## Talks earlier this year

List of speakers (in alphabetical order)
1. Goro Akagi, Traveling wave dynamics for a one-dimensional constrained Allen-Cahn equation , Monday, 13 September 2021.
2. Serena Dipierro, Boundary behaviour of nonlocal minimal surfaces, Monday, 22 February 2021.
3. Yihong Du, Spreading rate for the Fisher-KPP nonlocal diffusion equation with free boundary , Monday, 31 May 2021.
4. Ilaria Fragala, Concavity properties of solutions to Robin problems , Monday, 5 July 2021.
5. Rupert Frank, Blow-up of solutions of critical elliptic equations in three dimensions , Monday, 10 May 2021.
6. Nicola Garofalo, A heat equation approach to some problems in conformal geometry., Monday, 19 July 2021.
7. François Hamel, Symmetry properties for the Euler equations and semilinear elliptic equations, Monday, 12 April 2021.
8. In-Jee Jung, Ill-posedness for incompressible fluid models at critical Sobolev regularity, Monday, 26 April 2021.
9. Yoshiyuki Kagei, Stability and bifurcation analysis of the compressible Navier-Stokes equations, Monday, 8 March 2021.
10. Derek Kielty, Degeneration of the spectral gap with negative Robin parameter., Monday, 30 August 2021.
11. Wei-Xi Li, Gevrey well-posedness of the 3D Prandtl equations without Structural Assumption , Monday, 26 July 2021.
12. Tao Luo, Estimates and geometry for a free surface problem of fluids with heat-conductivity , Monday, 17 May 2021.
13. Yasunori Maekawa, Recent progress on the Prandtl boundary layer expansion for viscous incompressible flows , Monday, 14 June 2021.
14. Shuang Miao, On the free boundary hard phase fluid in Minkowski space-time, Monday, 15 February 2021.
15. Tatsuya Miura, Li-Yau type inequality for curves and applications, Monday, 6 September 2021.
16. Toru Nogayama, Maximal regularity in Besov-Morrey spaces and its application to Keller-Segel System, Monday, 12 July 2021.
17. Aaron Palmer, Hidden Convexity in problem of Nonlinear Elasticity, Monday, 22 March 2021.
18. Adam Sikora, Square functions and Riesz transforms on a class of non-doubling manifolds , Monday, 21 June 2021.
19. Mitsuru Sugimoto, A constructive approach to semilinear wave equations, Monday, 25 January 2021.
20. Emmanuel Trélat, Spectral analysis of sub-Riemannian Laplacians and Weyl measure, 8, February 2021.
21. Rongchan Zhu, Large $$N$$ Limit of the $$O(N)$$ Linear Sigma Model via Stochastic Quantization, Monday, 25 January 2021.

Monday, 13 September 2021.
Professor @ Tohoku University, Japan.
Goro Akagi received a Ph.D from Waseda University in 2004 and his former supervisor is Prof. Mitsuharu Otani. He has been studying evolution equations, variational problems, nonlinear diffusion equations, Allen-Cahn and Cahn-Hilliard equations, and so on. He has been a full professor of Mathematical Institute and Graduate School of Science, Tohoku University since Apr 2016.

#### Slides to the talk

Traveling wave dynamics for a one-dimensional constrained Allen-Cahn equation

This talk concerns a one-dimensional Allen-Cahn equation on the whole line with the positive-part function, which constrains the growth of each solution to be non-decreasing. We shall discuss traveling wave dynamics, which has been well studied for classical Allen-Cahn equations, for the constrained one. More precisely, we shall start with constructing a one-parameter family of "degenerate" traveling wave solutions (identified when coinciding up to translation) and investigate their properties. Furthermore, the traveling wave dynamics turns out to be relevant to a free boundary problem with a peculiar motion equation for the boundary through an analysis on a regularity issue for the constrained Allen-Cahn equation, and then, such a viewpoint enables us to prove exponential stability of degenerate traveling waves with some basin of attraction, although they are unstable in a usual sense. This talk is based on a joint work with Christian Kuehn (Muenchen) and Ken-Ichi Nakamura (Kanazawa).

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Monday, 6 September 2021.
Assoc. Professor @ Tokyo Institute of Technology, Japan.
Tatsuya Miura obtained his PhD from the University of Tokyo in 2017. After Postdoctoral Fellow at the Max Planck Institute for Mathematics in the Sciences around 2018, he was appointed as Assistant Professor in 2019 at the Tokyo Institute of Technology, where he got promoted to Associate Professor in 2021. He was awarded the MSJ Takebe Katahiro Prize for Encouragement of Young Researchers in 2017, the JSPS Ikushi Prize in 2018, and the Inoue Research Award for Young Scientists in 2019.

#### Slides to the talk

Li-Yau type inequality for curves and applications

A classical result of Li-Yau asserts an optimal relation between the bending energy (also known as the Willmore energy) and multiplicity of a closed surface in Euclidean space, and is used as a fundamental tool for many studies. In this talk we obtain an analogue for curves in a general form, and observe new phenomena due to low dimensionality. We also discuss its applications to elastic flows, networks, and knots, in particular resolving an open problem posed by Dall'Acqua-Novaga-Pluda.

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Monday, 30 August 2021.
PhD Student @ University of Illinois Urbana-Champaign, United States.
Derek Kielty is a sixth-year Ph.D. student at the University of Illinois Urbana-Champaign advised by Richard Laugesen. Most recently he has worked on various problems related to spectra gaps of the Robin Laplacian and Schrodinger operators.

#### Slides to the talk

Degeneration of the spectral gap with negative Robin parameter.

The spectral gap of the Neumann and Dirichlet Laplacians are each known to have a sharp positive lower bound among convex domains of a given diameter. Between these cases, for each positive value of the Robin parameter an analogous sharp lower bound on the spectral gap is conjectured. In this talk we show the extension of this conjecture to negative Robin parameters fails by proving that the spectral gap of double cone domains are exponentially small, for each fixed parameter value.

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Monday, 26 July 2021.
Professor @ Wuhan University, China.
Wei-Xi Li is currently a professor at Wuhan University. He received PHD in 2008 from Wuhan University. The research of W.-X.Li is microlocal analysis and its application in kinetic and fluid mechanics equations, with special focus on mathematical analysis of boundary layer system and kinetic equations, spectral analysis of Fokker-Planck operator and related models.

#### Slides to the talk

Gevrey well-posedness of the 3D Prandtl equations without Structural Assumption

We establish the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption. The proof combines in a novel way a new cancellation in the system with some of the old ideas to overcome the difficulty of the loss of derivatives in the system.This shows that the three dimensional instabilities in the system leading to ill-posedness are not worse than the two dimensional ones. Joint work with Nader Masmoudi and Tong Yang.

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Monday, 19 July 2021.
Professor of Mathematics @ University of Padova, Italy.
Nicola Garofalo is Professor of Mathematics at the University of Padova. He received his Ph.D. in Mathematics from the University of Minnesota in 1987 under the supervision of Eugene Fabes. He has held positions at the University of Bologna, Northwestern University and Purdue University. He has been invited visiting professor at the Institute H. Poincare’, Paris, invited visiting fellow at the I. Newton Institute for Mathematical Sciences, Cambridge, and invited visiting professor at the Mittag-Leffler Institute, Stockholm. He has also been Distinguished Visiting Professor of Mathematics at the Ohio State University, Visiting Professor of Mathematics at The Johns Hopkins University and Visiting Professor of Mathematics at the University of Maryland. He has received uninterrupted funding from the US National Science Foundation during the years 1989-2013, and was the recipient of the 2012 Ruth and Joel Spira Award for excellence in graduate teaching.

#### Slides to the talk

A heat equation approach to some problems in conformal geometry.

The Heisenberg group plays an ubiquitous role in analysis, geometry and mathematical physics. Such Lie group is equipped with a natural second order pdo L, the real part of the Kohn-Spencer sublaplacian, that is hypoelliptic (but fails to be elliptic at every point). It is of interest to study two different families of fractional powers of L, L^s and L_s, and their so-called extension problems. While the former has a purely analytical content, the pseudodifferential operators L_s play a critical role in conformal CR geometry. In this self-contained talk I plan to show that, notwithstanding their substantial differences, these two classes of nonlocal operators can be treated in a unified way by a systematic use of the heat equation and suitable modifications of the latter. Such approach leads to some intertwining formulas related to conformal geometry that are instrumental in inverting the relevant nonlocal operators, as well as in constructing explicit solutions of some nonlocal Yamabe problems. This is joint work with G. Tralli.

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Monday, 12 July 2021.
Postdoctoral fellow @ Tokyo Metropolitan University, Japan
Toru Nogayama obtained his PhD @ Tokyo Metropolitan University, Japan, in 2021. He was a JSPS Research fellow DC @ Tokyo Metropolitan University in 2020. Since 2021, he is a JSPS Research fellow PD @ Tokyo Metropolitan University.

#### Slides to the talk

Maximal regularity in Besov-Morrey spaces and its application to Keller-Segel system

Maximal regularity is an important tool in the theory of nonlinear differential equations. Furthermore, from the viewpoint of harmonic analysis, it is also a very interesting subject. The maximal regularity for parabolic equations is established within the general framework on Banach spaces that satisfy the unconditional martingale differences (called UMD). Meanwhile, we need to treat differently the maximal regularity on Banach spaces which do not satisfy UMD. In particular, it is well known that non-reflexive Banach spaces are not UMD. In this talk, we discuss the maximal regularity of the heat equation in Besov--Morrey spaces which are not reflexive. As an application, we present the well-posedness results for the two dimensional Keller--Segel system.

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Monday, 5 July 2021.
Professor @ Politecnico di Milano, Italy.
1999: Researcher at Politecnico di Milano; 2000 PhD in Mathematics at University of Pisa (Advisor G. Buttazzo); 2016-today: Full professor at Politecnico di Milano. Main fields of interest: Calculus of variations, shape optimization, PDEs. Author of 80 publications on peer-reviewed journals.

#### Slides to the talk

Concavity properties of solutions to Robin problems

We present some recent results about the log-concavity of the Robin ground state and the power concavity of the Robin torsion function on convex domains in the Euclidean N-dimensional space. The results are valid for smooth domains, and for sufficiently large value of the parameter: the threshold for concavity turns out to depend on the space dimension and on the geometry of the domain, precisely on its diameter and boundary curvatures. In particular, this answers positively a conjecture set in 2020 by Andrews-Clutterbuck-Hauer. Anyhow, many related open questions remain open.

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Monday, 28 June 2021.
Professor @ Chung-Ang University, Seoul, Korea
Professor Hong received his PhD in Mathematics from Brown University, United States, in 2013. After this, he accepted a position as Instructor at the University of Texas at Austin, where he stayed until 2016. From 2016 until February 2018, he was a Postdoctoral researcher at Yonsei University Seoul, South Korea. Since March 2018, Hong is Assistant Professor at Chung-Ang University in Seoul.

#### Slides to the talk

On the Korteweg-de Vries limit for the Fermi-Pasta-Ulam system

The Fermi-Pasta-Ulam (FPU) system is a simple nonlinear dynamical lattice model describing a one-dimensional chain of vibrating strings with nearest neighbor interactions. This model was introduced by Fermi, Pasta and Ulam in 1955. It was anticipated at that time that chaotic nonlinear interactions would lead to thermalization. Surprisingly however, numerical simulations showed the opposite behavior – it exhibited quasi-periodic motions. This phenomena is known as the FPU paradox. This puzzle has been solved by Zabusky and Kruskal by discovering a formal convergence to the Kortewegde Vries equation, and the convergence has been rigorously justified. We revisit this convergence problem, and show how to put it into the dispersive PDE framework. This talk is based on joint work with Chulkwang Kwak and Changhun Yang.

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Monday, 14 June 2021.
Associate Professor @ Macquarie University, Australia.
Adam Sikora obtained his PhD from the Polish Academy of Science in 1994. Between 1995 and 2001 he was a visiting fellow at the Australian National University. Then he worked as Assistant Professor @ New Mexico State University (United States) before joining Macquarie University in 2009.

#### Slides to the talk

Spreading rate for the Fisher-KPP nonlocal diffusion equation with free boundary

We consider a class of manifolds $$\mathcal{M}$$ obtained by taking the connected sum of a finite number of $$N$$-dimensional Riemannian manifolds of the form $$(\mathbb{R}^{n_i}, \delta) \times (\mathcal{M}_i, g)$$, where $$\mathcal{M}_i$$ is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions $$n_i$$ are not all equal. This means that the ends have different `asymptotic dimensions', and implies that the Riemannian manifold $$\mathcal{M}$$ is not a doubling space.We completely describe the range of exponents $$p$$ for which the Riesz transform and vertical square function on $$\mathcal{M}$$ are bounded operators on $$L^p(\mathcal{M})$$.

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Monday, 31 May 2021.
Professor @ University of New England, Australia.
Yihong obtained his PhD in 1988 from Shandong University (China). After spending two years at Shandong University as a Lecturer (1988-90), and one year at Heriot-Watt University, UK,(1990-91), he joined UNE in 1991, as a postdoctoral research fellow (1991-92) working with Prof. E.N. Dancer. He became a Lecturer at UNE in 1993 and was promoted to a Professor in 2008. While at UNE, he visited Heriot-Watt University (UK), the Chinese Academy of Science (1999, 2001), the University of Tokyo and Waseda University (2006), and the Institute of Mathematics and Applications at the University of Minnesota (2012).

#### Slides to the talk

Spreading rate for the Fisher-KPP nonlocal diffusion equation with free boundary

Propagation has been modelled by reaction-diffusion equations since the pioneering works of Fisher and Kolmogorov-Peterovski-Piskunov (KPP). Much new developments have been achieved in the past several decades on the modelling of propagation, with traveling wave and related solutions playing a central role. In this talk, I will report some recent results obtained with several collaborators on the Fisher-KPP equation with free boundary and "nonlocal diffusion". A key feature of this nonlocal equation is that the propagation may or may not be determined by traveling wave solutions. There is a threshold condition on the kernel function which determines whether the propagation rate is linear or superlinear in time, also known as accelerated spreading in the latter case, where the rate of spreading is not determined by traveling waves. For some typical kernel functions, sharp estimates of the spreading rate will be presented.

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Monday, 17 May 2021.
Professor @ City University of Hong Kong.
Professor Luo received his PhD from Chinese Academy of Sciences in 1995 under the supervision of Prof. Ling Hsiao. He held positions at Georgetown University and University of Michigan before joining the City University of Hong Kong in 2016, where he is holding a position of professor.

#### Slides to the talk

Estimates and geometry for a free surface problem of fluids with heat-conductivity

In this talk, I shall discuss results on the estimates for a free surface problem of highly subsonic heat-conducting inviscid flow. The issues on the loss of derivatives and the coupling of interior solutions and evolving boundary geometry will be addressed.

This talk is based on a joint work with Prof. Huihui Zeng at Tsinghua University.

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Monday, 10 May 2021.
Professor @ Caltech University, United States, and @ Ludwig Maximilian University of Munich, Germany.
He defended his PhD thesis in 2007 @ the Royal Institute of Technology in Stockholm under the supervision of Ari Laptev. After a post-doctoral internship and assistant professorship @ Princeton, in 2013 he became professor @ Caltech and in 2016 at LMU Munich.

#### Slides to the talk

Blow-up of solutions of critical elliptic equations in three dimensions

We describe the asymptotic behavior of positive solutions $$u_\varepsilon$$ of the equation $-\Delta u + au = 3\,u^{5-\varepsilon}\qquad \textrm{in $$\Omega\subseteq\,\mathbb{R}^3$$}$ with a homogeneous Dirichlet boundary condition. The function $$a$$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $$u_\varepsilon$$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $-\Delta u + (a+\varepsilon V) u = 3\,u^5\qquad\textrm{ in $$\Omega$$.}$ For the variational problem corresponding to the latter problem we also obtain precise energy asymptotics and a detailed description of the blow-up behavior of almost minimizers (but not necessarily minimizers or solutions).

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Monday, 26 April 2021.
Assistant Professor @ Seoul National University, South Korea.
Professor Jung received his PhD in 2017 from Princeton University, United States, under the supervision of Professor Yakov Sinai. He became a Research Fellow @ the Korean Institute of Advanced Study (KIAS) in 2017, where he was promoted to KIAS Fellow in 2019. Since March 2023, Jung is Assistant Professor @ Seoul National University.

#### Slides to the talk

Ill-posedness for incompressible fluid models at critical Sobolev regularity

We consider the incompressible fluid equations including the Euler and SQG equations in critical Sobolev spaces, which are Sobolev spaces with the same scaling as the Lipschitz norm of the velocity. We show that initial value problem for the equations are ill-posed at critial regularity.

Such an ill-posedness result can be used to prove enhanced dissipation for the dissipative counterpart. The proof relied on the Key Lemma of Kiselev and Sverak, which allows to compute the main term of the velocity gradient.

This is based on joint works with Tarek Elgindi, Tsuyoshi Yoneda, and Junha Kim.

#### Video of the talk not yet released.

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Monday, 22 February 2021.
Professor @ Aix-Marseille Université, France
Professor Hamel received his PhD in 1996 @ Sorbonne University (Ex-Paris VI) under the supervision of Henri Berestycki. Before he had the privilege to be a student @ the École Normale Supçrieure de Paris (ENS), later to be a researcher @ CNRS from 1995 until 2001. Since 2001, François Hamel is professor @ Aix-Marseille Université. He was a Visiting professor @ MIT, United States and member of the Institut Universitaire de France from 2009 until 2014.

#### Slides to the talk

Symmetry properties for the Euler equations and semilinear elliptic equations

In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane.

The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function.

The talk in based on joint work with N. Nadirashvili.

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Professor and Head of the Department of Mathematics and Statistics @ the University of Western Australia.
Serena Dipierro took her PhD in Mathematical Analysis @ the International School for Advanced Studies (SISSA, Trieste) in 2012. After PostDoc positions @ the Universidad de Chile and University of Edinburgh, and a Humboldt Fellowship, she held permanent positions @ the University of Melbourne and the Università di Milano. In August 2018 she moved @ the University of Western Australia, where she is now Professor and Head of the Department of Mathematics and Statistics.

#### Slides to the talk

Boundary behaviour of nonlocal minimal surfaces

In this talk we present a peculiar behaviour of nonlocal minimal surfaces (i.e. local minimisers of a nonlocal perimeter functional), namely the capacity, and the strong tendency, of adhering to the boundary of the reference domain. This characteristic is in contrast not only with the boundary behaviour of classical minimal surfaces but also with the pattern produced by solutions of linear equations. We will discuss this phenomenon and present some recent results.

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Monday, 8 March 2021.
Professor @ Tokyo Institute of Technology.
Professor Kagei received his Doctor of Science from Hiroshima University in 1994. He was research fellow of the Alexander von Humboldt Foundation from 1997 until 1998 @ the University of Bayreuth, Germany. In 1998, Kagei becam Associate Professor @ Kyushu University in Japan, where he got promoted to Professor in 2006. Since 2019, Kagei is Professor @ the Tokyo Institute of Technology. He was awarded in 1998 with the prestigeous MSJ Takebe Katahiro Prize of the Mathematical Society of Japan and in 2012 with the Analysis Prize of the Mathematical Society of Japan.
Stability and bifurcation analysis of the compressible Navier-Stokes equations

The compressible Navier-Stokes equation, which is the basic equation for compressible viscous fluids, is classified as a quasi-linear hyperbolic-parabolic system. Due to the hyperbolic and parabolic aspects of the sysytem, solutions exhbit iinteresting behavior. In this talk, I will review the stability and bifurcation analysis of the compressible Navier-Stokes equation, and state some recent results on a related bifurcation phenomenon and singular limit problem.

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Monday, 15 February 2021.
Professor @ Wuhan University, China.
Professor Miao received his PhD in July 2013 @ the University of Chinese Academy of Sciences under the supervision of Prof. Demetrios Christodoulou and Prof. Ping Zhang. Before he accepted the professorship @ Wuhan University, he held several postdoc positions @ University of Michigan, @ ETH Zurich, and @ EPFL in Switzland.

#### Slides to the talk

On the free boundary hard phase fluid in Minkowski space-time

I will present a recent work on the free boundary hard phase fluid model with Minkowski background. The hard phase model is an idealized model for a relativistic fluid where the sound speed approaches the speed of light. This work consists of two results: First, we prove the well-posedness of this model in Sobolev spaces. Second, we give a rigorous justification of the non-relativistic limit for this model as the speed of light approaches infinity.

This is joint work with Sohrab Shahshahani and Sijue Wu.

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Monday, 22 March 2021.
Postdoctoral Fellow @ University of British Columbia, Canada.
Palmer received his PhD in mathematics @ Cornell University in 2016 under the supervision of Tim Healey on a topic of nonlinear elasticity. Currently, he is working as a postdoctoral fellow @ the University of British Columbia jointly with Nassif Ghoussoub and Young-Heon Kim on topics related to optimal transport.

#### Slides to the talk

Hidden Convexity in problem of Nonlinear Elasticity

Problems in elasticity / solid mechanics have been essential to the development of the theory of partial differential equations and Calculus of Variations. However, fundamental questions regarding the regularity of equilibria in nonlinear elasticity remain largely unanswered. The theory of optimal transport has had a lot of recent success in the analysis of nonlinear PDE, ranging from the Monge-Ampere equation to kinetic gas equations.

In a recent work, we introduced techniques from optimal transport to problems of nonlinear elasticity and uncover a hidden convexity. I will discuss the connection between optimal transport and nonlinear elasticity, and our results on when equilibria in nonlinear elasticity correspond to minimizers of an optimal transport problem.

This is joint work with Nassif Ghoussoub and Hugo Lavenant.

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Monday, 25 January 2021.
Professor @ Nagoya University, Japan
Professor Sugimoto was a Research Associate @ University of Tsukuba in 1987-90 and received his Ph.D. there in 1992. He worked @ Osaka University from 1990 until 2008, and is now a Professor @ Nagoya University since 2008.

#### Slides to the talk

A constructive approach to semilinear wave equations

In this talk, I will explain a new attempt to construct self-similar solutions to semilinear wave equations with power nonlinearity. The existence of self-similar solutions to the same equations has been already established by Pecher (2000), Kato-Ozawa (2003), etc. based on the standard fixed point theorem. We will rediscuss it by a constructive approach using the theory of hypergeometric differential equations.

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Monday, 8 February 2021.
Head of the Laboratoire Jacques-Louis Lions and Professor @ Sarbonne Université, CNRS, Université de Paris, France
Emmanuel Trélat got his PhD in 2000 in optimal control and sub-Riemannian geometry, under the supervision of Bernard Bonnard, at Dijon University, France. He became an assistant professor at Paris-Sud University (Orsay) in 2001, then professor at Orléans University in 2006, and he moved to Pierre et Marie Curie University (now named Sorbonne University) at Paris in 2011. He is the head of Laboratoire Jacques-Louis Lions.

#### Slides to the talk

Spectral analysis of sub-Riemannian Laplacians and Weyl measure

In collaboraton with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are selfadjoint hypoelliptic operators satisfying the Hörmander condition.

Thanks to the knowledge of the small-time asymptotics of heat kernels in a neighborhood of the diagonal, we establish the local and microlocal Weyl law.

When the Lie bracket configuration is regular enough (equiregular case), the Weyl law resembles that of the Riemannian case. But in the singular case (e.g., Baouendi-Grushin, MarVnet) the Wey law reveals much more complexity. In turn, we derive quantum ergodicity properties in some sub-Riemannian cases.

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Monday, 15 March 2021.
Professor @ the Beijing Institute of Technology.
Rongchan Zhu got her Phd in 2021 @ Chinese Academy of Science and Bielefeld University. Now, she is professor @ the Beijing Institute of Technology.

#### Slides to the talk

Large $$N$$ Limit of the $$O(N)$$ Linear Sigma Model via Stochastic Quantization

In this talk, I will discuss large $$N$$ limits of a coupled system of $$N$$ interacting $$\Phi^4$$ equations posed over $$\mathbb{T}^{d}$$ for $$d=1,2,3$$, known as the $$O(N)$$ linear sigma model. Uniform in $$N$$ bounds on the dynamics are established, allowing us to show convergence to a mean-field singular SPDE, also proved to be globally well-posed. Moreover, I show tightness of the invariant measures in the large $$N$$ limit. For large enough mass, they converge to the (massive) Gaussian free field, the unique invariant measure of the mean-field dynamics, at a rate of order $$1/\sqrt{N}$$ with respect to the Wasserstein distance.

I will also consider fluctuations and obtain tightness results for certain $$O(N)$$ invariant observables, along with an exact description of the limiting correlations in $$d=1,2$$.

This talk is based on joint work with Hao Shen, Scott Smith and Xiangchan Zhu.