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Documents  Teleman, Andrei | enregistrements trouvés : 25

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Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

37Cxx ; 37Jxx ; 53D25 ; 53D40 ; 53D42

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I will discuss work in progress aimed towards defining contact homology using "virtual" holomorphic curve counting techniques.

37J10 ; 53D35 ; 53D40 ; 53D42 ; 53D45 ; 57R17

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Outreach;Mathematics Education and Popularization of Mathematics

Dusa McDuff is the Helen Lyttle Kimmel '42 Professor of Mathematics at Barnard College. At Barnard, she currently teaches "Calculus I", "Perspectives in Mathematics" and courses in geometry and topology.
Professor McDuff gained her early teaching experience at the University of York (U.K.), the University of Warwick (U.K.) and MIT. In 1978, she joined the faculty of the Department of Mathematics at SUNY Stony Brook, where she was awarded the title of Distinguished Professor in 1998.
Professor McDuff has honorary doctorates from the University of Edinburgh, the University of York, the University of Strasbourg and the University of St Andrews. She is a fellow of the Royal Society, a member of the National Academy of Sciences, a member of the American Philosophical Society, and an honorary fellow of Girton College, Cambridge.
She has received the Satter Prize from the American Mathematical Society and the Outstanding Woman Scientist Award from AWIS (Association for Women in Science).
Professor McDuff's service to the mathematical community has been extensive. She is particularly interested in issues connected with the position of women in mathematics, and currently serves on the MSRI Board of Trustees. Together with Dietmar Salamon, she has written several foundational books on symplectic topology as well as many research articles.
Dusa McDuff is the Helen Lyttle Kimmel '42 Professor of Mathematics at Barnard College. At Barnard, she currently teaches "Calculus I", "Perspectives in Mathematics" and courses in geometry and topology.
Professor McDuff gained her early teaching experience at the University of York (U.K.), the University of Warwick (U.K.) and MIT. In 1978, she joined the faculty of the Department of Mathematics at SUNY Stony Brook, where she was awarded the ...

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Algebraic and Complex Geometry

The remodeling conjecture proposed by Bouchard-Klemm-Marino-Pasquetti relates Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-orbifold to Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. It can be viewed as a version of all genus open-closed mirror symmetry. In this talk, I will describe results on this conjecture based on joint work with Bohan Fang and Zhengyu Zong.

14J33 ; 14N35

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Outreach;Mathematics Education and Popularization of Mathematics

François Lalonde, Professor at the Mathematics and Statistics Department of the Université de Montréal, was named Director of the Centre de recherches mathématiques (CRM) on September 14, 2004. The CRM is the first institute of research in mathematical sciences founded in Canada in 1969.
A member of the Royal Society of Canada since 1997, François Lalonde's research is mainly in the field of Symplectic geometry and topology. From 1996 to 2000, he directed the Institut des sciences mathématiques (ISM), a consortium of six Québec universities (Montréal, McGill, UQAM, Concordia, Laval and Sherbrooke). In this capacity, he developed the Institute by putting in place measures furthering the place of Montréal, and Québec as a whole, as a North American centre of excellence in mathematical research and training.
Mr. Lalonde was also the Founder and Director of the Centre interuniversitaire de recherche en géométrie différentielle et en topologie (CIRGET) which gathers together the best geometers and topologists from UQAM, McGill, Montreal and Concordia universities.
A mathematician and physicist by training, François Lalonde holds a Doctorat d’Etat (1985) from Orsay Center in Paris, in the field of differential topology. He was a Killam Research Fellowship recipient in 2000-2002 and holds a Canada Research Chair in the field of Symplectic Geometry and Topology. He is member of the editorial committees of the Canadian Journal of Mathematics and of the Canadian Bulletin of Mathematics. Member of the scientific committee of the First Canada-France congress in 2004 and plenary speaker at the First Canada-China congress in 1999, his works in collaboration with Dusa McDuff were presented in her plenary address at the ICM in 1998. He is an invited speaker at the ICM 2006.
CIRM - Chaire Jean-Morlet 2015 - Aix-Marseille Université
François Lalonde, Professor at the Mathematics and Statistics Department of the Université de Montréal, was named Director of the Centre de recherches mathématiques (CRM) on September 14, 2004. The CRM is the first institute of research in mathematical sciences founded in Canada in 1969.
A member of the Royal Society of Canada since 1997, François Lalonde's research is mainly in the field of Symplectic geometry and topology. From 1996 to 2000, ...

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Algebraic and Complex Geometry;Mathematical Physics

In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT’s, encoded in a one parameter family of Frobenius algebras, which we will construct.

14D20 ; 14H60 ; 57R56 ; 81T40 ; 14F05 ; 14H10 ; 22E46 ; 81T45

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Algebraic and Complex Geometry;Mathematical Physics

In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt on the asymptotics of the natural $L^2$-metric $G_{L^2}$ on the moduli space $\mathcal{M}$ of rank-2 Higgs bundles over a Riemann surface $\Sigma$ as given by the set of solutions to the so-called self-duality equations
$\begin{cases}
&0 = \bar{\partial}_A \Phi \\
& 0 = F_A + [ \Phi \wedge \Phi^*]
\end{cases}$
for a unitary connection $A$ and a Higgs field $\Phi$ on $\Sigma$. I will show that on the regular part of the Hitchin fibration ($A$, $\Phi$) $\rightarrow$ det $\Phi$ this metric is well-approximated by the semiflat metric $G_{sf}$ coming from the completely integrable system on $\mathcal{M}$. This also reveals the asymptotically conic structure of $G_{L^2}$, with (generic) fibres of the above fibration being asymptotically flat tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore and Neitzke. Its proof is based on a detailed understanding of the ends structure of $\mathcal{M}$. The analytic methods used there in addition yield a complete asymptotic expansion of the difference $G_{L^2} − G_{sf}$ between the two metrics.
In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt on the asymptotics of the natural $L^2$-metric $G_{L^2}$ on the moduli space $\mathcal{M}$ of rank-2 Higgs bundles over a Riemann surface $\Sigma$ as given by the set of solutions to the so-called self-duality equations
$\begin{cases}
&0 = \bar{\partial}_A \Phi \\
& 0 = F_A + [ \Phi \wedge \Phi^*]
\end{cases}$
for a unitary connection $A$ and a ...

53C07 ; 53C26 ; 53D18 ; 14H60 ; 14D20

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Geometry

The degenerate special Lagrangian equation governs geodesics in the space of positive Lagrangians. Existence of such geodesics has implications for uniqueness and existence of special Lagrangians. It also yields lower bounds on the cardinality of Lagrangian intersec- tions related to the strong Arnold conjecture. An overview of what is known about the existence problem will be given. The talk is based on joint work with A. Yuval and with Y. Rubinstein. The degenerate special Lagrangian equation governs geodesics in the space of positive Lagrangians. Existence of such geodesics has implications for uniqueness and existence of special Lagrangians. It also yields lower bounds on the cardinality of Lagrangian intersec- tions related to the strong Arnold conjecture. An overview of what is known about the existence problem will be given. The talk is based on joint work with A. Yuval and with Y. ...

53D12 ; 53C22

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Algebraic and Complex Geometry

I will talk about joint work during the recent years with Amin Gholampour, Richard Thomas and Yukinobu Toda, on proving the modularity property of the generating series of certain DT invariants of torsion sheaves with two dimensional support in ambient threefolds. More specifically, I will talk about algebraic-geometric proof of S-duality conjecture in superstring theory made formerly by physicists: Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingrediants to prove modularity of the above DT invariants over the quintic threefold. I will talk about joint work during the recent years with Amin Gholampour, Richard Thomas and Yukinobu Toda, on proving the modularity property of the generating series of certain DT invariants of torsion sheaves with two dimensional support in ambient threefolds. More specifically, I will talk about algebraic-geometric proof of S-duality conjecture in superstring theory made formerly by physicists: Gaiotto, Strominger, Yin, regarding the ...

14J30 ; 14N35 ; 81T30

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Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

37Cxx ; 37Jxx ; 53D25 ; 53D40 ; 53D42

... Lire [+]

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Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

37Cxx ; 37Jxx ; 53D25 ; 53D40 ; 53D42

... Lire [+]

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Theory of persistence modules is a rapidly developing field lying on the borderline between algebra, geometry and topology. It provides a very useful viewpoint at Morse theory, and at the same time is one of the cornerstones of topological data analysis. In the course I'll review foundations of this theory and focus on its applications to symplectic topology. In parts, the course is based on a recent work with Egor Shelukhin arXiv:1412.8277

37Cxx ; 37Jxx ; 53D25 ; 53D40 ; 53D42

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Geometry;Algebraic and Complex Geometry

I will report on aspects of work with Sheridan and Ganatra in which we show how homo- logical mirror symmetry for Calabi-Yau manifolds implies equality of Yukawa couplings on the A- and B-sides. On the A-side, these couplings are generating functions for genus-zero GW invariants. On the B-side, one has a degenerating family of CY manifolds, and the couplings are fiberwise integrals involving a holomorphic volume form. We show that the Fukaya category implicitly "knows" the correct normalization of this volume form, as well as the mirror map. I will report on aspects of work with Sheridan and Ganatra in which we show how homo- logical mirror symmetry for Calabi-Yau manifolds implies equality of Yukawa couplings on the A- and B-sides. On the A-side, these couplings are generating functions for genus-zero GW invariants. On the B-side, one has a degenerating family of CY manifolds, and the couplings are fiberwise integrals involving a holomorphic volume form. We show that the Fukaya ...

53D37 ; 14J33

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Algebraic and Complex Geometry;Mathematical Physics

I will review a conjecture (joint work with Davide Gaiotto and Greg Moore) which gives a description of the hyperkähler metric on the moduli space of Higgs bundles, and recent joint work with David Dumas which has given evidence that the conjecture is true in the case of $SL(2)$-Higgs bundles.

32Q20 ; 53C07 ; 53C55 ; 53C26 ; 81T13 ; 81T60

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Given a smooth cobordism with an almost complex structure, one can ask whether it is realized as a Liouville cobordism, that is, an exact symplectic manifold whose primitive induces a contact structure on the boundary. We show that this is always the case, as long as the positive and negative boundaries are both nonempty. The contact structure on the negative boundary will always be overtwisted in this construction, but for dimensions larger than 4 we show that the positive boundary can be chosen to have any given contact structure. In dimension 4 we show that this cannot always be the case, due to obstructions from gauge theory. Given a smooth cobordism with an almost complex structure, one can ask whether it is realized as a Liouville cobordism, that is, an exact symplectic manifold whose primitive induces a contact structure on the boundary. We show that this is always the case, as long as the positive and negative boundaries are both nonempty. The contact structure on the negative boundary will always be overtwisted in this construction, but for dimensions larger ...

53D05 ; 53D10 ; 53D35

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Algebraic and Complex Geometry

A Gushel-Mukai variety is a Fano variety of coindex 3, Picard number 1, and degree 10. I will discuss classification of these Fano varieties, their moduli spaces, and their relation to EPW sextics. This is a joint work with Olivier Debarre.

14H10 ; 14J45 ; 14E08

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Algebraic and Complex Geometry

Following Grothendieck’s vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, I explain how to define the motive of certain algebraic stacks. I will then focus on defining and studying the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers. If there is time, I will give a conjectural formula for this motive, and explain how this follows from a conjecture on the intersection theory of certain Quot schemes. This is joint work with Simon Pepin Lehalleur. Following Grothendieck’s vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, I explain how to define the motive of certain algebraic stacks. I will then focus on defining and studying the motive of the moduli stack of vector bundles on a smooth ...

14A20 ; 14C25 ; 14C15 ; 14D23 ; 14F42 ; 14H60 ; 18E30 ; 19E15

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Algebraic and Complex Geometry

We will overview some conjectures on the mixed Hodge structure of character varieties in the framework of non-abelian Hodge theory on a Riemann surface. Then we introduce and study toric analogues of these spaces, in particular we prove that the toric character variety retracts to its core, the zero fiber of the toric Hitchin map, that its cohomology is Hodge-Tate and satisfies curious Hard Lefschetz, as well as the purity conjecture. We will indicate how these shed light on the $P=W$ conjecture in the toric case as well as for general character varieties. This is based on joint work with Nick Proudfoot. We will overview some conjectures on the mixed Hodge structure of character varieties in the framework of non-abelian Hodge theory on a Riemann surface. Then we introduce and study toric analogues of these spaces, in particular we prove that the toric character variety retracts to its core, the zero fiber of the toric Hitchin map, that its cohomology is Hodge-Tate and satisfies curious Hard Lefschetz, as well as the purity conjecture. We will ...

14H60 ; 14C30 ; 14J32 ; 14M25

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Algebraic and Complex Geometry

The absolute Galois group of the rational numbers acts on the various flavours (profinite, prounipotent, pro-$\ell$) of the fundamental group of a smooth projective curve over the rationals. The image of the corresponding homomorphism normalizes the image of the profinite mapping class group in the automorphism group of the geometric fundamental group of the curve. The image of the Galois action modulo these “geometric automorphisms” is independent of the curve. A basic problem is to determine this image. This talk is a report on a joint project with Francis Brown whose goal is to understand the image mod geometric automorphisms in the prounipotent case. Standard arguments reduce the problem to one in genus 1, where one can approach the problem by studying the periods of iterated integrals of modular forms and their relation to multiple zeta values. The absolute Galois group of the rational numbers acts on the various flavours (profinite, prounipotent, pro-$\ell$) of the fundamental group of a smooth projective curve over the rationals. The image of the corresponding homomorphism normalizes the image of the profinite mapping class group in the automorphism group of the geometric fundamental group of the curve. The image of the Galois action modulo these “geometric automorphisms” is ...

14H30 ; 14H52 ; 11M32

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Algebraic and Complex Geometry

Hitchin’s equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmu ̈ller theory, and the geometric Langlands correspondence. In this talk, I’ll describe what solutions of SL(n, C)-Hitchin’s equations “near the ends” of the moduli space look like, and the resulting compactification of the Hitchin moduli space. Wild Hitchin moduli spaces are an important ingredient in this construction. This construction generalizes Mazzeo-Swoboda-Weiss-Witt’s construction of SL(2, C)-solutions of Hitchin’s equations where the Higgs field is “simple.” Hitchin’s equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmu ̈ller theory, and the geometric Langlands correspondence. In this talk, I’ll describe what solutions of SL(n, C)-Hitchin’s equations “near the ends” of the moduli space look like, and the resulting compactification of the Hitchin moduli space. Wild Hitchin moduli spaces are an important ...

14D20 ; 14D21 ; 14H70 ; 14H60 ; 14K25 ; 14P25 ; 53C07 ; 53D50 ; 53D30 ; 81T45 ; 81T15

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