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Lectures
Seminar information archive ～04/25｜Next seminar｜Future seminars 04/26～
Seminar information archive
2020/01/09
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Javier Fresan (Ecole Polytechnique)
Nori motives over function fields and period functions (ENGLISH)
Javier Fresan (Ecole Polytechnique)
Nori motives over function fields and period functions (ENGLISH)
[ Abstract ]
Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs
algebraic relations between period functions.
This last part is based on an ongoing work with Peter Jossen.
Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs
algebraic relations between period functions.
This last part is based on an ongoing work with Peter Jossen.
2018/09/21
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Laurent Fargues (CNRS, Institut Mathématique de Jussieu)
On the geometry of some p-adic period domains (ENGLISH)
Laurent Fargues (CNRS, Institut Mathématique de Jussieu)
On the geometry of some p-adic period domains (ENGLISH)
[ Abstract ]
p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction. The talk will be mainly introductory, presenting the objects showing up in this theorem. This is joint work with Miaofen Chen and Xu Shen.
p-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction. The talk will be mainly introductory, presenting the objects showing up in this theorem. This is joint work with Miaofen Chen and Xu Shen.
2018/09/18
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Alexander Beilinson (University of Chicago)
Topological epsilon-factors (ENGLISH)
Alexander Beilinson (University of Chicago)
Topological epsilon-factors (ENGLISH)
[ Abstract ]
I will explain (following mostly my old article arXiv:0610055) how the Kashiwara-Shapira Morse theory construction of the characteristic cycle of a constructible R-sheaf can be refined to yield the cycle with coefficients in the K-theory spectrum K(R). The construction can be viewed as a topological analog of the arithmetic theory of epsilon-factors.
I will explain (following mostly my old article arXiv:0610055) how the Kashiwara-Shapira Morse theory construction of the characteristic cycle of a constructible R-sheaf can be refined to yield the cycle with coefficients in the K-theory spectrum K(R). The construction can be viewed as a topological analog of the arithmetic theory of epsilon-factors.
2018/07/10
15:00-16:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Sam Nariman (Northwestern University)
On the moduli space of flat symplectic surface bundles
Sam Nariman (Northwestern University)
On the moduli space of flat symplectic surface bundles
[ Abstract ]
There are at least three different approaches to construct characteristic invariants of flat symplectic bundles. Reznikov generalized Chern-Weil theory for finite dimension Lie groups to the infinite dimensional group of symplectomorphisms. He constructed nontrivial invariants of symplectic bundles whose fibers are diffeomorphic to complex projective spaces. Kontsevich used formal symplectic geometry to build interesting classes that are not yet known to be nontrivial. Also for surface bundles whose holonomy groups preserve the symplectic form, Kotschick and Morita used the flux homomorphism to construct many nontrivial stable classes.
In this talk, we introduce infinite loop spaces whose cohomolgy groups describe the stable characteristic invariants of symplectic flat surface bundles. As an application, we give a homotopy theoretic description of
Kotschick and Morita's classes and prove a result about codimension 2 foliations that implies the nontriviality of KM classes.
There are at least three different approaches to construct characteristic invariants of flat symplectic bundles. Reznikov generalized Chern-Weil theory for finite dimension Lie groups to the infinite dimensional group of symplectomorphisms. He constructed nontrivial invariants of symplectic bundles whose fibers are diffeomorphic to complex projective spaces. Kontsevich used formal symplectic geometry to build interesting classes that are not yet known to be nontrivial. Also for surface bundles whose holonomy groups preserve the symplectic form, Kotschick and Morita used the flux homomorphism to construct many nontrivial stable classes.
In this talk, we introduce infinite loop spaces whose cohomolgy groups describe the stable characteristic invariants of symplectic flat surface bundles. As an application, we give a homotopy theoretic description of
Kotschick and Morita's classes and prove a result about codimension 2 foliations that implies the nontriviality of KM classes.
2018/06/22
16:00-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Michael Harrison (Lehigh University)
Fibrations of R^3 by oriented lines
Michael Harrison (Lehigh University)
Fibrations of R^3 by oriented lines
[ Abstract ]
Is it possible to cover 3-dimensional space by a collection of lines, such that no two lines intersect and no two lines are parallel? More precisely, does there exist a fibration of R^3 by pairwise skew lines? We give some examples and provide a complete topological classification of such objects, by exhibiting a deformation retract from the space of skew fibrations of R^3 to its subspace of Hopf fibrations. As a corollary of the proof we obtain Gluck and Warner's classification of great circle fibrations of S^3. We continue with some recent results regarding contact structures on R^3 which are naturally induced by skew fibrations. Finally, we discuss fibrations of R^3 which may contain parallel fibers, and discuss when such objects induce contact structures.
Is it possible to cover 3-dimensional space by a collection of lines, such that no two lines intersect and no two lines are parallel? More precisely, does there exist a fibration of R^3 by pairwise skew lines? We give some examples and provide a complete topological classification of such objects, by exhibiting a deformation retract from the space of skew fibrations of R^3 to its subspace of Hopf fibrations. As a corollary of the proof we obtain Gluck and Warner's classification of great circle fibrations of S^3. We continue with some recent results regarding contact structures on R^3 which are naturally induced by skew fibrations. Finally, we discuss fibrations of R^3 which may contain parallel fibers, and discuss when such objects induce contact structures.
2018/06/12
15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Alexander Kupers (Harvard University)
Cellular E_2-algebras and the unstable homology of mapping class groups
Alexander Kupers (Harvard University)
Cellular E_2-algebras and the unstable homology of mapping class groups
[ Abstract ]
We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.
We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new "multiplicative" approach to homological stability -- in contrast to the "additive" one due to Quillen -- which has the advantage of providing information outside of the stable range.
2018/05/11
13:00-14:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Alex Youcis (University of California, Berkeley)
The Langlands-Kottwitz method for deformation spaces of Hodge type
Alex Youcis (University of California, Berkeley)
The Langlands-Kottwitz method for deformation spaces of Hodge type
[ Abstract ]
Cohomology of global Shimura varieties is an object of universal importance in the Langlands program. Given a Shimura datum (G,X) and a (sufficiently nice) representation ¥xi of G, one obtains an l-adic sheaf F_{¥xi,l} on Sh(G,X) with a G(A_f)-structure. Thus, in the standard way, the cohomology group H^*(Sh(G,X),F_¥xi) has an admissible action of Gal(¥overline{E}/E) ¥times G(A_f), where E=E(G,X) is the reflex field of (G,X). Extending work of Kottwitz, Scholze, and others we discuss a method for computing the traces of this action, more specifically of an element ¥tau ¥times g where ¥tau ¥in W_{E_¥mathfrak{p}} for some prime ¥mathfrak{p} of E dividing p and g ¥in G(A_f^p) ¥times G(Z_p), in terms of a weighted point count on the Shimura variety's special fiber, as well as the traces of various local Shimura varieties over E_¥mathfrak{p}, at least in the case when (G,X) is a abelian-type Shimura datum unramified at p.
Cohomology of global Shimura varieties is an object of universal importance in the Langlands program. Given a Shimura datum (G,X) and a (sufficiently nice) representation ¥xi of G, one obtains an l-adic sheaf F_{¥xi,l} on Sh(G,X) with a G(A_f)-structure. Thus, in the standard way, the cohomology group H^*(Sh(G,X),F_¥xi) has an admissible action of Gal(¥overline{E}/E) ¥times G(A_f), where E=E(G,X) is the reflex field of (G,X). Extending work of Kottwitz, Scholze, and others we discuss a method for computing the traces of this action, more specifically of an element ¥tau ¥times g where ¥tau ¥in W_{E_¥mathfrak{p}} for some prime ¥mathfrak{p} of E dividing p and g ¥in G(A_f^p) ¥times G(Z_p), in terms of a weighted point count on the Shimura variety's special fiber, as well as the traces of various local Shimura varieties over E_¥mathfrak{p}, at least in the case when (G,X) is a abelian-type Shimura datum unramified at p.
2018/05/10
11:00-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Alexander Bertoloni Meli (University of California, Berkeley)
The Cohomology of Rapoport-Zink Spaces of EL-Type
Alexander Bertoloni Meli (University of California, Berkeley)
The Cohomology of Rapoport-Zink Spaces of EL-Type
[ Abstract ]
I will discuss Rapoport-Zink spaces of EL-type and how to explicitly compute a certain variant of their cohomology in terms of the local Langlands correspondence for general linear groups. I will then show how this computation can be used to resolve certain cases of a conjecture of Harris.
I will discuss Rapoport-Zink spaces of EL-type and how to explicitly compute a certain variant of their cohomology in terms of the local Langlands correspondence for general linear groups. I will then show how this computation can be used to resolve certain cases of a conjecture of Harris.
2018/05/08
13:00-14:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Sander Mack-Crane (University of California, Berkeley)
Langlands-Rapoport for the Modular Curve
Sander Mack-Crane (University of California, Berkeley)
Langlands-Rapoport for the Modular Curve
[ Abstract ]
We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.
We discuss a concrete version of the Langlands-Rapoport conjecture in the case of the modular curve, and use this case to illuminate some of the more abstract features of the Langlands-Rapoport conjecture for general (abelian type) Shimura varieties.
2018/03/13
10:00-11:00 Room #126 (Graduate School of Math. Sci. Bldg.)
2018/03/09
13:30-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Luc Illusie
Sliced nearby cycles and duality, after W. Zheng (ENGLISH)
Luc Illusie
Sliced nearby cycles and duality, after W. Zheng (ENGLISH)
[ Abstract ]
In the early 1980's Gabber proved duality for nearby cycles and, by a different method, Beilinson proved duality for vanishing cycles in the strictly local case (up to a twist of the inertia action on the tame part). Recently W. Zheng found a simple proof of a result, conjectured by Deligne, which implies them both, and extended it over finite dimensional excellent bases. I will explain the main ideas of his work, which relies on new developments, due to him, of Deligne's theory of fibered and oriented products.
In the early 1980's Gabber proved duality for nearby cycles and, by a different method, Beilinson proved duality for vanishing cycles in the strictly local case (up to a twist of the inertia action on the tame part). Recently W. Zheng found a simple proof of a result, conjectured by Deligne, which implies them both, and extended it over finite dimensional excellent bases. I will explain the main ideas of his work, which relies on new developments, due to him, of Deligne's theory of fibered and oriented products.
2017/10/25
11:00-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Ahmed Abbes (CNRS/IHES)
On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)
Ahmed Abbes (CNRS/IHES)
On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)
[ Abstract ]
In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).
In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).
2017/10/17
17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Guoniu Han (Université de Strasbourg/CNRS)
Integer partitions and hook length formulas (ENGLISH)
www-irma.u-strasbg.fr/~guoniu/
Guoniu Han (Université de Strasbourg/CNRS)
Integer partitions and hook length formulas (ENGLISH)
[ Abstract ]
Integer partitions were first studied by Euler.
The Ferrers diagram of an integer partition is a very useful tool for
visualizing partitions. A Ferrers diagram is turned into a Young tableau
by filling each cell with a unique integer satisfying some conditions.
The number of Young tableaux is given by the famous hook length formula,
discovered by Frame-Robinson-Thrall.
In this talk, we introduce the hook length expansion technique and
explain how to find old and new hook length formulas for integer
partitions. In particular, we derive an expansion formula for the
powers of the Euler Product in terms of hook lengths, which is also
discovered by Nekrasov-Okounkov and Westburg. We obtain an extension
by adding two more parameters. It appears to be a discrete
interpolation between the Macdonald identities and the generating
function for t-cores. Several other summations involving hook length,
in particular, the Okada-Panova formula, will also be discussed.
[ Reference URL ]Integer partitions were first studied by Euler.
The Ferrers diagram of an integer partition is a very useful tool for
visualizing partitions. A Ferrers diagram is turned into a Young tableau
by filling each cell with a unique integer satisfying some conditions.
The number of Young tableaux is given by the famous hook length formula,
discovered by Frame-Robinson-Thrall.
In this talk, we introduce the hook length expansion technique and
explain how to find old and new hook length formulas for integer
partitions. In particular, we derive an expansion formula for the
powers of the Euler Product in terms of hook lengths, which is also
discovered by Nekrasov-Okounkov and Westburg. We obtain an extension
by adding two more parameters. It appears to be a discrete
interpolation between the Macdonald identities and the generating
function for t-cores. Several other summations involving hook length,
in particular, the Okada-Panova formula, will also be discussed.
www-irma.u-strasbg.fr/~guoniu/
2017/10/11
11:00-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Ahmed Abbes (CNRS/IHES)
On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)
Ahmed Abbes (CNRS/IHES)
On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)
[ Abstract ]
In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).
In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros).
2017/09/11
15:30-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Jean Zinn-Justin (CEA Saclay)
3D field theories with Chern-Simons term for large N in the Weyl gauge (ENGLISH)
Jean Zinn-Justin (CEA Saclay)
3D field theories with Chern-Simons term for large N in the Weyl gauge (ENGLISH)
[ Abstract ]
ADS/CFT correspondance has led to a number of conjectures concerning, conformal invariant, U(N) symmetric 3D field theories with Chern-Simons term for N large. An example is boson-fermion duality. This has prompted a number of calculations to shed extra light on the ADS/CFT correspondance.
We study here the example of gauge invariant fermion matter coupled to a Chern-Simons term. In contrast with previous calculations, which employ the light-cone gauge, we use the more conventional temporal gauge. We calculate several gauge invariant correlation functions. We consider general massive matter and determine the conditions for conformal invariance. We compare massless results with previous calculations, providing a check of gauge independence.
We examine also the possibility of spontaneous breaking of scale invariance and show that this requires the addition of an auxiliary scalar field.
Our method is based on field integral and steepest descent. The saddle point equations involve non-local fields and take the form of a set of integral equations that we solve exactly.
ADS/CFT correspondance has led to a number of conjectures concerning, conformal invariant, U(N) symmetric 3D field theories with Chern-Simons term for N large. An example is boson-fermion duality. This has prompted a number of calculations to shed extra light on the ADS/CFT correspondance.
We study here the example of gauge invariant fermion matter coupled to a Chern-Simons term. In contrast with previous calculations, which employ the light-cone gauge, we use the more conventional temporal gauge. We calculate several gauge invariant correlation functions. We consider general massive matter and determine the conditions for conformal invariance. We compare massless results with previous calculations, providing a check of gauge independence.
We examine also the possibility of spontaneous breaking of scale invariance and show that this requires the addition of an auxiliary scalar field.
Our method is based on field integral and steepest descent. The saddle point equations involve non-local fields and take the form of a set of integral equations that we solve exactly.
2017/05/23
17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Frédéric Jouhet (Université Claude Bernard Lyon 1 / Institut Camille Jordan)
Enumeration of fully commutative elements in classical Coxeter groups (English)
http://math.univ-lyon1.fr/homes-www/jouhet/
Frédéric Jouhet (Université Claude Bernard Lyon 1 / Institut Camille Jordan)
Enumeration of fully commutative elements in classical Coxeter groups (English)
[ Abstract ]
An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. They index naturally a basis of the (generalized) Temperley-Lieb algebra associated with W. In this talk, focusing on the (affine) type A, I will describe how to
enumerate these elements according to their Coxeter length, in all classical finite and affine Coxeter groups. The methods, which generalize previous work of Stembridge,
involve many combinatorial objects, such as heaps, walks, or parallelogram
polyominoes. This talk is based on joint works with R. Biagioli, M. Bousquet-Mélou and
P. Nadeau.
[ Reference URL ]An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. They index naturally a basis of the (generalized) Temperley-Lieb algebra associated with W. In this talk, focusing on the (affine) type A, I will describe how to
enumerate these elements according to their Coxeter length, in all classical finite and affine Coxeter groups. The methods, which generalize previous work of Stembridge,
involve many combinatorial objects, such as heaps, walks, or parallelogram
polyominoes. This talk is based on joint works with R. Biagioli, M. Bousquet-Mélou and
P. Nadeau.
http://math.univ-lyon1.fr/homes-www/jouhet/
2015/07/28
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Vincent Alberge (Université de Strasbourg)
Convergence of some horocyclic deformations to the Gardiner-Masur
boundary of Teichmueller space. (ENGLISH)
Vincent Alberge (Université de Strasbourg)
Convergence of some horocyclic deformations to the Gardiner-Masur
boundary of Teichmueller space. (ENGLISH)
[ Abstract ]
It is well known that a point of the Teichmueller space and a measured foliation determine an isometric embedding of the hyperbolic disc to the Teichmueller space equipped with the so-called Teichmueller metric. In this talk, we will consider the image by this embedding of a particular horocycle whose points will be called an horocyclic deformation. To be more precise, we will be interested in the closure of this subset in the Gardiner-Masur compactification. As the embedding of the disc does not admit a continuous extension to boundaries, we cannot say that the boundary of the set of horocyclic deformations consists of one point.
However, according to Miyachi's results, we will see that it is the case if the given foliation is either a simple closed curve or a uniquely ergodic foliation.
It is well known that a point of the Teichmueller space and a measured foliation determine an isometric embedding of the hyperbolic disc to the Teichmueller space equipped with the so-called Teichmueller metric. In this talk, we will consider the image by this embedding of a particular horocycle whose points will be called an horocyclic deformation. To be more precise, we will be interested in the closure of this subset in the Gardiner-Masur compactification. As the embedding of the disc does not admit a continuous extension to boundaries, we cannot say that the boundary of the set of horocyclic deformations consists of one point.
However, according to Miyachi's results, we will see that it is the case if the given foliation is either a simple closed curve or a uniquely ergodic foliation.
2015/05/21
16:00-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Gunnar Carlsson (Stanford University, Ayasdi INC)
The Shape of Data
(ENGLISH)
http://faculty.ms.u-tokyo.ac.jp/Carlsson.html
Gunnar Carlsson (Stanford University, Ayasdi INC)
The Shape of Data
(ENGLISH)
[ Abstract ]
There is a tremendous amount of attention being paid to the notion of
"Big Data". In many situations, however, the problem is not so much the
size of the data but rather its complexity. This observation shows that
it is now important to find methods for representing complex data in a
compressed and understandable fashion. Representing data by shapes
turns out to be useful in many situations, and therefore topology, the
mathematical sub discipline which studies shape, becomes quite
relevant. There is now a collection of methods based on topology for
analyzing complex data, and in this talk we will discuss these methods,
with numerous examples.
[ Reference URL ]There is a tremendous amount of attention being paid to the notion of
"Big Data". In many situations, however, the problem is not so much the
size of the data but rather its complexity. This observation shows that
it is now important to find methods for representing complex data in a
compressed and understandable fashion. Representing data by shapes
turns out to be useful in many situations, and therefore topology, the
mathematical sub discipline which studies shape, becomes quite
relevant. There is now a collection of methods based on topology for
analyzing complex data, and in this talk we will discuss these methods,
with numerous examples.
http://faculty.ms.u-tokyo.ac.jp/Carlsson.html
2014/12/03
16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Xavier Cabre (ICREA and UPC, Barcelona)
New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)
Xavier Cabre (ICREA and UPC, Barcelona)
New isoperimetric inequalities with densities, part II: Detailed proofs and related works (ENGLISH)
[ Abstract ]
This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.
In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.
More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.
This is a sequel to the Tuesday Analysis Seminar on December 2 by the same speaker.
In joint works with X. Ros-Oton and J. Serra, the study of the regularity of stable solutions to reaction-diffusion problems has led us to certain Sobolev and isoperimetric inequalities with weights. We will present our results in these new isoperimetric inequalities with the best constant, that we establish via the ABP method.
More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (or densities) in open convex cones of R^n. Our results apply to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Surprisingly, even that our weights are not radially symmetric, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient. As a particular case of our results, we provide with new proofs of classical results such as the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella. Furthermore, we also study the anisotropic isoperimetric problem for the same class of weights and we prove that the Wulff shape always minimizes the anisotropic weighted perimeter under the weighted volume constraint.
2014/11/26
16:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Mykhaylo Shkolnikov (Princeton University)
Intertwinings, wave equations and beta ensembles (ENGLISH)
Mykhaylo Shkolnikov (Princeton University)
Intertwinings, wave equations and beta ensembles (ENGLISH)
[ Abstract ]
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. Based on joint works with Vadim Gorin and Soumik Pal.
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. Based on joint works with Vadim Gorin and Soumik Pal.
2014/09/04
12:10-13:00 Room #470 (Graduate School of Math. Sci. Bldg.)
Samuli Siltanen (University of Helsinki, Finland)
"X-ray imaging of moving objects" (ENGLISH)
Samuli Siltanen (University of Helsinki, Finland)
"X-ray imaging of moving objects" (ENGLISH)
2014/06/10
14:40-16:10 Room #056 (Graduate School of Math. Sci. Bldg.)
Sergei Duzhin (Steklov Institute of Mathematics)
Bipartite knots (ENGLISH)
Sergei Duzhin (Steklov Institute of Mathematics)
Bipartite knots (ENGLISH)
[ Abstract ]
We give a solution to a part of Problem 1.60 in Kirby's list of open
problems in topology thus proving a conjecture raised in 1987 by
J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,
that is, a plane diagram that has an even number of crossings which can be
split into pairs that look like a simple braid on two strands with two
crossings. The conjecture was that there exist knots that do not have such
diagrams. I will prove this fact using higher Alexander ideals.
This talk is based on a joint work with my student M.Shkolnikov
We give a solution to a part of Problem 1.60 in Kirby's list of open
problems in topology thus proving a conjecture raised in 1987 by
J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,
that is, a plane diagram that has an even number of crossings which can be
split into pairs that look like a simple braid on two strands with two
crossings. The conjecture was that there exist knots that do not have such
diagrams. I will prove this fact using higher Alexander ideals.
This talk is based on a joint work with my student M.Shkolnikov
2014/05/15
16:30-17:30 Room #050 (Graduate School of Math. Sci. Bldg.)
Cédric Villani (Université de Lyon, Institut Henri Poincaré)
Synthetic theory of Ricci curvature
― When Monge, Riemann and Boltzmann meet ― (ENGLISH)
http://faculty.ms.u-tokyo.ac.jp/Villani.html
Cédric Villani (Université de Lyon, Institut Henri Poincaré)
Synthetic theory of Ricci curvature
― When Monge, Riemann and Boltzmann meet ― (ENGLISH)
[ Abstract ]
Optimal transport theory, non-Euclidean geometry and statistical physics met fifteen years ago with the discovery that Ricci curvature can be studied quantitatively thanks to entropy and
Monge-Kantorovich transport.
This unexpected encounter was very fruitful, leading to progress in each of these fields.
[ Reference URL ]Optimal transport theory, non-Euclidean geometry and statistical physics met fifteen years ago with the discovery that Ricci curvature can be studied quantitatively thanks to entropy and
Monge-Kantorovich transport.
This unexpected encounter was very fruitful, leading to progress in each of these fields.
http://faculty.ms.u-tokyo.ac.jp/Villani.html
2014/03/13
10:15-11:45 Room #470 (Graduate School of Math. Sci. Bldg.)
Michele Triestino (Ecole Normale Superieure de Lyon)
Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)
Michele Triestino (Ecole Normale Superieure de Lyon)
Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)
[ Abstract ]
By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).
The nature of this measure has been mostly investigated in connection to representation theory.
For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.
By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).
The nature of this measure has been mostly investigated in connection to representation theory.
For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.
2014/03/12
10:15-11:45 Room #470 (Graduate School of Math. Sci. Bldg.)
Michele Triestino (Ecole Normale Superieure de Lyon)
Invariant distributions for circle diffeomorphisms of
irrational rotation number and low regularity (ENGLISH)
Michele Triestino (Ecole Normale Superieure de Lyon)
Invariant distributions for circle diffeomorphisms of
irrational rotation number and low regularity (ENGLISH)
[ Abstract ]
The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.
Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.
We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.
The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.
Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.
We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.
