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Upcoming Colloquia & Seminars

  1. Harmonic Analysis Seminar: On boundary value problems for parabolic equations with time-dependent measurable coefficients

    Location: MSRI: Simons Auditorium
    Speakers: Pascal Auscher (Université de Paris XI)

    We will explain the proof of a Carleson measure estimate on solutions of parabolic equations with real measurable time-dependent coefficients that implies that the parabolic measure is an $A_\infty$ weight.
    This corresponds to the parabolic analog of a recent result by Hofmann, Kenig, Mayboroda and Pipher for elliptic equations. Our proof even simplifies theirs. As is well known, the $A_\infty$ property implies that   $L^p$ Dirichlet problem is well-posed. An important ingredient of the proof is a Kato square root property for parabolic operators on the boundary, which can be seen as a consequence of certain square function estimates applicable to Neumann and regularity problems.  All this is  joint work with Moritz Egert and Kaj Nyström.

    Updated on Apr 19, 2017 01:24 PM PDT
  2. Analytic Number Theory Graduate Student Seminar

    Location: MSRI: Baker Board Room
    Speakers: Corina Panda (California Institute of Technology), Vinay Viswanathan (University of Bristol)

    4:00pm: Vinay Kumaraswamy

    Title: On correlations between class numbers of imaginary quadratic fields

    Abstract: Let h(-d) denote the class number of the imaginary quadratic field Q(\sqrt{-d}).  Moments of class numbers​ have been studied in the past, and are well understood. In this talk, I will speak about obtaining an asymptotic formula for the shifted sum \sum_{d \leq X} h(-d)h(-d-l), where l is a positive integer; the proof makes use of the smooth delta-symbol.
    4:30pm: Corina Panda
    A generalization of a theorem of Hecke for SL_2(F_p) to fundamental discriminants
    ​ ​
    Let p > 3 be an odd prime, p ≡ 3 mod 4 and let π+, π− be the pair of cuspidal representations of SL2(Fp). It is well known by Hecke that the difference mπ+ − mπ− in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ(p), equals the class number h(−p) of the imaginary quadratic field Q(\sqrt(−p)). We extend this result to all fundamental discriminants −D of imaginary quadratic fields Q( \sqrt(−D)). The proof uses the holomorphic Lefschetz number.
    Updated on Apr 20, 2017 12:13 PM PDT
  3. Harmonic Analysis Graduate Student Seminar: Rough path theory and Harmonic Analysis

    Location: MSRI: Baker Board Room
    Speakers: Gennady Uraltsev (Rheinische Friedrich-Wilhelms-Universität Bonn)

    In this talk we will cover the basics of Rough Path theory. We will make several examples of how these results can be used in harmonic analysis problems (e.g. proving a variational non-linear Hausdorff Young inequality) and, maybe we will see how harmonic analysis results come into the theory of rough paths.

    Updated on Apr 20, 2017 02:40 PM PDT
  4. Analytic Number Theory Seminar: Trace Inequalities and Non-vanishing of L-functions

    Location: MSRI: Simons Auditorium
    Speakers: Dinakar Ramakrishnan (California Institute of Technology)
    Let f be a holomorphic newform of prime level N, weight 2 and trivial character, for example one associated to a elliptic curve E over Q. For any imaginary quadratic field K of discriminant -D in which N is inert, and an ideal class character \chi of K, one is led to the ubiquitous Rankin-Selberg L-function L(s, f x g_\chi), where g_\chi is the modular form of level D associated to \chi by Hecke. It is well known that the central value L(1/2, f x g_\chi) is non-zero for "many" (D, \chi), which is a consequence of equidistribution of special points. The object of this talk is to indicate how to derive a strengthening of this, namely that if we fix an f as above together with a finite number of even Dirichlet characters \eta_1, ..., \eta_r, then one can find many (D,\chi) for which one has the simultaneous non-vanishing of L(1/2, (f.\eta_j) x g_\chi) for all j. The additional ingredient used here is an inequality of traces for tori relative to a non-abelian twist.
    The talk will hopefully be accessible to a variety of mathematicians.
    Updated on Apr 20, 2017 12:17 PM PDT
  5. ANT Postdoc Seminar: Integer partitions and restricted partition functions

    Location: MSRI: Simons Auditorium
    Speakers: Ayla Gafni (University of Rochester)

    The theory of integer partitions is a rich subject that lives in the intersection of number theory and combinatorics.  In this colloquium-style talk, I will go through a brief history of partitions and the various tools used to study them, along with connections to Waring's problem and other topics in additive number theory.  I will then state some results about counting partitions in which the parts are restricted to various subsets of the integers (e.g., primes, squares, arithmetic progressions).  

    Updated on Apr 21, 2017 11:26 AM PDT
  6. HA Postdoc Seminar: Variational Methods for a Two-Phase Free Boundary Problem For Harmonic Measure (Colloquium Talk)

    Location: MSRI: Simons Auditorium
    Speakers: Max Engelstein (Massachusetts Institute of Technology)

    There are lots of very good techniques for studying the regularity of a minimizer of some functional (think harmonic functions, minimal surfaces etc). But what if you want to study something that doesn't minimize a functional? We will show how GMT and harmonic analysis can help us use shiny tools from the calculus of variations in a non-variational setting. Some of what we will talk about is joint work with Matthew Badger and Tatiana Toro. 

    Updated on Apr 19, 2017 12:30 PM PDT
  7. Joint ANT & HA Seminar: Polynomial congruences: Some light entertainment

    Location: MSRI: Simons Auditorium
    Speakers: James Wright (University of British Columbia)

    Exponential sums over Z or Z^d are basic objects in Analytic Number Theory and oscillatory integrals over R or R^d are basic objects in Harmonic Analysis. These objects are quite different; for oscillatory integrals over R, a single continuum of scales is often sufficient for the analysis whereas for exponential sums over Z, every prime p gives rise to a family of scales {p^k}, all needed in the analysis. Nevertheless if one fixes the prime p and carries out the analysis at the corresponding scales (e.g.  by examining exponential sums over Z/p^k Z,  k=1,2,3,...) then the analogy to oscillatory integrals in euclidean settings is uncanny.

    We will illustrate this in the simple setting of polynomial congruences and formulate some problems in elementary number theory in a way that harmonic analysts can appreciate and be able to use their prior acquired intuition. 

    Updated on Apr 20, 2017 12:15 PM PDT
  8. Harmonic Analysis Seminar

    Location: MSRI: Baker Board Room
    Created on Apr 10, 2017 04:28 PM PDT
  9. Harmonic Analysis Seminar: On the HRT Conjecture

    Location: MSRI: Baker Board Room
    Speakers: Kasso Okoudjou (University of Maryland)

    Given a non-zero square integrable function $g$ and a subset  $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \R^2$,  let $$\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N.$$  The Heil-Ramanathan-Topiwala (HRT) Conjecture asks whether  $\mathcal{G}(g, \Lambda)$ is linearly independent. For the last two decades, very little progress has been made in settling the conjecture. In the first part of the talk, I will give an overview of the state of the conjecture. I will then describe a small variation of the conjecture that asks the following question: Suppose that the HRT conjecture holds for a given $g\in L^{2}(\R)$ and a given set $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \R^2$. Give a characterization of all points $(a, b)\in \R^2\setminus \Lambda$ such that the conjecture remains true for the same function $g$ and the new set of point $\Lambda_1=\Lambda\cup\{(a, b)\}$. If time permits I will illustrate this approach for the cases  $N=4$, and  $5$ and when $g$ is a real-valued function. 

    Updated on Apr 20, 2017 12:24 PM PDT
  10. Harmonic Analysis Seminar: Haar expansions in Sobolev spaces

    Location: MSRI: Simons Auditorium
    Speakers: Andreas Seeger (Technische Universität Darmstadt)

    Consider expansions with respect to the Haar system on the real line, for functions  in  Sobolev spaces with small smoothness parameter. We report on work with Gustavo Garrig\'os and with Tino Ullrich and answer the following questions: Is the Haar system a Schauder basis?  Is it an unconditional basis?  We discuss  the boundedness of multiplier transformations and other quantitative versions of these questions.

    Updated on Apr 21, 2017 11:28 AM PDT
  11. HA Postdoc Seminar

    Location: MSRI: Simons Auditorium
    Updated on Apr 19, 2017 12:31 PM PDT
  12. Harmonic Analysis Seminar: A Sharp Divergence Theorem in Rough Domains and Applications

    Location: MSRI: Simons Auditorium
    Speakers: Marius Mitrea (University of Missouri)

    Arguably, one of the most basic results in analysis is Gauss' Divergence Theorem. Its original formulation involves mildly regular domains and sufficiently smooth vector fields (typically both of class C^1), though applications to rougher settings have prompted various generalizations. One famous extension, due to De Giorgi and Federer, lowers the regularity assumptions on the underlying domain to a mere local finite perimeter condition. While geometrically this is in the nature of best-possible, the De Giorgi- Federer theorem still asks that the intervening vector field has Lipschitz components. The latter assumption is, however, unreasonably strong, both from the point of view of the very formulation of the Divergence Formula, and its applications to PDE's which often involve much less regular functions. In my talk I will discuss a refinement which addresses this crucial issue, through the use of tools and techniques from Harmonic Analysis (Whitney decompositions, weighted isoperimetric inequalities, non-tangential maximal operators). In particular, this sharpened form of the Divergence Theorem yields a variety of refined results, from the nature of the Green function, to the behavior of singular integral operators in very general domains.

    Updated on Apr 18, 2017 11:01 AM PDT
  13. MSRI/Pseudorandomness seminar

    Location: MSRI: Simons Auditorium
    Updated on Feb 16, 2017 02:37 PM PST
  14. ANT Postdoc Seminar

    Location: MSRI: Simons Auditorium
    Created on Feb 02, 2017 12:03 PM PST
  15. HA Postdoc Seminar

    Location: MSRI: Simons Auditorium
    Created on Feb 02, 2017 12:05 PM PST
  16. Joint ANT & HA Seminar

    Location: MSRI: Simons Auditorium
    Created on Feb 02, 2017 12:01 PM PST