This is an advanced workshop in harmonic analysis with lecture series' on four different themes. For registration, please follow instructions available at
https://www.atmschools.org/school/2019/NCMW/ha

**Speaker:** Luca Fanelli (SAPIENZA Università di Roma, Italy)

**Description:** First set of lectures will be on **Uniform Resolvent Estimates for Laplace and Dirac operators and applications**. The plan of these lectures is as follows:

**i)** Introduction, non self-adjoint Hamiltonians and eigenvalue problems. The Birman-Schwinger Principle and absence of eigenvalues, as a consequence of uniform resolvent estimates for the free Hamiltonian. An overview of recent results in the non-relativistic case (Frank, Seiringer, Simon et al).

**ii)** The uniform Sobolev Estimate for the resolvent of the free Schrödinger Hamiltonian by Kenig-Ruiz-Sogge. Proof, and failure of the estimate for the Dirac Hamiltonian.

**iii)** A new Agmon-Hörmander-type estimate for the free Dirac operators. As a consequence, absence of eigenvalues for Dirac operators with almost scaling invariant complex potentials. The results are obtained in collaboration with P. D’Ancona, and N. Schiavone (Roma Sapienza).

**Speaker:** Bent Orsted (Aarhus University, Denmark)

**Description:** Second set of lectures will be on **Branching problems for unitary representations
and Poisson transforms**. An interesting problem studied in the literature is to investigate how certain
unitary representations of semi-simple Lie groups decompose when restricted to subgroups.
This is known as Branching problem and it leads to structures that shed new light on certain
elliptic boundary value problems. For example, for the groups O(1,n+1) and U(1,n+1) the branching theory
leads to the well known Poisson transforms associated to symmetric spaces. Studying this problem in the
general context one is led to Symmetric Breaking operators of Kobayashi and Speh providing new examples
of Poisson transforms.

**Speaker:** Sanghyuk Lee (Seoul National University, South Korea)

**Description:** Third set of lectures will be on **Bochner-Riesz conjecture and related problems**.
The Bochner-Riesz conjecture is an attempt to understand the convergence of
multidimensional Fourier series and integral, which is one of the most fundamental questions in the
classical harmonic analysis. Though the conjecture is settled in 2-dimensions,
it is still open in higher dimensions. As is well known to experts in this field,
the progresses in the Bochner-Riesz conjecture are closely tied to those of Fourier restriction problems,
where we have witnessed rapid developments in recent years. Bilinear and multilinear generalizations
of linearestimates turned out to be the most efficient tools. In this series of lectures we will cover
developments in the Bochner-Riesz conjecture and its related problems in connection
with the Fourier restriction problem.

**Speaker:** Tao Mei (Baylor University, Texas, USA)

**Description:** Fourth set of lectures will be on **Noncommutative Harmonic Analysis**, as per the following details:

**i)** Positive definite functions and group C* algebra: This will be on Haagerup's characterization of completely (positive) bounded multipliers on group C* algebras. References 1, 2, 3.

**ii)** Semigroup-BMO spaces: Junge-Mei's BMO interpolation theory in the noncommutative setting, References 4, 5.

**iii)** Schatten p classes and noncommutative Lp spaces: Introduction of noncommutative Lp spaces. Reference 6.

**iv)** Hilbert transform and Lp Fourier Multipliers: Mei-Ricard's free Hilbert transforms and Junge-Mei-Parcet's Lp fourier mutlipliers theory through cocycles. References 7, 8.

**v)** Free Group von Neuman algebras and Connes' quantum differential forms. Ref 3, 9.

**References:**

**1.** Haagerup, Uffe; An example of a nonnuclear C∗-algebra, which has the metric approximation property. Invent. Math. 50 (1978/79), no. 3, 279–293.

**2.** Haagerup, Uffe; Knudby, Søren A Lévy-Khinchin formula for free groups. Proc. Amer. Math. Soc. 143 (2015), no. 4, 1477–1489.

**3.** Brown, Nathanial P.; Ozawa, Narutaka C∗-algebras and finite-dimensional approximations. Graduate Studies in Mathematics, 88.

**4.** Junge, M.; Mei, T. BMO spaces associated with semigroups of operators. Math. Ann. 352 (2012), no. 3, 691–743.

**5.** Mei, Tao Tent spaces associated with semigroups of operators. J. Funct. Anal. 255 (2008), no. 12, 3356–3406.

**6.** Gilles Pisier and Quanhua Xu, Non-commutative Lp-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517.

**7.** Junge, Marius; Mei, Tao; Parcet, Javier Smooth Fourier multipliers on group von Neumann algebras. Geom. Funct. Anal. 24 (2014), no. 6, 1913–1980.

**8.** Mei, Tao; Ricard, Éric Free Hilbert transforms. Duke Math. J. 166 (2017), no. 11, 2153–2182.

**9.** Connes, Alain Noncommutative geometry. Academic Press, Inc., San Diego, CA, 1994. xiv+661 pp. ISBN: 0-12-185860-X

#### List of selected outstaion participants:

SID | Name | | SID | Name |

28332 | Mr. Pritam Ganguly | | 30226 | Ms. Surbhi |

28909 | Mr Partha Sarathi Patra | | 30229 | Mr Manoj Kumar |

29043 | Mr. Jayanta Sarkar | | 30234 | Mr. Arup Kumar Maity |

29096 | Mr. Yasser K.T | | 30246 | Mr. Ryu Jae Hyeon |

29181 | Mr Muna Naik | | 30261 | Mr. Salman Ashraf |

29291 | Mr. Ramesh Manna | | 30288 | Ms. Hyerim Ko |

29376 | Mr. Santanu Debnath | | 30289 | Dr. Chuhee Cho |

29492 | Mr. Arun Kumar Bhardwaj | | 30303 | Mr Tapendu Rana |

29559 | Mr. Shubham Rameshsingh Bais | | 30304 | Mr. Ankit Bhojak |

29722 | Mr. Md Nurul Molla | | 30307 | Mr. Rajesh Kumar Singh |

29798 | Mr. Dhiraj Patel | | 30308 | Dr. Kanailal Mahato |

29849 | Mr. Anoop V P | | 30390 | Mr. Sumit Kumar Rano |

30031 | Mr Somnath Ghosh | | 30399 | Mr. Santi Ranjan Das |

30061 | Mr. Rupak Kumar Dalai | | 30413 | Mr. Md Hasan Ali Biswas |

30091 | Mr. Duranta Chutia | | 30419 | Mr. Yehyun Kwon |

30115 | Mr Shyam Swarup Mondal | | 30431 | Dr. Anupam Gumber |

30136 | Mr. Naman Kumar | | 30439 | Mr. Shubham Namdeo |

30163 | Mr Rattan Lal Lal | | 30466 | Mr. Abhishek Ghosh |

30208 | Mr. Sewook Oh | | 30531 | Dr. Eunhee Jeong |

30214 | Mr Juyoung Lee | | 30538 | Dr. Choiti Bandyopadhyay |

#### List of selected local participants:

SID | Name | | SID | Name |

29015 | Mr Kalachand Shuin | | 29307 | Dr. Neha Choubey |

29027 | Mr Riju Basak | | 30381 | Mr. Muzammil Khan |

29236 | Prof Manisha Udit Jain | | 30462 | Mr. Sonu Kurmi |

#### List of invited participants:

Name | Affiliation | | Name | Affiliation |

Ashisha Kumar | IIT Indore | | Pratyoosh Kumar | IIT Guwahati |

E.K. Narayanan | IISc Bangalore | | Radoun Daher | University of hassan II, Casablanca, Morocco |

Ionnis Parissis | Universidad del País Vasco, Spain | | Sajith G | Sullamussalam Science College |

Joachim Toft | Linnaeus University, Sweden | | Sanjay P K | NIT Calicut |

Jotsaroop Kaur | IISER Mohali | | Sanjoy Pusti | IIT Bombay |

Kangwei Li | Tianjin University, China | | Sayan Bagchi | IISER Kolkata |

Lakshmi Lavanya | IISER Tirupati | | Senthil Raani K S | IISER Berhampur |

Luz Roncal | BCAM- Basque Center for Applied Mathematics, Spain | | Shobha Madan | IIT Goa |

Mithun Bhowmik | IIT Bombay | | Sundaram Thangavelu | IISc Bangalore |

Oscar Blasco | Universidad de Valencia, Spain | | Suparna Sen | Calcutta University |

P K Ratnakumar | HRI Allahabad | | Venku Naidu Dogga | IIT Hyderabad |

Parasar Mohanty | IIT Kanpur | | Vijay kumar Sohani | IIT Indore |