Past

In this talk I will explain how a subfactor (ie an inclusion of type II_1 factors) give rise to a diagrammatic algebra called the Temperley-Lieb-Jones algebra. We will observe the connection between the index of the subfactor, and the TLJ algebra. In the TLJ algebra setting, we will observe that indices below four are discrete, while any number above four can be an index.

Computational science is facing several major challenges with rapidly changing highly complex heterogeneous computer architectures. To meet these challenges and yield fast and efficient performance, solvers need to be easily portable. Algebraic multigrid (AMG) methods have great potential to achieve good performance, since they have shown excellent numerical scalability for a variety of problems. However, their implementation on emerging computer architectures, which favor structure, presents new challenges. To face these difficulties, we have considered modularization of AMG, that is breaking AMG components into smaller kernels to improve portability as well as the development of new algorithms to replace components that are not suitable for GPUs. Another way to achieve performance on accelerators is to increase structure in algorithms. This talk will discuss new algorithmic developments, including a new class of interpolation operators that consists of simple matrix operations for unstructured AMG and efforts to develop a semi-structured AMG method.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Soft elastic interfaces can strongly deform under the influence of external forces, and can even exhibit elastic singularities. Here we discuss two cases where such singularities occur. First, we describe surface creases that form under compression (or swelling) of an elastic medium. Second, we consider the elastocapillary ridges that form when a soft substrate is wetted by a liquid drop. Analytical descriptions are presented and compared to experiments. We reveal that, like for liquid interfaces, the surface tension of the solid is a key factor in shaping the surface, and determines the nature of the singularity.

The Elekes-Szabó's theorem says very roughly that if a complex irreducible subvariety V of X*Y*Z has ''too many'' intersection with cartesian products of finite sets, then V is in correspondence with the graph of multiplication of an algebraic group G. It was noticed by Breuillard and Wang that the algebraic group G must be abelian. There is a constraint for the finite sets witnessing ''many'' intersections with V, namely a condition called in general position, which plays a key role in forcing the group to be abelian.  In this talk, I will present a result which shows that in the case of the graph of complex algebraic groups, with a weaker general position assumption, nilpotent groups will appear. More precisely, for a connected complex algebraic group G the following are equivalent:

1. The graph of G has ''many'' intersections with finite sets in weak general position;

2. G is nilpotent;

3. The ultrapower of G has a pseudofinite coarse approixmate subgroup in weak general position.

Surprisingly, the proof of the direction from 2 to 3 invokes some form of generic Mordell-Lang theorem for commutative complex algebraic groups.

This is joint work with Martin Bays and Jan Dobrowolski.

The large cardinal strength of the Axiom of Determinacy when enhanced with the hypothesis that all sets of reals are universally Baire is known to be much stronger than the Axiom of Determinacy itself. In fact, Sargsyan conjectured it to be as strong as the existence of a cardinal that is both a limit of Woodin cardinals and a limit of strong cardinals. Larson, Sargsyan and Wilson showed that this would be optimal via a generalization of Woodin's derived model construction. We will discuss a new translation procedure for hybrid mice extending work of Steel, Zhu and Sargsyan and use this to prove Sargsyan's conjecture.

Abstract. The aim of the course is to present several results on extensions of functions. Among the most important are Kirszbraun's and Whitney's theorems.
They provide powerful technical tools in many problems of analysis. One way to view these theorems is that they show that there exists an interpolation
of data with certain properties. In this context they are useful in computer science, e.g. in clustering of data (see e.g. [26, 23]) and in dimension reduction (see e.g. [15]).

1. Syllabus
Lecture 1. McShane's theorem [25], Kirszbraun's theorem [18, 31, 35], Kneser- Poulsen conjecture [19, 29, 16].
Lecture 2. Whitney's covering and associated partition of unity, Whitney's ex-tension theorem [37, 12, 33].
Lecture 3. Whitney's theorem { minimal Lipschitz extensions [22].
Lecture 4. Ball's extension theorem, Markov type and cotype [6].

2. Required mathematical background
Markov chains, Hilbert spaces, Banach spaces, metric spaces, Zorn lemma

3. Reading list
The reading list consists of all the papers cited above, lecture notes [27], and parts of books [36, 8].

4. Assesment
Students will be encouraged to give a short talk on a topic related to the content of the course. Suggested topics include:
(1) Brehm's theorem [10],
(2) continuity of Kirszbraun's extension theorem [20],
(3) Kirszbraun's theorem for Alexandrov spaces [21, 1],
(4) two-dimensional Kneser-Poulsen conjecture [9],
(5) origami [11],
(6) absolutely minimising Lipschitz extensions and innity Laplacian [17, 32,
34, 2, 3, 5, 4],
(7) Fenchel duality and Fitzpatrich functions [30, 7],
(8) sharp form of Whitney's extension theorem [13],
(9) Whitney's extension theorem for Cm [14],
(10) Markov type and cotype calculation [27, 6, 28], 

(11) extending Lipschitz functions via random metric partitions [24, 27].

References
1. S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov meets Kirszbraun, 2017.
2. G. Aronsson, Minimization problems for the functional supx F(x; f(x); f0(x)), Ark. Mat. 6 (1965), no. 1, 33{53.
3. , Minimization problems for the functional supx F(x; f(x); f0(x))(ii), Ark. Mat. 6 (1966), no. 4-5, 409{431.
4. , Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967), no. 6, 551{561.
5. , Minimization problems for the functional supx F(x; f(x); f0(x))(iii), Ark. Mat. 7 (1969), no. 6, 509{512.
6. K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric & Functional Analysis GAFA 2 (1992), no. 2, 137{172.
7. H. Bauschke, Fenchel duality, Fitzpatrick functions and the extension of rmly nonexpansive mappings, Proceedings of the American Mathematical Society 135 (2007), no. 1, 135{139. MR 2280182
8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium publications (American Mathematical Society) ; v. 48, American Mathematical Society, Providence, R.I., 2000 (eng).
9. K. Bezdek and R. Connelly, Pushing disks apart { the Kneser-Poulsen conjecture in the plane, Journal fur die reine und angewandte Mathematik (2002), no. 553, 221 { 236.
10. U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187{193. MR 642266
11. B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, Journal de Mathematiques Pures et Appliques 90 (2008), no. 1, 66 { 81.
12. L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions; Rev. ed., Textbooks in mathematics, ch. 6, CRC Press, Oakville, 2015.
13. C. L. Feerman, A sharp form of Whitney's extension theorem, Annals of Mathematics 161 (2005), no. 1, 509{577. MR 2150391
14. , Whitney's extension problem for Cm, Annals of Mathematics 164 (2006), no. 1, 313{359. MR 2233850
15. L.-A. Gottlieb and R. Krauthgamer, A nonlinear approach to dimension reduction, Weizmann Institute of Science.
16. M. Gromov, Monotonicity of the volume of intersection of balls, Geometrical Aspects of Functional Analysis (Berlin, Heidelberg) (J. Lindenstrauss and V. D. Milman, eds.), Springer Berlin Heidelberg, 1987, pp. 1{4.
17. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 123 (1993), no. 1, 51{74.
18. M. Kirszbraun,  Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae 22 (1934), no. 1, 77{108 (ger).
19. M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenma, Archiv der Mathematik 6 (1955), no. 5, 382{390.
20. E. Kopecka, Bootstrapping Kirszbraun's extension theorem, Fund. Math. 217 (2012), no. 1, 13{19. MR 2914919
21. U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geometric & Functional Analysis GAFA 7 (1997), no. 3, 535{560. MR 1466337
22. E. Le Gruyer, Minimal Lipschitz extensions to dierentiable functions dened on a Hilbert space, Geometric and Functional Analysis 19 (2009), no. 4, 1101{1118. MR 2570317
23. J. Lee, Jl lemma and Kirszbraun's extension theorem, 2020, Sublinear Algorithms for Big Data Lectues Notes, Brown University.
24. J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones mathematicae 160 (2005), no. 1, 59{95.
25. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837{842. MR 1562984
26. A. Naor, Probabilistic clustering of high dimensional norms, pp. 690{709. 

27. , Metric embeddings and Lipschitz extensions, Princeton University, Lecture Notes, 2015.
28. A. Naor, Y. Peres, O. Schramm, and S. Sheeld, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165{197.
29. E. T. Poulsen, Problem 10, Mathematica Scandinavica 2 (1954), 346.
30. S. Reich and S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun{Valentine extension theorem, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2657{2660. MR 2146211
31. I. J. Schoenberg, On a Theorem of Kirzbraun and Valentine, The American Mathematical Monthly 60 (1953), no. 9, 620{622. MR 0058232
32. S. Sheeld and C. K. Smart, Vector-valued optimal Lipschitz extensions, Communications on Pure and Applied Mathematics 65 (2012), no. 1, 128{154. MR 2846639
33. E. Stein, Singular integrals and dierentiability properties of functions, ch. 6, Princeton University Press, 1970.
34. P. V. Than, Extensions lipschitziennes minimales, Ph.D. thesis, INSA de Rennes, 2015.
35. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83{93. MR 0011702
36. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 84, Springer-Verlag, Berlin, 1975 (eng).
37. H. Whitney, Analytic extensions of dierentiable functions dened in closed sets, Transactions of the American Mathematical Society 36 (1934), no. 1, 63{89. MR 1501735 

University of Oxford, Mathematical Institute and St John's College, Oxford, United Kingdom
E-mail address: @email

There has been a lot of interest in recent years in understanding the multifractality of characteristic polynomials of random matrices. In this talk I shall consider the study of moments of moments from the probabilistic perspective of Gaussian multiplicative chaos, and in particular establish exact asymptotics for the so-called critical-subcritical regime in the context of large Haar-distributed unitary matrices. This is based on a joint work with Jon Keating.

We study the set $L(G)$ of cycle lengths that appear in a sparse binomial random graph $G(n,c/n)$ and in a random $d$-regular graph $G_{n,d}$. We show in particular that for most values of $c$, for $G$ drawn from $G(n,c/n)$ the set $L(G)$ contains typically an interval $[\omega(1), (1-o(1))L_{\max}(G)]$, where $L_{\max}(G)$ is the length of a longest cycle (the circumference) of $G$. For the case of random $d$-regular graphs, $d\geq 3$ fixed, we obtain an accurate asymptotic estimate for the probability of $L(G)$ to contain a full interval $[k,n]$ for a fixed $k\geq 3$. Similar results are obtained also for the supercritical case $G(n,(1+\epsilon)/n)$, and for random directed graphs.
A joint work with Yahav Alon and Eyal Lubetzky.

Many systems exhibit complex temporal dynamics due to the presence of different processes taking place simultaneously. Temporal networks provide a framework to describe the time-resolve interactions between components of a system. An important task when investigating such systems is to extract a simplified view of the temporal network, which can be done via dynamic community detection or clustering. Several works have generalized existing community detection methods for static networks to temporal networks, but they usually rely on temporal aggregation over time windows, the assumption of an underlying stationary process, or sequences of different stationary epochs. Here, we derive a method based on a dynamical process evolving on the temporal network and restricted by its activation pattern that allows to consider the full temporal information of the system. Our method allows dynamics that do not necessarily reach a steady state, or follow a sequence of stationary states. Our framework encompasses several well-known heuristics as special cases. We show that our method provides a natural way to disentangle the different natural dynamical scales present in a system. We demonstrate our method abilities on synthetic and real-world examples.

arXiv link: https://arxiv.org/abs/2101.06131

Percolation models were originally introduced to describe the propagation of a fluid in a random medium. In dimension two, the percolation properties of a model are encoded by so-called crossing probabilities (probabilities that certain rectangles are crossed from left to right). In the eighties, Russo, Seymour and Welsh obtained general bounds on crossing probabilities for Bernoulli percolation (the most studied percolation model, where edges of a lattice are independently erased with some given probability $1-p$). These inequalities rapidly became central tools to analyze the critical behavior of the model.
In this talk I will present a new result which extends the Russo-Seymour-Welsh theory to general percolation measures satisfying two properties: symmetry and positive correlation. This is a joint work with Laurin Köhler-Schindler.

I will be first introducing the Polyakov loop space formalism to
gauge theories. I will also discuss how the Polyakov loop space is modified
by Planck scale corrections.  The gauge theory will be deformed by the
Planck length as the minimum measurable length in the background spacetime.
This deformation will in turn deform the Polyakov loops space. It will be
observed that this deformation can have important consequences for
non-abelian monopoles in gauge theories.

Abstract. The aim of the course is to present several results on extensions of functions. Among the most important are Kirszbraun's and Whitney's theorems.
They provide powerful technical tools in many problems of analysis. One way to view these theorems is that they show that there exists an interpolation
of data with certain properties. In this context they are useful in computer science, e.g. in clustering of data (see e.g. [26, 23]) and in dimension reduction (see e.g. [15]).

1. Syllabus
Lecture 1. McShane's theorem [25], Kirszbraun's theorem [18, 31, 35], Kneser- Poulsen conjecture [19, 29, 16].
Lecture 2. Whitney's covering and associated partition of unity, Whitney's ex-tension theorem [37, 12, 33].
Lecture 3. Whitney's theorem { minimal Lipschitz extensions [22].
Lecture 4. Ball's extension theorem, Markov type and cotype [6].

2. Required mathematical background
Markov chains, Hilbert spaces, Banach spaces, metric spaces, Zorn lemma

3. Reading list
The reading list consists of all the papers cited above, lecture notes [27], and parts of books [36, 8].

4. Assesment
Students will be encouraged to give a short talk on a topic related to the content of the course. Suggested topics include:
(1) Brehm's theorem [10],
(2) continuity of Kirszbraun's extension theorem [20],
(3) Kirszbraun's theorem for Alexandrov spaces [21, 1],
(4) two-dimensional Kneser-Poulsen conjecture [9],
(5) origami [11],
(6) absolutely minimising Lipschitz extensions and innity Laplacian [17, 32,
34, 2, 3, 5, 4],
(7) Fenchel duality and Fitzpatrich functions [30, 7],
(8) sharp form of Whitney's extension theorem [13],
(9) Whitney's extension theorem for Cm [14],
(10) Markov type and cotype calculation [27, 6, 28], 

(11) extending Lipschitz functions via random metric partitions [24, 27].

References
1. S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov meets Kirszbraun, 2017.
2. G. Aronsson, Minimization problems for the functional supx F(x; f(x); f0(x)), Ark. Mat. 6 (1965), no. 1, 33{53.
3. , Minimization problems for the functional supx F(x; f(x); f0(x))(ii), Ark. Mat. 6 (1966), no. 4-5, 409{431.
4. , Extension of functions satisfying lipschitz conditions, Ark. Mat. 6 (1967), no. 6, 551{561.
5. , Minimization problems for the functional supx F(x; f(x); f0(x))(iii), Ark. Mat. 7 (1969), no. 6, 509{512.
6. K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric & Functional Analysis GAFA 2 (1992), no. 2, 137{172.
7. H. Bauschke, Fenchel duality, Fitzpatrick functions and the extension of rmly nonexpansive mappings, Proceedings of the American Mathematical Society 135 (2007), no. 1, 135{139. MR 2280182
8. Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, Colloquium publications (American Mathematical Society) ; v. 48, American Mathematical Society, Providence, R.I., 2000 (eng).
9. K. Bezdek and R. Connelly, Pushing disks apart { the Kneser-Poulsen conjecture in the plane, Journal fur die reine und angewandte Mathematik (2002), no. 553, 221 { 236.
10. U. Brehm, Extensions of distance reducing mappings to piecewise congruent mappings on Rm, J. Geom. 16 (1981), no. 2, 187{193. MR 642266
11. B. Dacorogna, P. Marcellini, and E. Paolini, Lipschitz-continuous local isometric immersions: rigid maps and origami, Journal de Mathematiques Pures et Appliques 90 (2008), no. 1, 66 { 81.
12. L. C. Evans and R. F. Gariepy, Measure theory and ne properties of functions; Rev. ed., Textbooks in mathematics, ch. 6, CRC Press, Oakville, 2015.
13. C. L. Feerman, A sharp form of Whitney's extension theorem, Annals of Mathematics 161 (2005), no. 1, 509{577. MR 2150391
14. , Whitney's extension problem for Cm, Annals of Mathematics 164 (2006), no. 1, 313{359. MR 2233850
15. L.-A. Gottlieb and R. Krauthgamer, A nonlinear approach to dimension reduction, Weizmann Institute of Science.
16. M. Gromov, Monotonicity of the volume of intersection of balls, Geometrical Aspects of Functional Analysis (Berlin, Heidelberg) (J. Lindenstrauss and V. D. Milman, eds.), Springer Berlin Heidelberg, 1987, pp. 1{4.
17. R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis 123 (1993), no. 1, 51{74.
18. M. Kirszbraun,  Uber die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Mathematicae 22 (1934), no. 1, 77{108 (ger).
19. M. Kneser, Einige Bemerkungen uber das Minkowskische Flachenma, Archiv der Mathematik 6 (1955), no. 5, 382{390.
20. E. Kopecka, Bootstrapping Kirszbraun's extension theorem, Fund. Math. 217 (2012), no. 1, 13{19. MR 2914919
21. U. Lang and V. Schroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geometric & Functional Analysis GAFA 7 (1997), no. 3, 535{560. MR 1466337
22. E. Le Gruyer, Minimal Lipschitz extensions to dierentiable functions dened on a Hilbert space, Geometric and Functional Analysis 19 (2009), no. 4, 1101{1118. MR 2570317
23. J. Lee, Jl lemma and Kirszbraun's extension theorem, 2020, Sublinear Algorithms for Big Data Lectues Notes, Brown University.
24. J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones mathematicae 160 (2005), no. 1, 59{95.
25. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837{842. MR 1562984
26. A. Naor, Probabilistic clustering of high dimensional norms, pp. 690{709. 

27. , Metric embeddings and Lipschitz extensions, Princeton University, Lecture Notes, 2015.
28. A. Naor, Y. Peres, O. Schramm, and S. Sheeld, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165{197.
29. E. T. Poulsen, Problem 10, Mathematica Scandinavica 2 (1954), 346.
30. S. Reich and S. Simons, Fenchel duality, Fitzpatrick functions and the Kirszbraun{Valentine extension theorem, Proceedings of the American Mathematical Society 133 (2005), no. 9, 2657{2660. MR 2146211
31. I. J. Schoenberg, On a Theorem of Kirzbraun and Valentine, The American Mathematical Monthly 60 (1953), no. 9, 620{622. MR 0058232
32. S. Sheeld and C. K. Smart, Vector-valued optimal Lipschitz extensions, Communications on Pure and Applied Mathematics 65 (2012), no. 1, 128{154. MR 2846639
33. E. Stein, Singular integrals and dierentiability properties of functions, ch. 6, Princeton University Press, 1970.
34. P. V. Than, Extensions lipschitziennes minimales, Ph.D. thesis, INSA de Rennes, 2015.
35. F. A. Valentine, A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945), 83{93. MR 0011702
36. J. H. Wells and L. R. Williams, Embeddings and extensions in analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 84, Springer-Verlag, Berlin, 1975 (eng).
37. H. Whitney, Analytic extensions of dierentiable functions dened in closed sets, Transactions of the American Mathematical Society 36 (1934), no. 1, 63{89. MR 1501735 

University of Oxford, Mathematical Institute and St John's College, Oxford, United Kingdom
E-mail address: @email

Multiple zeta values, originally considered by Euler, generalise the Riemann zeta function to multiple variables. While values of the Riemann zeta function at odd positive integers are conjectured to be algebraically independent, multiple zeta values satisfy many algebraic and linear relations, even forming a Q-algebra. While families of well understood relations are known, such as the associator relations and double shuffle relations, they only conjecturally span all algebraic relations. As multiple zeta values arise as the periods of mixed Tate motives, we obtain further algebraic structures, which have been exploited to provide spanning sets by Brown. In this talk we will aim to define a new set of relations, known to be complete in low block degree.

To achieve this, we will first review the necessary algebraic set up, focusing particularly on the motivic Lie algebra associated to the thrice punctured projective line. We then introduce a new filtration on the algebra of (motivic) multiple zeta values, called the block filtration, based on the work of Charlton. By considering the associated graded algebra, we quickly obtain a new family of graded motivic relations, which can be shown to span all algebraic relations in low block degree. We will also touch on some conjectural ungraded `lifts' of these relations, and if we have time, compare to similar approaches using the depth filtration.

We consider a reaction-diffusion equation of the type ∂tψ=∂2xψ+V(ψ)+λσ(ψ)W˙on (0,∞)×?, subject to a "nice" initial value and periodic boundary, where ?=[−1,1] and W˙ denotes space-time white noise. The reaction term V:ℝ→ℝ belongs to a large family of functions that includes Fisher--KPP nonlinearities [V(x)=x(1−x)] as well as Allen-Cahn potentials [V(x)=x(1−x)(1+x)], the multiplicative nonlinearity σ:ℝ→ℝ is non random and Lipschitz continuous, and λ>0 is a non-random number that measures the strength of the effect of the noise W˙. The principal finding of this paper is that: (i) When λ is sufficiently large, the above equation has a unique invariant measure; and (ii) When λ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.

This is based on joint work with Carl Mueller (Univ. Rochester) and Kunwoo Kim (POSTECH, S. Korea).

We will discuss recent developments of the theory of a-contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows, like the compressible Euler equation.

In the one dimensional configuration, the Bressan theory shows that small BV solutions are stable under small BV perturbations (together with a technical condition known as bounded variations on space-like curve).

The theory of a-contraction allows to extend the Bressan theory to a weak/BV stability result allowing wild perturbations fulfilling only the so-called strong trace property. Especially, it shows that the technical condition of BV on space-like curve is not needed. (joint work with Sam Krupa and Geng Chen). 

We will show several applications of the theory of a-contraction with shifts on the barotropic Navier-Stokes equation. Together with Moon-Jin Kang and Yi wang, we proved the conjecture of Matsumura (first mentioned in 1986). It consists in proving the time asymptotic stability of composite waves of viscous shocks and rarefactions. 

Together with Moon-Jin Kang, we proved also that inviscid shocks of the Euler equation, are stable among the family of inviscid limits of Navier-Stokes equation (Inventiones 2021). This stability result holds in the class of wild perturbations of inviscid limits, without any regularity restriction (not even strong trace property). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem.

This is obtained thanks to a stability result at the level of Navier-Stokes, which is uniform with respect to the viscosity, allowing asymptotically infinitely large perturbations (JEMS 2021).

A first multi D result of stability of contact discontinuities without shear, in the class of inviscid limit of Fourier-Navier-Stokes, shows that the same property is true for some situations even in multi D (joint work with Moon-jin Kang and Yi Wang). 

I will discuss recent progress on understanding the tmf-based Adams spectral sequence, where tmf = topological modular forms.  The idea is to generalize the work of Mahowald and others in the context of bo-resolutions.  The work I will discuss is joint with Prasit Bhattacharya, Dominic Culver, and J.D. Quigley.

(joint with Indranil Biswas, Jacques Hurtubise, Sean Lawton, arXiv:2104.05589)

Let $G$ be either a compact Lie group or a reductive Lie group. Let $\pi$ be the fundamental group of a 2-manifold (possibly with boundary).
We can define a character variety by ${\rm Hom}(\pi, G)/G$, where $G$ acts by conjugation.

We explore the mappings between character varieties that are induced  by mappings between surfaces. It is shown that these mappings are generally Poisson.

In some cases, we explicitly calculate the Poisson bi-vector.

I will discuss a precise connection between renormalization group (RG) and quantum chaos. Every RG flow between two conformal fixed points can be described in terms of the dynamics of Nambu-Goldstone bosons of broken symmetries. The theory of Nambu-Goldstone bosons can be viewed as a theory in anti-de Sitter space with the flat space limit. This enables an equivalent formulation of these 4d RG flows in terms of spectral deformations of a generalized free CFT in 3d. This approach provides a precise relation between C-functions associated with 4d RG flows and certain out-of-time-order correlators that diagnose chaos in 3d. As an application, I will show that the 3d chaos bound imposes constraints on the low energy effective action associated with unitary RG flows in 4d with a broken continuous global symmetry in the UV. These bounds, among other things, imply that the proof of the 4d a-theorem remains valid even when additional global symmetries are broken.

Over the last several years, it has been shown that black hole microstate level statistics in various models of 2D gravity are encoded in wormhole amplitudes.  These statistics quantitatively agree with predictions of random matrix theory for chaotic quantum systems; this behavior is realized since the 2D theories in question are dual to matrix models.  But what about black hole microstate statistics for Einstein gravity in 3D and higher spacetime dimensions, and ultimately in non-perturbative string theory?  We will discuss progress in these directions.  In 3D, we compute a wormhole amplitude that encodes the energy level statistics of BTZ black holes.  In 4D and higher, we find analogous wormholes which appear to encode the level statistics of small black holes just above threshold.  Finally, we study analogous Euclidean wormholes in the low-energy limit of type IIB string theory; we provide evidence that they encode the level statistics of small black holes just above threshold in AdS5 x S5.  Remarkably, these wormholes appear to be stable in appropriate regimes, and dominate over brane-anti-brane nucleation processes in the computation of black hole microstate statistics.

In this work we present a thorough study of the theoretical properties and devise efficient algorithms for the persistent Laplacian, an extension of the standard combinatorial Laplacian to the setting of simplicial pairs: pairs of simplicial complexes related by an inclusion, which was recently introduced by Wang, Nguyen, and Wei. 

In analogy with the non-persistent case, we establish that the nullity of the q-th persistent Laplacian equals the q-th persistent Betti number of any given simplicial pair which provides an interesting connection between spectral graph theory and TDA. 

We further exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix. This relation permits us to uncover a link with the notion of effective resistance from network circuit theory and leads to a persistent version of the Cheeger inequality.

This relationship also leads to a novel and fundamentally different algorithm for computing the persistent Betti number for a pair of simplicial complexes which can be significantly more efficient than standard algorithms. 

The last years have seen an immense increase in high-throughput and high-resolution technologies for experimental observation as well as
high-performance techniques to simulate molecular systems at a microscopic level, resulting in vast and ever-increasing amounts of high-dimensional data.
However, experiments provide only a partial view of macromolecular processes and are limited in their temporal and spatial resolution. On the other hand,
atomistic simulations are still not able to sample the conformation space of large complexes, thus leaving significant gaps in our ability to study
molecular processes at a biologically relevant scale. We present our efforts to bridge these gaps, by exploiting the available data and using state-of-the-art
machine-learning methods to design optimal coarse models for complex macromolecular systems. We show that it is possible to define simplified
molecular models to reproduce the essential information contained both in microscopic simulation and experimental measurements.

How to calculate the minimal number of summands of a nonzero polynomial in a given polynomial ideal? In this talk, we first discuss the roots of this question in computational algebra. Afterwards, we switch to the viewpoint of a commutative algebraist. In particular, we see that classical tools from this field, such as primary decomposition or the Castelnuovo–Mumford regularity, fail to provide a solution to this problem. Finally, we discuss a concrete example: A standard determinantal ideal generated by $t$-minors does not contain any polynomials with fewer than $t!/2$ terms.