Probability Theory and Statistics Research and Its Application- Study with Professors at University of Cambridge & University of Oxford

1、Program Background

Probability Theory and Statistics are an extremely important branch of mathematics. It studies accidental and random phenomena and how to extract meaningful information from data. The two disciplines are rapidly becoming the foundation for understanding the world we live in, with the increasing application of engineering, physics, biology and data science. The application of probability statistics theory is currently developing within the social sciences, particularly economics, where there are a large number of probability statistics methods aimed at studying optimal decision making and stable economic growth.

2、Program Description

This project will provide an introduction to a world-class probability theory and statistics program, include the basic concepts of probability distribution, expectations, independence, conditional expectations, Markov chains, maximum likelihood estimation, confidence intervals and hypothesis testing. At the end of this project, students will submit their research report and show their results.

Suggested Future Research Fields:

• Solving and studying the autocorrelation problem in linear regression modeling
• Linear regression (housing price prediction model)
• Processing and analysis of deformation monitoring data based on unary linear regression

3. Eligibility

• High school/college student
• Students who are interested in mathematics, statistics, data science, computer science, economics, psychology. A knowledge of calculus is a plus

4、Professor Introduction - Professor Ciubotaru

Academic Positions

• September 2017– . Professor of Mathematics, University of Oxford.
• July 2019– . Diana Brown Fellow and Tutor in Pure Maths, Somerville College, Oxford.
• September 2014– June 2019 . Tutorial Fellow at Somerville College, Oxford.
• September 2014–August 2017. Associate Professor of Pure Mathematics, University of Oxford.
• July 2011–August 2014. Associate Professor of Mathematics, University of Utah.
• July 2007–June 2011. Assistant Professor of Mathematics, University of Utah.
• July 2004–June 2007. C.L.E. Moore Instructor, Massachusetts Institute of Technology

Teaching Awards

• MPLS Individual Teaching Award (2017), Oxford.
• M.A. Oxford (2014), degree by resolution.
• Faculty Undergraduate Teaching Award (2011), Department of Mathematics, University of Utah.
• Graduate Student Teaching Award (2003), Department of Mathematics, Cornell University

Research interests

Representation theory of reductive Lie groups: the unitary dual, local Langlands correspondence, affine Hecke algebras, Coxeter groups, Dirac operators in representation theory.

Papers Preprints/Published

• 1. Weyl groups, the Dirac inequality, and isolated unitary unramified representations, arXiv:2011. (2020), 18 pages.
• 2. Deformations of unitary Howe dual pairs, with H. De Bie, M. De Martino, R. Oste, arXiv:2009.05412 (2020), 26 pages.
• 3. The nonabelian Fourier transform for elliptic unipotent representations of exceptional p-adic groups, arXiv:2006.13540 (2020), 27 pages.
• 4. Symplectic Dirac operators for Lie algebras and graded Hecke algebras, with M. De Martino and P. Meyer, arXiv:2003 (2020), 26 pages.
• 5. Hermitian forms for affine Hecke algebras, with D. Barbasch, arXiv:1312.3316v2 (2015), 29 pages.
• 6. Cocenters of p-adic groups III: elliptic and rigid cocenters, with X. He, Peking Math. Journal, online first, DOI 10.1007/s42543-020-00027-1, 30 pages.
• 7. Dirac induction for rational Cherednik algebras, with M. De Martino, IMRN 17 (2020), 5155–5214.
• 8. The Dunkl-Cherednik defomation of a Howe duality, with M. De Martino, J. Algebra 560 (2020), 914–959.
• 9. Star operations for affine Hecke algebras, with D. Barbasch, Representation Theory, Automorphic Forms, and Complex Geometry: A Tribute to Wilfried Schmid, International Press Boston, 2019, 107–137.
• 10. On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras, with V. Heiermann, Israel Journal Math, 231 (2019), no. 1, 379–417.
• 11. An Euler-Poincar´e formula for a depth zero Bernstein projector, with D. Barbasch and A. Moy, Representation Theory 23 (2019), 154–187.
• 12. Types and unitary representations of reductive p-adic groups, Invent. Math. 213 (2018), no. 1, 237–269.
• 13. One-W-type modules for rational Cherednik algebra and cuspidal two-sided cells, Bull. Inst. Math. Acad. Sinica, 13 (2018), no. 1, 1–29.
• 14. Cocenters and representations of affine Hecke algebras, with X. He, Jour. Eur. Math. Soc. 19 (2017), no. 10, 3143–3177.
• 15. A uniform classification of discrete series representations of affine Hecke algebras, with E. Opdam, Algebra and Number Theory 11, no. 5 (2017), 1089–1134. 2 of 9
• 16. On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups, with E. Opdam, 18 pages, “Representation Theory, Number Theory, and Invariant Theory: In Honor of Roger Howe on the Occasion of His 70th Birthday”, Progr. Math. 323, Birkh¨auser (2017), 87–113.
• 17. Dirac cohomology for symplectic reflection algebras, Selecta Math. 22 (2016), no. 1, 111–144.
• 18. Ladder representations of GL(n, Qp), with D. Barbasch, Representations of Reductive Groups: in honor of the 60th birthday of David A. Vogan, Jr., Progr. Math. 312, Birkh¨auser (2016), 117–137.
• 19. The cocenter of graded affine Hecke algebra and the density theorem, with X. He, in J. Pure Appl. Algebra 220 (2016), no. 1, 382–410.
• 20. Green polynomials, elliptic pairings, and the extended Dirac operator, with X. He, Adv. Math. 283 (2015), 1–50.
• 21. Formal degrees of unipotent discrete series representations and the exotic Fourier transform, with E. Opdam, Proc. London Math. Soc. 110 (2015), no. 3, 615–646.
• 22. Dirac cohomology of one-W-type representations, with A. Moy, Proc. Amer. Math. Soc. 143 (2015), no. 3, 1001–1013.
• 23. Unitary Hecke modules with nonzero Dirac cohomology, with D. Barbasch, Symmetry in Representation Theory and Its Applications: in honor of Nolan Wallach, Progress in Mathematics, Birkh¨auser 257 (2015), 1–20.
• 24. Special unipotent representations, with P. Trapa, in appendix (6 pages) to “Small representations, string instantons, and Fourier modes of Eisenstein series” by M. B. Green, S. D. Miller, and P. Vanhove, J. Number Theory 146 (2015), 187–309.
• 25. Algebraic and analytic Dirac induction for graded affine Hecke algebras, with E. Opdam and P. Trapa, J. Inst. Math. Jussieu 13 (2014), no. 3, 447–486.
• 26. Unitary equivalences for reductive p-adic groups, with D. Barbasch, Amer. J. Math. 135 (2013), no. 6, 1633–1674.
• 27. Characters of Springer representations on elliptic conjugacy classes, with P. Trapa, Duke Math. J. 162 (2013), no. 2, 201–223.
• 28. Dirac cohomology for graded affine Hecke algebras, with D. Barbasch and P. Trapa, Acta Math. 202 (2012), no. 2, 197–227.
• 29. Spin representations of Weyl groups and Springer’s correspondence, J. Reine Angew. Math. 671 (2012), 199–222.
• 30. On characters and formal degrees for classical affine Hecke algebras, with M. Kato and S. Kato, Invent. Math. 187 no. 3 (2012), 589–635.
• 31. Duality for GL(n, R), GL(n, Qp), and the degenerate affine Hecke algebra for gl(n), with P. Trapa, Amer. J. Math. 134 (2012), 1–30.
• 32. Regular orbits of symmetric subgroups on partial flag varieties, with K. Nishiyama and P. Trapa, Representation Theory, Complex Analysis, and Integral Geometry, Birkh¨auser (2012), 61–86.
• 33. Tempered modules in exotic Deligne-Langlands correspondence, with S. Kato, Adv. Math. 226, issue 2 (2011), 1538–1590.
• 34. Functors for unitary representations of real classical groups and affine Hecke algebras, with P. Trapa, Adv. Math. 227 (2011), no. 4, 1585–1611.
• 35. Reducibility of generic unipotent standard modules, with D. Barbasch, J. Lie Theory 21 (2011), no. 4, 837–846. 3 of 9
• 36. Ramanujan bigraphs arising from p-adic SU(3), with C. Ballantine, Proc. Amer. Math. Soc. 139 (2011), no. 6, 1939–1953.
• 37. Whittaker unitary dual for affine graded Hecke algebras of type E, with D. Barbasch, Compositio Math. 145, issue 6 (2009), 1563–1616.
• 38. On unitary unipotent representations of p-adic groups and affine Hecke algebras with unequal parameters, Represent. Theory 12 (2008), 453–498.
• 39. Multiplicity matrices for the affine graded Hecke algebra, J. Algebra 320 (2008), 3950–3983.
• 40. Unitarizable minimal principal series of reductive groups, with D. Barbasch and A. Pantano, Contemp. Math., 472, Amer. Math. Soc., 2008, 63–136.
• 41. Unitary I-spherical representations for split p-adic E6, Represent. Theory 10 (2006), 435–480.
• 42. Spherical unitary principal series, with D. Barbasch, Pure Appl. Math. Q. 1 (2005), no. 4, 755–789.
• 43. The unitary I-spherical dual of split p-adic F4, Represent. Theory 9 (2005), 94–137.

5、The World’s top University – 5. University of Oxford

University of Oxford (Oxford for short), located in Oxford, England, is the world's leading public research University, using college federalism.

The exact date of the university's founding is unknown, but the earliest documented date for teaching is 1096, and it grew rapidly in 1167, thanks to the support of the English royal family. The University of Oxford is the oldest university in the English-speaking world and the second oldest surviving institution of higher education in the world. The university has produced a number of epoch-leading scientific masters and trained a large number of epoch-making art masters and heads of state, including 28 British prime ministers and dozens of heads of state and political and business leaders around the world. The University of Oxford has a lofty academic status and extensive influence in mathematics, physics, medicine, law, business and other fields, and is recognized as one of the world's top institutions of higher education. Since 1902, Oxford University has also established rhodes Scholarships for undergraduates from all over the world. As of March 2019, there are 72 Nobel Prize winners (ninth in the world), 3 Fields Medal winners (20th in the world) and 6 Turing Prize winners (ninth in the world) among Oxford's alumni, professors and researchers.

The University of Oxford is ranked 1st in the world in 2019-20 by the QS World University Rankings, 4th in the World in 2020USNews World University Rankings, 5th in the world in 2019 Academic Rankings, 7th in the World in 2019 Times Higher Education World University Reputation Rankings. In particular, THE University of Oxford has been ranked number one in THE world for THE fourth consecutive year in 2017-20 in THE World University Rankings.

6、Syllabus

• 1. Discrete random variables: counting probabilities, expectation, examples
• 2. Continuous random variables: cumulative probability, distribution functions, examples
• 3. Conditional probability: Bayes' Rule, independence, examples
• 4. Variance: Chebyshev's Theorem, concentration of measure, examples
• 5. Weak law of large numbers, central limit theorem, examples
• 6. Important distributions, examples
• 7. Hypothesis testing, examples
• 8. Review and presentations

7、Schedule and Outcome

• ·Online group research learning + thesis guidance
• ·Recommendation letter from the professor
• · EI/CPCI/Scopus index international conference summary & Publication（can be used for application）
• ·Certificate of completion
• ·Academic evaluation report

8. Class Dates

For more information, please contact us.