine learning and computer vision, in addition to resource limited integrated circuits and FPGAs for real-time signal processing. Before joining the faculty at Maryland, he completed his PhD in Mathematics at UCLA, and was a research scientist at Rice University and Stanford University. He has been the recipient of several awards, including SIAM’s DiPrima Prize, a DARPA Young Faculty Award, and a Sloan Fellowship.

Abstract:

Two Teams of MFM students from the 2019 MFM Modeling Workshp will present their work to the Seminar. Each group will take a half hour to cover their topics, including Q & A

Yield Curve Construction: Calculating the present value of future cash flows is a crucial task for any trading desk. We will explore yield curve bootstrapping and interpolation techniques used to value financial instruments in a consistent framework. The pros and cons of each method will become apparent, and we will converge on an optimal technique that is consistent with major trading desks. Finally, we will see how the financial crisis changed yield curve construction, and fundamentally altered the way we think of the valuing future cash flows.

Valuation and Replication Strategies for Variance Derivatives:Variance derivatives have played a major role in the financial markets since the 1990s, as they provide pure exposure to volatility without other added effects. A primary instrument is variance swaps. In this project the participants will explore traditional theory of valuing variance swaps, and their continuous and discrete replication with vanilla derivatives. The connections among variance, volatility swaps, and VIX derivatives will be drawn, alongside examining non-parametric variance swap replication approaches.

Abstract:

In preparation for my oral exam, I will extend the existing connections between quantum groups and the study of spherical Whittaker models on metaplectic covering groups of GL(r,F), for F a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic n-fold cover of GL(r,F) a set of Yang-Baxter equations govern the behavior of the Whittaker functions and that these equations arise from a Drinfeld twist of a quantum affine Lie superalgebra. I will extend their results to all metaplectic covers of GL(r,F), showing that the same Yang-Baxter equations underlie the scattering matrix for the Whittaker functions over an nQ-fold metaplectic cover, where nQ is an integer determined by the cover, and that these equations arise from Drinfeld twists of quantum groups. (I will also define what most of these words mean.)

Abstract:

We review minuscule weights, their weight posets, and minuscule heaps, with the goal of eventually understanding Section 4 of the paper by Garver, Patrias and Thomas.

Abstract:

In this talk we discuss Vlasov-Poisson-Boltzmann system in bounded domains. Some recent results on regularity and large time behavior of solutions will be presented.

Abstract:

In experimental systems involving diffusing and convecting vapors that react to form solid particulates, a complex sequence of nucleation and growth reactions produces pulsing charged crystals, oscillating fronts, and patterns such as beautiful 3-dimensional structures that we call “microtornadoes”, “microstalagtites”, and “microhurricanes”. We will review the rich history of these experiments, starting with a counterdiffusional experiment that figures in the pioneering work on diffusion of Dalton, Graham, Fick, and Stefan. Mathematical analysis will progress from a reaction-diffusion model for the origins of the initial reaction zone, to an analysis of oscillations and particle size distributions, to a fluid dynamical model. The insights carry over to similar structures in protein crystallization experiments and the formation of periodic structures in plants.

Abstract:

In this talk, we consider decision-making problems under data uncertainty. In particular, we study a framework, called Wasserstein distributionally robust optimization, that aims to find a decision that hedges against a set of distributions that are close to some nominal distribution in Wasserstein metric. This framework, although being an infinite dimensional optimization, has a finite-dimensional tractable reduction in various data-driven settings by virtue of duality, and is closely related to many regularization problems in statistical learning. I will discuss the generalization error bound and asymptotic properties of such framework, and compare it with other distributional uncertainty sets including divergence-based and moment-based sets. If time permits, I will talk about an application of the framework to robust classification with limited information.

Rui Gao is an Assistant Professor of Information, Risk, and Operations Management at the McCombs School of Business at the University of Texas at Austin. He received his Ph.D. in Operations Research from Georgia Institute of Technology in 2018. His current research interests lie in the intersection of decision-making under uncertainty and statistical learning, as well as their applications in data analytics. His work has been recognized with several INFORMS prize, including finalist in Nicholson student paper competition in 2016, runner-up in the Computing Society student paper prize in 2017, and winner of the Data Mining best paper award in 2017.

Abstract:

In a recent beautiful but technical article, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums.

Abstract:

From "is networking really important" to "do hiring managers read cover letters", from "why should data exploration begin with a question" to "how to build a culture of data driven decision making", we have heard it all many, many times. But these are questions that warrant a reminder. And another. And another.

I hope to be that person and remind you of a few key strategies to employ in job search and best practices to avoid pitfalls in the business world. I will share failures and lessons from my experience in data management and analytics, as well as decode marketing function.

Sandhya Surapaneni (aka Sandy) works as a Global Marketing Analytics Leader at Stratasys, a pioneer in 3D printing technology. Her current focus is on building a foundation in analytics to integrate data-driven insights into every day business decisions that align with the company's long term strategy. Prior to joining Stratasys, Sandy worked as a Database Lead at BI Worldwide, building and supporting the data layer of several web applications. Sandy has a M.S in Computer Science from the University of Texas at Arlington and a M.S in Business Analytics from Carlson School of Management. Born and raised in India, she has been a Minneapolis resident for 13 years and enjoys its lakes and coffee shops.

Abstract:

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.

Abstract:

Lagrangian chaos refers to the chaotic behavior of
Lagrangian trajectories in a fluid. This chaotic behavior often
characterized by the dynamics having a positive Lyapunov exponent,
namely the property that initially close trajectories will separate at
an exponential rate after long time. In this talk we will consider a
variety of stochastically forced fluid models, including the 2
dimensional stochastic Navier-Stokes equations, and show that under
certain non-degeneracy conditions on the noise, the Lagrangian flow
possesses a positive Lyapunov exponent. The proof crucially uses the
theory of random dynamical systems as well as tools from Malliavin
calculus and control theory to satisfy a certain non-degeneracy
criterion originally due to Furstenberg. We will explore several
important consequences of a positive exponent for passive scalars
advected by the fluid. Most importantly, we show that these velocity
fields are almost sure exponential mixers. Specifically, using the
positive Lyapunov exponent, we obtain almost sure exponential decay in
time of passive scalars in any negative Sobolev norm through a
detailed study of ergodicity of the two point Lagrangian motion. This
work is joint with Jacob Bedrossian and Alex Blumenthal.

Abstract:

Serre and Swinnerton-Dyer reproved many congruence relations around 1970 for the Ramanujan tau function by reinterpreting the relations in terms of l-adic representations. We will give an introduction to this topic.

Abstract:

We describe how the Jordan form data of quiver representations interact with reflection functors, continuing with the paper of Garver, Patrias and Thomas: Minuscule reverse plane partitions via quiver representations.

Abstract:

In the absence of analytical reconstruction methods, numerical solutions of nonlinear inverse problems have been mostly based on least-square formulations where solutions are sought by minimizing the $L^2$ difference between model predictions and measured data. We present here recent computational studies of some nonlinear inverse problems where quadratic Wasserstein distances are used to measure the discrepancy between model predictions and measured data. Related numerical and theoretical issues will be discussed.

Abstract:

Modern massive datasets pose an enormous computational burden to practitioners. Distributed computation has emerged as a universal approach to ease the burden: Datasets are partitioned over machines, which compute locally, and communicate short messages. Distributed data also arises due to privacy reasons, such as with medical databases. It is important to study how to do statistical inference and machine learning in a distributed setting. In this talk, we present results about one-step parameter averaging in statistical linear models under data parallelism. We do linear regression on each machine, and take a weighted average of the parameters. How much do we lose compared to doing linear regression on the full data? Here we study the performance loss in estimation error, test error, and confidence interval length in high dimensions, where the number of parameters is comparable to the training data size. We discover several key phenomena. First, averaging is not optimal, and we find the exact performance loss. Second, different problems are affected differently by the distributed framework. Estimation error and confidence interval length increases a lot, while prediction error increases much less. These results match numerical simulations and a data analysis example. To derive these results, we rely on recent results from random matrix theory, where we also develop a new calculus of deterministic equivalents as a tool of broader interest.

Edgar Dobriban is an assistant professor of statistics at the Wharton School of the University of Pennsylvania. He obtained his PhD in Statistics in 2017 from Stanford University, advised by David Donoho, and his undergraduate degree in mathematics from Princeton University in 2012. His research interests are in developing statistical methods and theory for large-scale data analysis.

Abstract:

Two Teams of MFM students from the 2019 MFM Modeling Workshop will present their work to the Seminar. Each group will take a half hour to cover their topics, including Q & A

Machine Learning in Equity Classification: This MFM modeling workshop team worked with various machine learning classification models with the goal of classifying equities via well-known quantitative factors such as Value and Momentum. The classification was supervised, utilizing a novel ETF dataset which was supplemented extensively. The team worked in Python, especially the Scikit Learn module. They will present their project to seminar attendees

Smart Beta Investing in Commodities: Ever since the first stocks and bonds were issued by the Dutch East India Company (VOC), investors have tried to understand what drives returns. Smart Beta strategies have gained popularity lately by offering the potential for better-than-market returns with better-defined risks, especially after the recognition in 2008 that multi-asset classes can experience severe losses at the same time despite their apparent intrinsic differences. Smart beta strategies can take many different forms, with a variety of objectives. They can simply aim at reducing risks (the “risk-based approach”) or enhancing return through exposure to systematic factors (the “factor-based” approach). In commodities investing, alternative index movement was born from frustration with the inherent biases of conventional indices. For example, negative commodity “roll yields” can erode returns by as much as 50%. This team explored the opportunities of constructing a commodity investment portfolio that uses different smart beta approaches to seek enhancing returns and risk reduction. Factors like curve, value, and momentum were examined in the back-test.

Abstract:

I was late. I only decided to find a job in industry in the 4th year of my graduate study even though I never had a job outside of academia. The one question that I kept asking was: how do you get a job?

It was scary because I did't know what to do. But I ended up getting a job. And today, I think I'm doing fine.

If you are about to start on the path to industry but you feel that you are full of questions and doubts, then this talk may be for you. In this talk, I will show you two things:

1. my experience of going into industry, and
2. how to thrive in it.

You will walk away with guidelines and tips that I learned from my mistakes and observations. It will help you save time and avoid frustrations in the process.

Chang currently works as a data scientist at Lowe's. He entered into the data science field as an intern for a baseball team, the Tampa Bay Rays. He got his Ph.D in Mathematics from Vanderbilt University in 2017 under Alex Powell. Besides math and data, Chang has become a soul food advocate and survives on barbecue and hot chicken. You can learn more about his experience outside of academia at his blog: https://changhsinlee.com and his YouTube channel: https://youtube.com/c/softwareforscience

Abstract:

Bernstein-Sato polynomials are fundamental in D-module theory. For
example, they are the main finiteness ingredient in the construction of
nearby cycles. We will present a positive characteristic analogue of the
Bernstein-Sato polynomials. After diving in the world of characteristic
p D-modules, we shall consider how our construction varies with the
prime p. This turns out to be related to questions of Hodge theory and
Poisson homology.

Abstract:

Multi-scale kinetic equations can be compressed: in certain regimes, the Boltzmann equation is asymptotically equivalent to the Euler equations, and the radiative transfer equation is asymptotically equivalent to the diffusion equation. A lot of detailed information is lost when a system passes to the limit. In linear algebra, it is equivalent to a system being of low rank. I will discuss such transition and how it affects the computation: mainly, in the forward regime, inserting low-rankness could greatly advances the computation, while in the inverse regime, the system being of low rank typically makes the problems significantly harder.

Abstract:

A quiver Grassmannian is a variety parametrizing subrepresentations of a given quiver representation. Reineke has shown that all projective varieties can be realized as quiver Grassmannians. In this talk, I will study a class of smooth projective varieties arising as quiver Grassmannians for (truncated) preprojective representations of an n-Kronecker quiver, i.e. a quiver with two vertices and n parallel arrows between them. The main result I will present is a recursive construction of cell decompositions for these quiver Grassmannians motivated by the theory of rank two cluster algebras. If there is time I will discuss a combinatorial labeling of the cells by which their dimensions may conjecturally be directly computed. This is a report on joint work with Thorsten Weist.

Abstract:

Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.

Abstract:

In this talk, we will be concerned with the average values of certain random functionals of the path of a stochastic process during a given time period. High-order asymptotic characterizations of such values when the time period shrinks to 0 have a wide range of applications. In statistics, they are instrumental in establishing infill asymptotic properties of high-frequency based statistical methods of stochastic processes. In finance, they have been used as model selection and calibration tools based on near expiration option prices. In some Engineering problems, they also show up as a method to solve a problem in continuous time by looking at the analogous problem in discrete time and shrinking the time step to 0. These short-time asymptotic methods are especially useful in the study of complex models with jumps and stochastic volatility due to the lack of tractable formulas and efficient statistical and numerical procedures. In this talk, I will discuss some recent advances in the area and illustrate their broad relevance in several contexts.

Bio: Dr. Figueroa-López is a Professor in the Department of Mathematics and Statistics at Washington University in St. Louis. He currently serves as part of the executive committee and the chair of the statistics committee of the Department. Formerly he was Associate Professor of Statistics at Purdue University, where he served as Associate Director of the Computational Finance Program and as a member of the University Senate. Professor Figueroa’s ongoing research includes short-time asymptotics of jump-diffusion models, diffusion limits of Limit Order Book models, optimal limit order placement problems, market making via reinforced learning, and optimal tuning of high-frequency based econometric methods. He was awarded the NSF career award in 2012 and currently has two active NSF grants on the interplay of finance, statistics, and probability. He is an Associate Editor of the SIAM Journal on Financial Mathematics (SIFIN) and a former Associate Editor of Electronic Journal of Statistics.

Abstract:

We review the analogy between knots in 3-manifolds and prime ideals in number rings, and push it further to realize the Legendre symbol as the analogue of the (mod 2) linking number. This talk should be accessible even if you did knot attend seminar last week.

Abstract:

This talk is postponed to March 27.

Abstract:

A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field. This has applications to the equicharacteristic case of some conjectures on l-independence. I will also discuss the relationship with compatible wild ramification. This is joint work with Qing Lu.

Abstract:

Principal Components Analysis (PCA), a.k.a. subspace learning, is one of the most widely used dimension reduction techniques that attempts to find a low-dimensional subspace approximation of a given dataset. PCA is a solved problem when the observed data is relatively clean and lies in (or close to) a low-dimensional subspace. However, in many modern applications, the data are often either incomplete (missing data) or corrupted by outliers. Robust PCA refers to this harder problem of PCA in the presence of entry-wise outliers (sparse corruptions). An important example application is video analytics when slow-changing videos are corrupted by foreground occlusions, e.g., by moving vehicles or persons. For long data sequences, e.g., long surveillance videos, if one tries to use a single subspace to represent the entire sequence, the required subspace dimension may be too large. For such data, a better model is to assume that the data subspace can change with time, albeit gradually. This problem of tracking data lying in a slowly changing subspace, while being robust to additive sparse outliers is referred to as Robust Subspace Tracking (RST). While robust PCA has received a lot of attention in the last decade, its dynamic version, RST, was largely open until recently. In a recent body of work, we have introduced the first provably correct and practically usable online solution framework for RST that we call Recursive Projected Compressive Sensing (ReProCS). Our most recent work from ICML 2018 shows that a simple ReProCS-based algorithm provides a provably fast and nearly (delay and memory) optimal RST solution under mild assumptions: weakened standard robust PCA assumptions and subspace change that is slow enough compared to the smallest magnitude outlier entry. Our theoretical claims are also backed by extensive experimental evidence for two video applications.

In new work, we have looked at what can be called the ``Phaseless PCA’’ problem. This involves recovering a low-rank matrix from phaseless (magnitude-only) linear projections of each of its columns. It finds important applications in dynamic phaseless imaging applications, such as dynamic sub-diffraction imaging or Fourier ptychography, involving recovering a set of slowly-changing images that together form an approximately low-rank matrix (with each vectorized image being one column). We introduce a simple alternating minimization solution that can provable recover the low-rank matrix with sa

Abstract:

On the Monday of February 5th 2018, VIX Index (measure of expected future volatility) spiked by 116% to 37.3, and massive turbulence was observed across global financial markets. During the talk, we will review this event and discuss on the following topics:

• How did we get there? --- Low volatility environment and the great snake of risk 
• What happened? --- 2017’s hottest trades went wrong, and several funds and firms (!) were effectively wiped out
• What did we learn? – Positioning, behavior, and never underestimate the risk of financial markets (convexity to volatility and VaR calcs)

Abstract:

Internship/job searching is a multifaceted process that can summon many questions along the way! During this workshop, we will aim to address your most asked questions about getting started with your search, including:
- Where to find internships/jobs to apply for
- What to look for and how to read an internship/job posting
- What you can be doing now and over the summer to help prepare for your search.
There will be time to discuss more questions during the session, so come with your questions in hand!

Whitney Moore has been a Career Counselor in the CSE Career Center at the University of Minnesota since the fall of 2011. Her passion lies in living a life of positivity and inspiring others to do the same, particularly while navigating career and leadership development. In addition to her work at the U of M, Whitney is a past President of the Minnesota College and University Career Services Association (MCUCSA) and holds degrees from Gustavus Adolphus College and Minnesota State University, Mankato. Away from campus, Whitney is motivated by running with her dog, competing in triathlon, eating baked goods, and exploring MN with her husband. As an utter positivist, you can also find her musings on living with positivity on Instagram at @ThePositivistExplorer!