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Mathematics Department

Colloquium Series

Fall 2019

All talks are 3:45-4:45 p.m. in the Colloquium room (Chauvenet 110), unless otherwise specified.

Cookies will be served in the lecture room starting shortly before the talk.

  • Mar
    28
  • [POSTPONED] Complexity Chasms in Equation Solving
    J. Maurice Rojas
    Texas A&M / NSF

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    We discuss an important aspect about solving real polynomial equations that is often swept under the rug: Bit complexity. In particular, we review how it's easy to solve high degree equations with just two monomial terms (e.g., computing d-th roots), and then present new results on the case of more terms: An algorithm for solving trinomial equations in time polynomial in the input size, and explicit families of tetranomial equations provably not solvable in polynomial-time. We also discuss what this means for solving large systems of equations in many variables and smoothed complexity. This is joint work with Yuyu Zhu.
  • Mar
    21
  • [POSTPONED] Pitfalls and paradoxes in the history of probability theory
    Mike Shlesinger
    ONR

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    This lecture traces the history of probability theory from the throwing of bones, sticks, and dice to modern times. Early 18th century books, Jacob Bernouill's "The Art of Conjecturing" and Abraham DeMoivre's "The Doctrine of Chances" were rich with new mathematics, insight and gambling odds. Progress was often made by confronting paradoxes. The first of these confused probabilities with expectations and was explained in the Pascal-Fermat letters of 1654. The St. Petersburg Paradox involved a distribution with an infinite first moment, and Levy discovered a whole class of probabilities with infinite moments that have found a surprising utility in physics involving fractals. The Bertrand paradox involved measure theory for continuous probabilities, Poisson discovered that adding random variables need not always produce the Gaussian, and Daniel Bernoulli and D'Alembert argued over the probabilities for the safety of smallpox vaccinations. Using these and other anecdotes, this lecture discusses vignettes that have brought us to our modern probability theory.
  • Feb
    21
  • Quantitative Diagonalizability
    Time: 12:00 PM

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    A diagonalizable matrix has linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every matrix is a limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta in the operator norm of a matrix whose eigenvectors have condition number poly(n)/delta, confirming a conjecture of E. B. Davies. The proof is based adding a complex Gaussian perturbation to the matrix and studying its pseudospectrum. Finally, we mention a recent application of this result to numerical analysis, yielding the fastest known provable algorithm for diagonalizing an arbitrary matrix. Joint work with J. Banks, A. Kulkarni, S. Mukherjee, J. Garza Vargas.
  • Nov
    20
  • Why is lettuce so wrinkly?
    John Gemmer
    Wake Forest
    Time: 03:45 PM

    View Abstract

    Many patterns in Nature and industry arise from the system minimizing an appropriate energy. Examples range from the periodic rippling in hanging drapes to the six-fold symmetries observed in snowflakes. Torn plastic sheets and growing leaves provide striking examples of pattern forming systems which can transition from single wavelength geometries (leaves) to complex fractal like shapes (lettuce). These fractal like patterns seem to have many length scales - the same amount of extra detail can be seen when looking closer (“statistical self-similarity”). It is a mystery how such complex patterns could arise from energy minimization alone. In this talk I will address this puzzle by showing that such patterns naturally arise from the sheet adopting a hyperbolic non-Euclidean geometry. However, there are many different hyperbolic geometries that the growing leaf could select. I will show using techniques from analysis, differential geometry and numerical optimization that the fractal like patterns are indeed the natural minimizers for the system.
  • Nov
    13
  • Ergodic Theory: The Mathematics of Experiment, Probability and Time
    Time: 03:45 PM

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    An overview of the field of ergodic theory, originally developed to systematize the ideas underlying the scientific method of repeated experiment and observation, focusing on the interplay of probability and time. The presentation will assume very little in the way of background and everyone is encouraged to attend. Rudimentary knowledge of probability and some familiarity with matrices is all that is necessary. Towards the end, I will present (in a high-level form) some of my own results on the behavior of matrices in the probabilistic setting, specifically on the nature of discretizing matrix actions (as is necessary and commonly done when utilizing computers).
  • Oct
    30
  • Lagrange's Identity and Apportionment of the U.S. House of Representatives
    Tommy Wright
    US Census Bureau
    Time: 03:45 PM

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    For very general audiences, this talk makes use of an elementary result known as Lagrange's Identity to provide a bridge between an insightful motivation and an elementary derivation of the method of equal proportions. The method of equal proportions is the current method for apportioning the 435 seats in the U.S. House of Representatives among the 50 states, following each decennial census. We also present some historical comments about the first two methods of apportionment.
  • Oct
    23
  • Semi-infinite flag manifolds via nonsymmetric Macdonald polynomials
    Daniel Orr
    Virginia Tech
    Time: 03:45 PM

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    Homogeneous spaces for Lie groups are fundamental objects in (geometric) representation theory. The semi-infinite flag manifold Q is a particular infinite-dimensional version of the flag manifold X=G/B of a complex simple Lie group. The significance of Q arises in part from its relation to the following: (1) representation theory of affine Lie algebras, and (2) geometry of (quasi-)maps from a genus zero curve to X. Recent developments in the study of these related structures has in turn revealed that the (notoriously difficult) geometry of semi-infinite flag manifolds is encoded by concrete objects from algebraic combinatorics: nonsymmetric Macdonald polynomials. I will explain these developments in down-to-earth terms, using the example of G=SL(2) as a guide.
  • Oct
    16
  • Curvature, spinning tops and other counterintuitive phenomena
    Mark Levi
    Penn State
    Time: 03:45 PM

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    Two of the most counterintuitive gravity-defying effects in mechanics are the spinning top and the upside-down pendulum made stable by vibration of its pivot. I will give a geometrical explanation of these and related effects. It turns out that curvature plays a key role in both of these effects. I will also pose a related open problem of explaining the so-called “ponderomotive magnetism” discovered in the last couple of years.
  • Oct
    09
  • Practical Calculations for Matrix Groups
    Alexander Hulpke
    Colorado State University
    Time: 03:45 PM

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    Over the last few years, matrix group algorithms over finite fields have reached a certain maturity both in theory as well as in practical applicability. I will describe the underlying ideas, show how to use them in practice, and illustrate how one can extend these to work with matrix groups over residue class ring or the integers. No prior knowledge of group theoretic algorithms (or of group theory beyond an introductory algebra class) will be needed.
  • Sep
    25
  • Algebraic geometry and post-quantum cryptography
    Gretchen Matthews
    Virginia Tech
    Time: 03:45 PM

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    How do we store private information? How do we communicate information securely? Answers to these questions are changing as computational capabilities change. Addressing them is vital not only to our national security but also our everyday existence, impacting commerce, healthcare, and the ways we interact with one another. Quantum computing poses a threat to current encryption schemes, such as RSA and elliptic curve cryptography, which underpin nearly all digital transactions. Public key encryption as we know it succumbs to Shor’s Algorithm, making a replacement necessary. For this reason, the National Institute of Standards and Technology (NIST) has issued a call for cryptosystems which are post-quantum secure, meaning are resilient in the face of quantum algorithms. We share modernizations of McEliece’s 1978 code-based cryptosystems which are based on polynomials and provide robust post-quantum security for classical information. The McEliece cryptosystem utilizes error-correcting codes to keep private information secure, and its effectiveness depends on the properties of the underlying codes. In this talk, we consider the use of algebraic geometric (AG) codes in the McEliece cryptosystem and demonstrate modifications of traditional AG codes that yield superior performance.
  • Sep
    18
  • On Geodesic Triangles in the Hyperbolic Plane
    Rita Gitik
    U. Michigan
    Time: 03:45 PM

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    Let M be an orientable hyperbolic surface without boundary and let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c.
  • Sep
    11
  • Homotopy types as a foundation for mathematics
    Emily Riehl
    Johns Hopkins
    Time: 03:45 PM

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    The Curry-Howard correspondence formalizes an analogy between computer programs and mathematical proofs. This talk will introduce alternative foundations for mathematics animated by this analogy. The basic object is called a type, which can be simultaneously interpreted as something like a set or as something like a mathematical proposition. Homotopy type theory refers to the recent discovery that a type can also be interpreted as something like a topological space. We will discuss the implications of this homotopy theoretic interpretation for the so-called univalent foundations of mathematics.
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