Location: 384I. (Was in 380F in autumn 2014!)
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Tuesday, September 2, 11am-noon, 383N (unusual date/time/room):
Jan Dereziński (University of Warsaw)
``On the excitation spectrum of the Bose gas''
Abstract: One of the most beautiful chapters of theoretical physics is the
analysis of excitation spectrum of the Bose gas due to Bogoliubov. Clearly,
this analysis is nonrigorous. However it has been qualitatively confirmed by
experiments and underlies our understanding of one of the most remarkable
physical phenomena -- the superfluidity.
I will describe a rigorous result saying that, appropriately interpreted,
Bogoliubov's picture is correct in a certain limit that involves a large
number of particles and large volume. According to this result, in this
limit the joint spectrum of the many-body Schrödinger Hamiltonian and the
momentum has a special shape predicted by Bogoliubov.
I do not expect that everybody in the audience is familiar with the physics
jargon. The results and tools that I am going to describe can be understood
and appreciated purely in the context of spectral theory of operators on
Hilbert spaces.
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Friday, September 26, 1:05pm, 380W (unusual date/time/room):
Svitlana Mayboroda (Minnesota)
``Dirichlet problem for elliptic operators with rough coefficients and L-harmonic measure''
Abstract: Sharp estimates for the solutions to elliptic PDEs in $L^\infty$ in
terms of the corresponding norm of the boundary data follow directly
from the maximum principle. It holds on arbitrary domains for all
(real) second order divergence form elliptic operators $- \mathrm{div} A
\nabla$. The well-posedness of boundary problems in $L^p$, $p<\infty$,
is a far more intricate and challenging question, even in a
half-space. In particular, it is known that some smoothness of $A$ in
$t$, the transversal direction to the boundary, is needed.
In the present work we establish the well-posedness in $L^p$ of the Dirichlet problem for all divergence form elliptic equations with real (possibly non-symmetric) coefficients independent on the transversal direction to the boundary. Equivalently, we show that for all such operators the $L$-harmonic measure is quantifiably absolutely continuous with respect to the Lebesgue measure. The lack of smoothness and lack of symmetry in the coefficients defy most of the previously known methods. We introduce a new strategy and use the celebrated Kato problem estimate, adapted Hodge decomposition, square function/non-tangential maximal function bounds, epsilon-approximability and $A^\infty$ criteria for harmonic measure, among other tools.
This is joint work with S. Hofmann, C. Kenig, and J. Pipher.
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- 3:15-4:15pm, Room 380X:
Jonathan Pfaff (Stanford)
``Analytic torsion on locally symmetric spaces''
Abstract:
We will introduce the analytic torsion on
locally symmetric spaces and describe some of its asymptotic properties.
The talk will focus on the analytic aspects of the theory. In particular,
we will describe the torsion in the non-compact, finite-volume setting as
well as a gluing formula for the torsion in that case.
- 4:30-5:30pm, Room 380X:
Persi Diaconis (Stanford)
``Dirichlet Eigenvectors in Probability''
Abstract: Spectral theory (even microlocal analysis) has been very useful for
studying the long time behavior of recurrent Markov chains. The
quantitative theory of absorbing Markov chains is in its infancy. Here
there is a `quasi-stationary distribution' (the first Dirichlet
eigenfunction) and one may ask `How close are we to quasi-stationarity
if the process has not been absorbed up to time n?' One can also ask
about time to absorption (the top Dirichlet eigenvalue) and the
`shape' of the quasi-stationary distribution (McKeen-Vlasov
equation). In joint work with Laurent Miclo, we show how the Doob
transform produces a recurrent chain with upper and lower bounds for
rates of convergence.
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Monday, October 13:
Gerardo Mendoza (Temple University)
``BVP for first order elliptic wedge operators''
Abstract: Elliptic wedge operators on compact manifolds $\mathcal M$ with
fibered boundary are linear differential operators whose structure
resembles that of a regular differential operator written in
cylindrical coordinates. Such operators, in the elliptic case, behave
very much like regular elliptic operators. In this talk, an account of
joint work with T. Krainer, I will describe how to set up boundary
values problems for such operators in the first order case,
maintaining throughout a perspective closely paralleling that of
regular boundary value problems. The main points will be: a discussion
of boundary values, a needed generalization of the Douglis-Nirenberg
calculus and of anisotropic Sobolev spaces with varying regularity,
and finally, how to set up a boundary value problem with an
explanation of how the Lopatinski-Shapiro (or more generally, the APS)
condition is incorporated into the problem so as to give
well-posedness.
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Tuesday, October 14, 4pm, Room 380F (note unusual day):
Sijue Wu (Michigan)
``On water waves with angled crests''
Abstract: We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-trivial angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem, and we show that the two-dimensional water wave problem is solvable locally in time in this framework. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with $-\partial P/\partial n \ge 0$, instead of the strong Taylor stability criterion $-\partial P/\partial n \ge c > 0$. This work is partially joint with Rafe Kinsey.
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Monday, November 17, 4:10pm (ten minutes later than usual!):
Jared Wunsch (Northwestern)
``High frequency estimates for Helmholtz problems and applications to boundary integral equations''
Abstract: I will discuss joint work with Dean Baskin and Euan Spence on sharp high-frequency estimates for various Helmholtz boundary problems that are motivated by problems in numerical analysis.
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Friday, November 21, Bay Area Microlocal Analysis Seminar at
Berkeley:
- 2:30-3:30pm, 736 Evans Hall
Elon Lindenstrauss (Hebrew
University of Jerusalem/MSRI)
``Quantum ergodicity on the
sphere and averaging operators''
Abstract: The quantum ergodicity theorem of Shnirelman, Colin de Verdiere and Zelditch gives that for an orthonormal sequence of eigenfunctions of the laplacian on a compact manifold with ergodic geodesic, outside a density one subsequence, the eigenfunctions equidistribute.
The geodesic flow on the sphere is very much not ergodic, and indeed quantum ergodicity (QE) fails on the sphere for the standard sequence of spherical harmonics. On the other hand Zelditch has shown QE holds for a random orthonormal basis in this case.
We prove QE for joint eigenfunctions of laplacian and an averaging operator over a finite collection of rotations (with some restrictions). We also give a new approach to a QE theorem on graphs by Anantharaman and Le Masson.
Joint work with Shimon Brooks and Etienne Le Masson.
- 4:10-5pm, 736 Evans Hall
Nils Dencker (Lund)
``The
solvability and range of differential equations''
In the 50's, the consensus was that all linear PDEs were
solvable. Therefore it was a great surprise in 1957 when Hans Lewy
presented a complex vector field that is not solvable
anywhere. Hörmander then proved in 1960 that in fact linear partial
differential equations generically are not solvable. For nonsolvable
equations the range has infinite codimension, and Hörmander proved in 1963 that nonsolvable complex vector fields are determined by their range, up to right multiplication by functions.
We shall generalize this to nonsolvable systems of differential equations of constant characteristics and principal type, including scalar equations. We show that the ranges of these equations determine the Taylor expansions of the coefficients at minimal bicharacteristics, up to right composition by differential equations. The minimal bicharacteristics are the smallest sets on which the equation is not solvable. This is joint work with Jens Wittsten.
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Monday, December 1:
Raphael Ponge (Seoul/Berkeley)
``Noncommutative Geometry and Conformal Geometry''
Abstract: This will be report on a series of joint papers with Hang Wang (U.
Adelaide, Australia). The general aim of these papers is to use tools from
noncommutative geometry to study conformal geometry and some noncommutative
incarnations of conformal geometry. In this talk, I will present three main
results related to conformal geometry. The first main result is a
reformulation of the local index formula in the setting of
conformal-diffeomorphism invariant geometry, i.e., in settings of action of
group of diffeomorphism preserving a given conformal structure. The second
main result is the construction of a new family of global conformal
invariants taking into account the action of the group of
conformal-diffeomorphisms. These invariants are not of the same type as the
conformal invariants considered by Spyros Alexakis in his celebrated
solution of the Deser-Schwimmer conjecture. The third result is a version of
Vafa-Witten inequality in conformal geometry, i.e., we obtain a precise
control of the Vafa-Witten bound under conformal changes of metrics.
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Tuesday, January 13, 4:15, Room 384H (NOTE UNUSUAL DAY/TIME/ROOM):
Spyros Alexakis (Toronto)
``Reconstruction of motion in general relativity from emitted radiation''
Abstract: The problem of motion of bodies in general relativity dates back to the early days of the theory.
In the slow-motion approximation, the derivation of the equations of motion to first post-Newtonian order
is due to Einstein, Infeld and Hoffman, with much more precise approximations obtained since.
Motion also manifests itself via the emission of energy towards infinity, in the form of gravitational waves;
the latter prevents the existence of time-periodic solutions, which exist in Newtonian gravity.
Our main result is that any solution of the Einstein equations can (locally) be reconstructed from the radiation it emits towards future and past infinities.
A consequence is that any time-periodic (or non-radiative) solution of Einstein's equations must be stationary (thus, no motion). Joint work
with Volker Schlue.
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Monday, January 26:
Michał Wrochna (Grenoble, visiting Stanford)
``Quantum fields on curved spacetime: the microlocal point of
view'' (Note: this will be a series of about 4 lectures.)
Abstract:
Quantum fields are operator-valued solutions of hyperbolic PDEs,
subject to a list of physically motivated `axioms'. The study of
particle creation on curved spacetime and the seek for a theory that
would describe an interaction between quantum degrees of freedom and
geometry motivate the need for a detailed construction of quantum
states. Since the works of Radzikowski and Brunetti & Fredenhagen,
this problem is formulated in the language of microlocal analysis and
its connection to the theorems of Duistermaat and Hörmander on
distinguished parametrices has been made. In this series of talk I
intend to give an overview from physical motivation to detailed PDE
aspects and point towards the open problems.
I will first introduce the basic concepts of Quantum Field Theory and
spacetime geometry. In the case of scalar fields I will explain how
physical axioms lead to the problem of constructing solutions to the
Klein-Gordon equation with specific wave front set and positivity
properties, show how this amounts to finding a Feynman propagator and
review the existing approaches. I will focus on a method proposed in a
joint work with Gerard and dating back to Junker, where the
Klein-Gordon equation is diagonalized by means of pseudodifferential
operators. Next, I will focus on pure and thermal states, discuss
their role in the Unruh and Hawking effect and how one constructs the
corresponding solutions on (black hole) spacetimes with bifurcate
Killing horizons. The last part will serve as an invitation to
interacting QFT, including perspectives for renormalization by
regularized traces of pseudodifferential operators.
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Monday, February 2:
Vladislav Pukhnachev (Lavrentyev Institute of Hydrodynamics and Novosibirsk State University)
``Flow-through problem for the Navier-Stokes equations (the Jean
Leray problem)''
Abstract available!
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Monday, February 9:
Michał Wrochna (Grenoble, visiting Stanford)
``Quantum fields on curved spacetime: the microlocal point of
view'' (Note: this is part 2 of the lecture series.)
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Monday, February 23:
Michał Wrochna (Grenoble, visiting Stanford)
``Quantum fields on curved spacetime: the microlocal point of
view'' (Note: this is part 3 of the lecture series.)
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Tuesday, February 24, Bay Area Microlocal Analysis Seminar at
Berkeley:
- 2:10-3:00pm, 740 Evans Hall
Boaz Haberman (Berkeley)
``Recovering a gradient term from boundary measurements''
Abstract.
- 3:40-4:30pm, 740 Evans Hall
Michał Wrochna (Grenoble and
Stanford)
``Characteristic Cauchy data of positive-frequency
solutions of the wave equation''
Abstract.
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Monday, April 6:
Michał Wrochna (Grenoble, visiting Stanford)
``Quantum fields on curved spacetime: the microlocal point of
view'' (Note: this is part 4 of the lecture series.)
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Friday, April 17, at
Stanford:
- 3:15-4:15pm, Room 384I
Xuwen Zhu
(MIT/Stanford)
``Resolution of the canonical fiber metrics
for a Lefschetz fibration''
Abstract:
We consider the family of constant curvature fiber metrics
for a Lefschetz fibration with regular fibers of genus greater than
one. A result of Obitsu and Wolpert is refined by showing that on an
appropriate resolution of the total space, constructed by iterated
blow-up, this family is log-smooth, i.e. polyhomogeneous with integral
powers but possible multiplicities, at the preimage of the singular
fibers in terms of parameters of size comparable to the length of the
shrinking geodesic. This is joint work with Richard Melrose.
- 4:30-5:30pm, Room 384I
Semyon Dyatlov (MIT)
``Spectral gaps via additive combinatorics''
Abstract:
The spectral gap on a noncompact Riemannian manifold is an asymptotic strip
free of resonances (poles of the meromorphic continuation of the resolvent
of the Laplacian). The existence of such gap implies exponential decay of
linear waves, modulo a finite dimensional space; in a related case of
Pollicott--Ruelle resonances, a spectral gap gives an exponential remainder
in the prime geodesic theorem.
We study spectral gaps in the classical setting of convex co-compact
hyperbolic surfaces, where the trapped set is a fractal set of dimension
$2\delta + 1$. We obtain a spectral gap when $\delta=1/2$ (as well as for
some more general cases). Using a fractal uncertainty principle, we express
the size of this gap via an improved bound on the additive energy of the
limit set. This improved bound relies on the fractal structure of the limit
set, more precisely on its Ahlfors--David regularity, and makes it possible
to calculate the size of the gap for a given surface. Joint work with Joshua
Zahl.
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Monday, April 20, Room 380Y TBC (unusual room):
Ciprian Manolescu (UCLA)
``The equivalence of two Seiberg-Witten Floer homologies'' (Note:
this is part I of a series of two talks; talk 2 is in the geometry seminar.)
Abstract: The Seiberg-Witten equations are a system of elliptic PDEs that
one can write on a four-dimensional manifold. If Y is a three-manifold, the
Seiberg-Witten equations on R x Y can be used to define an invariant called
Seiberg-Witten Floer homology. I will discuss two different constructions of
this invariant: one (due to Kronheimer and Mrowka) based on finding generic
tame perturbations of the equations, and another one based on finite
dimensional approximation. I will also explain work in progress with Tye
Lidman proving the equivalence of these definitions. This equivalence can be
combined with the recently proved isomorphism between the Kronheimer-Mrowka
theory and its symplectic counterpart, Heegaard Floer homology. Putting the
two equivalences together, we obtain concrete applications to topology--for
example, a relation between two basic constructions of three-manifolds:
covering spaces and surgery.
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Tuesday, April 21, 4pm, Room 383N (unusual time/room):
Jonathan Luk (Cambridge)
``Null singularities in general relativity''
Abstract:
I will present some recent developments in constructing stable low-regularity solutions to the Einstein equations without any symmetry assumptions. These in particular include stable spacetimes which contain singularities propagating along null hypersurfaces. I will discuss some applications of these techniques for various problems in general relativity, including understanding the interaction of impulsive
gravitational waves, the formation of trapped surfaces and the weak
null singularities in the interior of black holes. This talk is based
on results obtained in collaboration with M. Dafermos and
I. Rodnianski.
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Monday, May 4:
Paul Feehan (Rutgers)
``Global existence and convergence of smooth solutions to
Yang-Mills
gradient flow over compact four-manifolds''
Abstract: We develop new results on global existence and convergence
of solutions to the gradient flow equation for the Yang-Mills energy
functional on a principal bundle, with compact Lie structure group,
over a closed, four-dimensional, Riemannian, smooth manifold,
including the following. If the initial connection is close enough to
a minimum of the Yang-Mills energy functional, in a norm or energy
sense, then the Yang-Mills gradient flow exists for all time and
converges to a Yang-Mills connection. If the initial connection is
allowed to have arbitrary energy but we restrict to the setting of a
Hermitian vector bundle over a compact, complex, Hermitian (but not
necessarily Kaehler) surface and the initial connection has curvature
of type (1,1), then the Yang-Mills gradient flow exists for all time,
though bubble singularities may (and in certain cases must) occur in
the limit as time tends to infinity. The
Lojasiewicz-Simon gradient inequality plays a crucial role in our
approach and we develop two versions of that inequality for the
Yang-Mills energy functional.