Organizers: S. Jim Gates (Physics), Amin Gholampour (Math), Tristan Hubsch (Physics, Howard Univ.), Jonathan Rosenberg (Math), Richard Wentworth (Math)
Other Faculty Participants: Paul Green (Math, emeritus), Konstantinos Koutrolikos (Physics)
When: Thursdays @ 3:30pm-4:30pm
Where: in MTH 1311, with online option here.

This interdisciplinary RIT will aim to foster interactions between mathematicians and physicists on topics of mutual interest.  It will roughly follow the example of a similar RIT from 2010-2011 and from the last many years. The topic for 2016-17 was mirror symmetry.  Topics since 2017-2018 are listed below to give you an idea of what sort of things we tend to cover.

It is not assumed that participants already be knowledgeable in both math and physics, just in some aspect of one or the other. Relevant math topics are differential geometry, representation theory, algebraic topology, and algebraic geometry. Relevant physics topics are classical and quantum field theories, and supersymmetry.

Students (advanced undergraduates or graduate students) who want to participate can get 1-3 credits as MATH 489 (undergrad) or MATH/AMSC 689 (graduate) if they wish, by contacting the organizers.

Topics and references for 2024-2025

The topic for sping 2025 is hyperbolic band theory.  Here are a few references:

  1.  A. J. Kollár, M. Fitzpatrick, A. A. Houck, Hyperbolic lattices in circuit quantum electrodynamics. Nature 571, 45–50 (2019). <https://arxiv.org/abs/1802.09549>
  2. A. J. Kollár, M. Fitzpatrick, P. Sarnak, A. A. Houck, Line-graph lattices: Euclidean and non-Euclidean flat bands, and implementations in circuit quantum electrodynamics. Commun. Math. Phys. 376, 1909–1956 (2020). <https://arxiv.org/abs/1902.02794>
  3. S. Yu, X. Piao, N. Park, Topological hyperbolic lattices. Phys. Rev. Lett. 125, 053901 (2020). <https://arxiv.org/abs/2003.07002>
  4. Maciejko, J., Rayan, S.: Automorphic Bloch theorems for hyperbolic lattices. Proc. Nat. Acad. Sci. USA 119, e2116869119 (2022). <https://arxiv.org/abs/2108.09314>
  5. J. Maciejko, S. Rayan, Hyperbolic band theory, Sci. Adv. 7 (36) (Sept. 2021), abe9170. <https://arxiv.org/abs/2008.05489>
  6. E. Kienzle, S. Rayan, Hyperbolic band theory through Higgs bundles, Adv. Math. 409 (2022) 108664. <https://arxiv.org/abs/2201.12689>
  7. K. Ikeda, S. Aoki, and Y. Matsuki, Hyperbolic band theory under magnetic field and
    Dirac cones on a higher genus surface, J. Phys.: Condens. Matter 33 485602. <https://arxiv.org/abs/2104.13314>
  8. A. Carey, K. Hannabuss, and V. Mathai, Quantum Hall effect on the hyperbolic plane in the presence of disorder, Lett. Math. Phys. 47 (1999), no. 3, 215–236. <https://arxiv.org/abs/math/9804128>
  9. K. Ikeda, Topological aspects of matter and Langlands program, Rev. Math. Phys. 36 (2024) 36 (2024) , No. 04, 2450005, <https://arxiv.org/abs/1812.11879>.
  10. A. Nagy and S. Rayan, On the Hyperbolic Bloch Transform, Ann. Henri Poincaré 25 (2024), 1713–1732, <https://arxiv.org/abs/2208.02749>.
  11. Boettcher, Igor, Gorshkov, Alexey V., Kollár, Alicia J., Maciejko, Joseph, Rayan, Steven, and Thomale, Ronny, Crystallography of hyperbolic lattices. Phys. Rev. B 105, 125118 (2022), <https://arxiv.org/abs/2105.01087>.
  12. Notes from the talk by Tristan Hubsch on classical band theory.

The topic for fall 2024 is Quantum groups, TQFTs, Chern-Simons, and various related topics.  Our hope is to eventually get to the paper A QFT for non-semisimple TQFT by Thomas Creutzig, Tudor Dimofte, Niklas Garner and Nathan Geer, ATMP 28 (2024), no. 1, pp. 161-405.  Here are a few references:

On quantum groups:

  1. J. Jantzen, Lectures on quantum groups, Amer. Math. Soc., 1996.
  2. J. Fröhlich and T. Kerler, Quantum Groups, Quantum Categories and Quantum Field Theory, Lecture Notes in Math., vol. 1542, Springer, 1993.  Available here.
  3. C. Kassel, Quantum Groups, Graduate Texts in Math., vol. 155, Springer, 1995.  Available here.
  4. S. Shnider and S. Sternberg, Quantum Groups, Graduate Texts in Math. Physics, International Press, 1993.
  5. P. Etingof and M. Semenyakin, A brief introduction to quantum groups, arXiv:2106.05252.
  6. P. Podleś and E. Müller, Introduction to quantum groups, arXiv:q-alg/9704002. (Published in Rev. Math. Phys. 10 (1998), 511-551.)
  7. A. Maes and A. van Daele, Notes on compact quantum groups, Nieuw Arch. Wisk. (4)16 (1998), no.1-2, 73–112. arXiv:math/9803122.
  8. J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837–934. available here.

On TQFTs:

  1. M. Atiyah, An introduction to topological quantum field theories, Turkish J. Math. 21 (1997), no. 1, 1–7. available here.
  2. M. Atiyah, Topological quantum field theories, Publ. Math. IHES, no. 68 (1988), 175–186 (1989). available here.
  3. B. Bakalov and A. Kirillov, Jr., Lectures on tensor categories and modular functors, Amer. Math. Soc., RI, 2001. preliminary version available here.
  4. Kursat Sozer and Alexis Virelizier, 3d TQFTs and 3-manifold invariants, to appear in Encyl. Math. Phys., arXiv:2401.10587, available here.
  5. V. Turaev, Homotopy Quantum Field Theory, EMS Tracts in Math, vol. 10, European Math. Soc. Publ. House, Zürich, 2010.
  6. R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Comm. Math. Phys. 129 (1990), 393-429, available here.
  7. K. Walker, TQFTs (incomplete notes), 2006, available here.
  8. L. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996), no. 5, 569–587.

On Chern-Simons:

  1. Dan Freed, Classical Chern-Simons Theory, part 1, arXiv:hep-th/9206021
  2. Dan Freed, Classical Chern-Simons Theory, part 2, available here
  3. Toshitake Kohno, Conformal Field Theory and Topology, Amer. Math. Soc., 2002.
  4. Toshitake Kohno, Local systems on configuration spaces, KZ connections and conformal blocks, Acta Mathematica Vietnamica 39, no. 4 (2014), 575--598.

Topics and references for 2023-2024

The topic for spring 2024 is Spectral networks, Chern-Simons, and the dilogarithm.  Here are a few basic references:

  1. Daniel Freed and Andrew Neitzke, 3d spectral networks and classical Chern–Simons theory, arXiv:2208.07420
  2. Daniel Freed and Andrew Neitzke, The dilogarithm and abelian Chern-Simons, arXiv:2006.12565
  3. Don Zagier, The Dilogarithm Function
  4. Dan Freed, Classical Chern-Simons Theory, part 1, arXiv:hep-th/9206021
  5. Dan Freed, Classical Chern-Simons Theory, part 2
  6. Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, arXiv:0807.4723
  7. Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke, Wall-crossing, Hitchin Systems, and the WKB Approximation, arXiv:0907.3987
  8. Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke, Spectral networks, arXiv:1204.4824 

The topic for fall 2023 is 3-dimensional mirror symmetry.  Here are a few basic references:

  1. Ben Webster and Philsang Yoo, 3-dimensional mirror symmetry, arXiv:2308.06191 (survey article).  There is a "glitzier" version in the AMS Notices.
  2. S. J. Gates, Jr., C. Hull, and M. Rocek, Twisted Multiplets and New Supersymmetric Nonlinear Sigma Models, Nucl. Phys. B 248 (1984) 157-186.
  3. S. J. Gates, Jr., Superspace Formulation of New Nonlinear Sigma Models, Nucl. Phys. B 238 (1984) 349-366.
  4. R. Brooks and S. J. Gates, Jr., Extended supersymmetry and superBF gauge theories, Nucl. Phys. B 432 (1994) 205-224, arXiv:hep-th/9407147.
  5. Notes by Konstantinos Koutrolikos on a physicist's view of 3D mirror symmetry.
  6. Ken Intriligator and Nathan Seiberg, Mirror Symmetry in Three Dimensional Gauge Theories, Phys. Lett. B 387 (1996) 513-519, arXiv:hep-th/9607207.
  7. Amihay Hanany and Edward Witten. Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997), no. 1-2, 152–190, arXiv:hep-th/9611230.
  8. Martin Gremm and Emanuel Katz, Mirror symmetry for 𝒩 = 1 QED in three dimensions, J. High Energy Phys. 2000, no. 2, Paper 8, 8 pp., arXiv:hep-th/9906020.
  9. and On mirror symmetry in three dimensional Abelian gauge theories, J. High Energy Phys. 1999, paper 021, arXiv:hep-th/9902033.
  10. Sergei Gukov and David Tong, D-Brane Probes of Special Holonomy Manifolds, and Dynamics of 𝒩 = 1 Three-Dimensional Gauge Theories, J. High Energy Phys. 2002, no. 4, Paper 50, 69 pp., arXiv:hep-th/0202126.
  11. Vadim Borokhov, Anton Kapustin, and Xinkai Wu, Monopole Operators and Mirror Symmetry in Three Dimensions, J. High Energy Phys. 2002, no. 12, Paper 044, arXiv:hep-th/0207074.
  12. Mathew Bullimore, Tudor Dimofte, Davide Gaiotto, The Coulomb Branch of 3d 𝒩=4 Theories, Commun. Math. Phys. (2017) 354-671, arXiv:1503.04817.
  13. Hiraku Nakajima, Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional 𝒩=4 gauge theories arXiv:1706.05154.
  14. Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩=4 gauge theories, I arXiv:1503.03676.
  15. Alexander Braverman, Michael Finkelberg, Hiraku Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional 𝒩=4 gauge theories, II, Adv. Theor. Math. Phys. 22 (2018) 1071-1147 arXiv:1601.03586.
  16. Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima, Line bundles on Coulomb branches, Adv. Theor. Math. Phys. 25 (2021), no.4, 957–993, arXiv:1805.11826.
  17. Constantin Teleman, The rôle of Coulomb branches in 2D gauge theory, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 11, 3497–3520, arXiv:1801.10124.
  18. Slides from talks by Henry Denson: Hyperkähler Manifolds and The Webster-Yoo Survey.
  19. Mina Aganagic and Andrei Okounkov, Elliptic stable envelopes, J. Amer. Math. Soc. 34 (2021), no. 1, 79–133, arXiv:1604.00423.

Topics and references for 2022-2023

The topic for fall 2022 is supersymmetry and representation theory. Here are a few basic references:

  1. Ursula Wichter, Designing supersymmetry.
  2. Yan X. Zhang, Adinkras for mathematicians
  3.  a quick bibliography for physicists at INSPIRE
  4. a book draft on "An Introduction to Supersymmetry Using Adinkras" by Charles Doran, Kevin Iga, and Ursula Whitcher.  not for public distribution yet, but  will be made available to participants
  5. S. J. Gates, et al., 4D, N = 1 Supersymmetry Genomics (I), JHEP (2009) 0912:008, https://arxiv.org/abs/0902.3830.

The topic for spring 2023 is topological quantum computing.  Here are a few basic references:

  1. Zhenghan Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics, vol. 112, 2010.
  2. Eric C. Rowell and Zhenghan Wang, Mathematics of Topological Quantum Computing, arXiv:1705.06206.
  3. Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Topological quantum computation, Bull. Amer. Math. Soc. 40 (2003), no. 1, 31–38.
  4. Alexei Kitaev, Quantum computations: algorithms and error correction, Russian Math. Surveys 52, no. 6 (1997), 1191–1249.
  5. Alexei Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303 (2003), 2–30.
  6. Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar_Das_Sarma, Non-Abelian Anyons and Topological Quantum Computation, Reviews of Modern Physics 80 (3) (2008), 1083–1159. arXiv:0707.1889.
  7. Videos on recent experimental detections of anyons: search on Youtube for videos by Steve Simon (Oxford) and by Adam Smith (Nottingham).

Topics and references for 2021-2022

The topic for spring 2022 is N=2 supersymmetry and integrable systems.  Here are a few basic references:

  1. Lectures of Andrew Neitzke from the 2nd PIMS Summer School on Algebraic Geometry in High Energy Physics, 2021.
  2. Colloquium lecture notes by Greg Moore, 2018.
  3. Nikita Nekrasov and Samson Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052.
  4. Nikita Nekrasov, Alexey Rosly, and Samson Shatashvili, Darboux coordinates, Yang-Yang functional, and gauge theory, arXiv:1103.3919.
  5. Lotte Hollands, Philipp Rüter, and Richard J. Szabo, A geometric recipe for twisted superpotentials, arXiv:2109.14699.
  6. Andrew Neitzke, PCMI lecture notes on BPS states and spectral networks, 2019.  (There are also YouTube videos of the original lectures.)
  7. Ron Donagi, Seiberg-Witten Integrable Systems, Algebraic geometry—Santa Cruz 1995, 3–43,
    Proc. Sympos. Pure Math., 62, Part 2, Amer. Math. Soc., Providence, RI, 1997.  arXiv:alg-geom/9705010.
  8. Slides from talks of Lutian Zhao: topological twists and EM duality.
  9. J. Labastida and M. Mariño, Topological quantum field theory and four-manifolds, Mathematical Physics Studies, 25. Springer, Dordrecht, 2005. ebook available free from the UM library.

The topic for fall 2021 is the Sachdev-Ye-Kitaev model.  Here are a few basic references:

  1. Juan Maldacena and Douglas Stanford, Comments on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016), arXiv:1604.07818.
  2. The Sachdev-Ye-Kitaev model, a summary and bibliography.
  3. Subir Sachdev, the SYK model.
  4. Vladimir Rosenhaus, An Introduction to the SYK Model, J. Phys. A: Math. Theor. 52, 323001 (2019), arXiv:1807.03334.
  5. Douglas Stanford and Edward Witten, Fermionic localization of the Schwarzian theory, J. High Energy Physics 2017 10, 008.

Topics and references for 2020-2021

Those interested in this RIT might also be interested in the Wales Mathematical Physics-Physical Mathematics Seminar and the Western Hemisphere Colloquium on Geometry and Physics.

The topic for Fall 2020 and the first half of Spring 2021 (up to Spring Break) is tropical geometry and its applications to physics. Here are a few basic references:

  1. Mark Gross, Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, 114, American Mathematical Society, Providence, RI, 2011, draft available here.
  2. Brief introduction to tropical geometry, by Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin, and Kristin Shaw, arXiv:1502.05950
  3. Geometry in the tropical limit by Ilia Itenberg and Grigory Mikhalkin, arXiv:1108.3111.
  4. (added 2023) Dhruv Ranganathan, Tropical Geometry Forwards and Backwards, AMS Notices, August 2023

Also, here are notes by Tristan Hubsch from his talk on March 11.

Starting April 2021 we plan to switch to N=4 SYM (super-Yang-Mills) theory and connections to the geometric Langlands program, following the 2009 Bourbaki talk by Frenkel, entitled Gauge Theory and Langlands Duality.  For current work on this topic, you might want to see the lecture by David Ben-Zvi at the Westerm Hemisphere Colloquium on Geometry and Physics.

Topics and references for 2019-2020

The topic for 2019-2020 is invertible TQFTs and symmetry-protected topological (SPT) phases of matter.  Here are a few basic references:

  1. M. Atiyah, An introduction to topological quantum field theories, Turkish J. Math. 21 (1997), no. 1, 1–7. available here.
  2. M. Atiyah, Topological quantum field theories, Publ. Math. IHES no. 68 (1988), 175–186 (1989). available here.
  3. Daniel Freed and Greg Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013), no. 8, 1927–2023, arXiv:1208.5055.
  4. D. Freed, M. Hopkins, J. Lurie, and C. Teleman, Topological quantum field theories from compact Lie groups. A celebration of the mathematical legacy of Raoul Bott, 367–403, CRM Proc. Lecture Notes, 50, Amer. Math. Soc., Providence, RI, 2010, arXiv:0905.0731.
  5. V. and T. Ivancevic, Undergraduate Lecture Notes in Topological Quantum Field Theory, arXiv:0810.0344.
  6. D. Freed and M. Hopkins, Reflection positivity and invertible topological phases, arXiv:1604.06527.
  7. D. Freed and M. Hopkins, Invertible phases of matter with spatial symmetry, arXiv:1901.06419.
  8. D. Gaiotto and T. Johnson-Freyd, Symmetry Protected Topological phases and Generalized Cohomology, arXiv:1712.07950.
  9. Wikipedia article on SPT order, with lots of references from the physics literature.
  10. A. Kapustin, Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology, arXiv:1403.1467.
  11. A. Kapustin and A. Turzillo, Equivariant topological quantum field theory and symmetry protected topological phases, J. High Energy Phys. 2017, no. 3, 006, arXiv:1504.01830.
  12. E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88, 035001 (2016), arXiv:1508.04715.
  13. D. Freed, Lectures on Field theory and topology. CBMS Regional Conference Series in Mathematics, 133. American Mathematical Society, Providence, RI, 2019.
  14. A. Kitaev, Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009). doi:10.1063/1.3149495, arXiv:0901.2686
  15. Jonathan Campbell, Homotopy Theoretic Classification of Symmetry Protected Phases, arXiv:1708.04264.
  16. E. Witten and K. Yonekura, Anomaly Inflow and the η-Invariant, arXiv:1909.08775
  17. Sri Tata, Notes on dimer models and Spin/Pin TQFTs
  18. En-Jui (Eric) Kuo, Notes on Kitaev's periodic table

Topics and references for 2018-2019

The topic for fall 2018 was the AdS/CFT correspondence. Here is a list of references to get started:

  1. Juan Maldacena, The gauge/gravity duality, arXiv:1106.6073.
  2. Horatiu Nastase, Introduction to AdS-CFT, arXiv:0712.0689.
  3. Makoto Natsuume, AdS/CFT Duality User Guide, arXiv:1409.3575.
  4. Sean A. Hartnoll, Andrew Lucas, Subir Sachdev, Holographic quantum matter, arXiv:1612.0732.
  5. Davide Gaiotto, Juan Maldacena, The gravity duals of N=2 superconformal field theories, arXiv:0904.4466.
  6. O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, Large N Field Theories, String Theory and Gravity, arXiv:hep-th/9905111.
  7. Juan Maldacena, The Large N limit of superconformal field theories and supergravity, arXiv:hep-th/9711200.
  8. James Lindesay and Leonard Susskind, The Holographic Universe, World Scientific, 2004.
  9. Jonas Probst, Applications of the Gauge/Gravity Duality, Ph.D. thesis, Oxford, 2018.
  10. Raman Sundrum, From Fixed Points to the Fifth Dimension, arXiv:1106.4501.
  11. Vladimir Rosenhaus, An introduction to the SYK model, arXiv:1807.03334.
  12. Gábor Sárosi, AdS2 holography and the SYK model, arXiv:1711.08482.

The topic for spring 2019 was Bridgeland stability. We started with a little background on algebraic geometry and the physics motivation, and then gave a quick introduction to triangulated categories, before getting to the main topic.  Here is a list of references:

  1. Emanuele Macrì and Benjamin Schmidt, Lectures on Bridgeland Stability, arXiv:1607.01262.
  2. Emanuele Macrì and Paolo Stellari, Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces, arXiv:1807.06169.
  3. François Charles, Conditions de stabilité et géométrie birationnelle [d'après Bridgeland, Bayer-Macrì, ...] (Bourbaki talk), arXiv:1901.02930.
  4. Arend Bayer, A tour to stability conditions on derived categories (lecture notes).
  5. Ciaran Meachan, Moduli of Bridgeland-Stable Objects, PhD thesis, Univ. of Edinburgh, 2012.
  6. Claudio Fontanari and Diletta Martinelli, Why should a birational geometer care about Bridgeland stability conditions?, arXiv:1605.04803.
  7. Dominic Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, arXiv:1401.4949.
  8. Daniel Huybrechts, Introduction to stability conditions, arXiv:1111.1745.
  9. Tom Bridgeland, Spaces of stability conditions, arXiv:math/0611510.
  10. Maxim Kontsevich and Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
  11. Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317-345.
  12. Michael R. Douglas, D-branes, categories and 𝒩=1 supersymmetry, J. Math. Phys. 42 (2001), no. 7, 2818–2843.
  13. Tom Bridgeland and Ivan Smith, Quadratic differentials as stability conditions, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 155–278.

Topics and references for 2017-2018

The topic for fall 2017 was topological states of matter. In no particular order, here is a list of references:

  1. Emil Prodan and Hermann Schulz-Baldes, "Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics", arXiv:1510.08724.
  2. Daniel Freed and Greg Moore, Twisted equivariant matter, Ann. Henri Poincaré 14 (2013), no. 8, 1927–2023, arXiv:1208.5055.
  3. A. Kitaev, Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009). doi:10.1063/1.3149495, arXiv:0901.2686
  4. F.D.M. Haldane, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the "parity anomaly". Phys. Rev. Lett. 61, 2015–2018 (1988).
  5. C.L. Kane and E.J. Mele, Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005), arXiv:cond-mat/0411737
  6. C.L. Kane and E.J. Mele, Z2 Topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005), arXiv:cond-mat/0506581.
  7. C.L. Kane and E.J. Mele, Topological Mirror Superconductivity, Phys. Rev. Lett. 111, 056403 (2013), arXiv:1303.4144.
  8. Jean Bellissard, Noncommutative Geometry and the Quantum Hall Effect, Proceedings of the International Conference of Mathematicians (Zürich 94), Birkhäuser (1995).
  9. J. Bellissard, A. van Elst, H. Schulz-Baldes, The Non Commutative Geometry of the Quantum Hall Effect (longer version of #8 above), arXiv:cond-mat/9411052.
  10. David Tong: Lectures on the Quantum Hall Effect, Univ. of Cambridge, arXiv:1606.06687.
  11. Edward Witten, Three Lectures On Topological Phases Of Matter, arXiv:1510.07698.
  12. Ralph M. Kaufmann, Dan Li, and Birgit Wehefritz-Kaufmann, Notes on topological insulators, Rev. Math. Phys., 28(10), 1630003, 2016, arXiv:1501.02874.
  13. Anton Akhmerov, Jay Sau, et al., Topological condensed matter, an online course.

The topic for spring 2018 was generalized geometry and its applications to physics (especially supersymmetry and string theory). We began with the first reference listed, Hitchin's notes. Here is a short list of basic references:

  1. N. Hitchin, Lectures on Generalized Geometry, arXiv:1008.0973.
  2. M. Zabzine, Lectures on Generalized Complex Geometry and Supersymmetry, arXiv:hep-th/0605148.
  3. Dimitrios Tsimpis, Generalized geometry lectures on type II backgrounds, arXiv:1606.08674.
  4. N. Hitchin, Generalized Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308, arXiv:math/0209099.
  5. M. Gualtieri, Generalized Kahler geometry, arXiv:1007.3485.
  6. G. Cavalcantri and M. Gualtieri, Generalized complex geometry and T-duality, arXiv:1106.1747.

Topics from previous years:

  1. Supermanifolds, topology, and integration --- a few references:
    • S. J. Gates, Ectoplasm has no topology, hep-th/9709104 and hep-th/9809056.
    • E. Witten, Notes On Supermanifolds and Integration, 1209.2199.
    • The Berezin integral.
    • S. J. Gates and G. Tartaglino-Mazzucchelli, Ectoplasm and superspace integration measure for 2D supergravity with four spinorial supercurrents, 0907.5264.
    • S. J. Gates and A. Morrison, A Derivation of an Off-Shell N = (2,2) Supergravity Chiral Projection Operator, 0901.4165.
  2. Dimensonal reduction in supersymmetry
    • For example, S. J. Gates and T. Hubsch, On dimensional extension of supersymmetry: From worldlines to worldsheets, 1104.0722, deals with reduction from 1+1 to 0+1 dimensions.
  3. Adinkras and combinatorics--- a few references:
    • Yan Zhang, The combinatorics of adinkras.
    • Yan Zhang, Adinkras for mathematicians, 1111.6055.
    • Greg Landweber, Bibliography on adinkras.
    • C. Doran, K. Iga, G. Landweber, and S. Mendez-Diez, Geometrization of N-extended 1-dimensional supersymmetry algebras, 1311.3736.
    • T. Hübsch and G.A. Katona, On the Construction and the Structure of Off-Shell Supermultiplet Quotients, Int. J. Mod. Phys. A27 (2012) 1250173, 1202.4342.
    • C.F. Doran, T. Hübsch, K.M. Iga and G.D. Landweber,  On General Off-Shell Representations of Worldline (1D) Supersymmetry, Symmetry 6 no. 1, (2014) 67–88, 1310.3258.
  4. Super-Riemann surfaces and physical applications
  5. The Haag-Łopuszański-Sohnius Theorem and its variants. This is the supersymmetric analogue of the better-known Coleman-Mandula Theorem.
  6. Mirror symmetry (2016-2017).  In the fall, we followed a somewhat ad hoc approach based on looking at a lot of examples (e.g., elliptic curves and the quintic Calabi-Yau).  In the spring, we followed the multi-author book published by AMS, Dirichlet Branes and Mirror Symmetry.  The preface and Chapter 1 can be downloaded from the AMS website; Chapter 2 is at arXiv:hep-th/0609042.  An electronic version of the whole book is available at http://www.claymath.org/publications/online-books

Detailed schedule posted below.

 

Organizers: Ricardo Nochetto, Wujun Zhang
When: Wednesdays @ 5pm-6pm, starting the first week of February
Where: room TBA
Subtitle: PDE theory and numerical analysis

Fully nonlinear second order elliptic PDEs arise naturally from differential geometry, stochastic control theory, optimal transport and other fields in science and engineering. In this RIT, we will discuss the concept of viscosity solutions and regularity theory of these PDEs. Some possible topics include:

  • fully nonlinear elliptic equations and viscosity solutions
  • Alexandroff-Bakelman-Pucci estimates
  • Harnack inequality and Hölder regularity
  • Uniqueness of solutions
  • W2,p estimates
  • C1, α estimates
  • C2, α estimates

In contrast to an extensive PDE literature, the numerical approximation reduces to a few papers. We would like to discuss some criteria for designing convergent numerical methods and some tools developed recently which are useful to obtain rates of convergence. Some possible topics include:

  • criterion for convergence of numerical methods
  • discrete version of the Alexandroff-Bakelman-Pucci estimate
  • finite element method for linear elliptic equations in non-divergence form
  • numerical method for Monge-Ampere equations

Location: Math 3206
Day: Wednesday (with occasional talks Friday)
Time: 3:15pm
Tea: 2:45pm in 3201
Organizers: Giovanni Forni, Harry Tamvakis, Konstantina Trivisa

2011 - 2012 Colloquium Schedule

Sept 14

Wonderful compactifications of groups as moduli spaces of principal bundles
Michael Thaddeus (Columbia University),

Sept 28

 An introduction to essential dimension
Patrick Brosnan(University of Maryland),

Oct 21

 Rigidity of group actions, cohomology and compactness
David Fisher (Indiana University, Bloomington),

Oct 28

 A Filtration of Open/Closed Topological Field Theory
Ezra Getzler (Northwestern University),

Nov 2

 Rational billiards and the SL(2,R) action on moduli space
Alex Eskin (University of Chicago),

Nov 9

 Why and how do we use wavelets to study turbulence?
Marie Farge(Directrice de Recherche CNRS)

Nov 16

 Mixed volume computation and solving polynomial systems
Tien-Yien Li (Michigan State University),

Dec 2

 Optimal and Practical Algebraic Solvers for Discretized PDEs - Aziz Lecture
Jinchao Xu (Pennsylvania State University),

Dec 7

 DMS to THRIVE
Sastry G. Pantula(National Science Foundation)

Feb 8

 On the rigidity of black holes - Douglis Lecture
Sergiu Klainerman(Princeton University )

Feb 17

 Product formulas for positive measures and applications - February Fourier Talks
Peter Jones(Yale University),

Feb 22

 Semismooth Newton Methods: Theory, Numerics and Applications - Aziz Lecture
Michael Hintermüller (Humboldt University, Berlin),

Feb 29

 On the size of the Navier - Stokes singular set
Walter Craig (McMaster University),

Mar 14

 De Giorgi methods applied to regularity issues in Fluid Mechanics
Alexis Vasseur (UT Austin )

Mar 30

 Birkhoff Normal Form and a problem of Herman - Dynamics Conference
Hakan Eliasson (University of Paris-6 and IAS),

Apr 25

 Lyapunov Functions: Towards an Aubry-Mather theory for homeomorphisms?
Albert Fathi (ENS-Lyon),

May 2

 Contractions of Lie Groups and Representation Theory
Nigel Higson (Penn State University), cancelled

May 9

 The Pfaffian-Grassmannian Derived Equivalence
Andrei Caldărăru (University of Wisconsin, Madison),

Organizers: Patrick Brosnan, Abba Gumel
When: Wednesday @ 3:15pm, Tea 2:45pm - 3:15 pm in room 3201
Where: Math 3206
From time to time special colloquia are held on other days, sometimes as part of conferences.
Other special colloquia are the Aziz Lectures and Avron Douglis Memorial Lectures.

This area includes information on research done in the department, seminars and conferences hosted by the department, as well as access to electronic research resources.

  1. RITs and Student Seminars
  2. Library Information
  3. Electronic Journals
  4. Worldwide Mathematics Research Resources

Subcategories

Conferences

Archives: F2011-S2012 F2012-S2013 F2013-S2014 F2014-S2015 F2015-S2016 F2016-S2017 F2017-S2018 F2018-S2019 F2019-S2020 F2020-S2021 F2021-S2022 F2022-S2023 F2023-S2024 

  • Orbit equivalence relations and the compact action realization problem

    Speaker: Alexander Kechris (CalTech) -

    When: Wed, August 28, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • Multilevel approximation of Gaussian random fields

    Speaker: Christoph Schwab (ETH, Zurich) - https://math.ethz.ch/research/applied-mathematics-numerical-analysis-scientific-computing/christoph-schwab.html

    When: Wed, September 4, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • Branching selection particle systems and the selection principle

    Speaker: Julien Berestycki (University of Oxford, Statistics Department) - https://www.stats.ox.ac.uk/~berestyc/

    When: Wed, September 18, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • TBA

    Speaker: Artur Avila (IMPA) -

    When: Wed, October 23, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • From Generative AI to Statistical Physics Through Harmonic Analysis

    Speaker: Stephane Mallat (College de France) - https://blog.umd.edu/nwc/fft/2024fft/

    When: Fri, October 25, 2024 - 4:00pm
    Where: Kirwan Hall 3206
  • Intrinsic Diophantine approximation on homogeneous spaces

    Speaker: Amos Nevo (Technion) -

    When: Wed, October 30, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • Ergodic and statistical properties of slowly chaotic systems

    Speaker: Adam Kanigowski (UMD) - https://akanigow.math.umd.edu/

    When: Wed, November 6, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • Forbidding Induced Subgraphs: Structure and Algorithms (Brin MRC Distinguished Lecture)

    Speaker: Maria Chudnovsky (Princeton University) - https://web.math.princeton.edu/~mchudnov/

    When: Wed, November 13, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • TBA

    Speaker: Alessio Figalli (ETH Zurich) - https://people.math.ethz.ch/~afigalli/

    When: Wed, November 20, 2024 - 3:30pm
    Where: Kirwan Hall 3206
  • Martin's Conjecture and order-preserving functions

    Speaker: Patrick Lutz (UC Berkeley) - https://math.berkeley.edu/~pglutz/

    When: Mon, December 9, 2024 - 3:30pm
    Where: Kirwan Hall 3206
  • Continuous spin systems: group synchronisation and topological phase transitions

    Speaker: Christophe Garban (University of Lyon) - https://math.univ-lyon1.fr/~garban/

    When: Wed, December 11, 2024 - 3:15pm
    Where: Kirwan Hall 3206
  • Mathematical and Historical Aspects of Breaking the Enigma Code

    Speaker: Zbigniew Blocki (Jagiellonian University and UMD) - https://gamma.im.uj.edu.pl/~blocki/

    When: Wed, February 5, 2025 - 3:15pm
    Where: Kirwan Hall 3206
  • Optimal dimensionality reduction

    Speaker: Albert Cohen ( Sorbonne University) - https://www.ljll.fr/cohen/

    When: Thu, February 13, 2025 - 3:15pm
    Where: Kirwan Hall 3206
  • Faculty meeting

    When: Wed, February 19, 2025 - 3:15pm
    Where:
  • On stabilisations of symplectic 4-manifolds

    Speaker: Amanda Hirschi (Sorbonne Université) - https://amandahirschi.com/

    When: Wed, February 26, 2025 - 3:15pm
    Where: Kirwan Hall 3206
  • AWM Talk - Katrin Trent

    Speaker: Katrin Tent (University of Münster) - https://www.uni-muenster.de/FB10srvi/persdb/MM-member.php?id=482

    When: Wed, March 5, 2025 - 3:15pm
    Where: Kirwan Hall 3206
  • A counterexample to Viterbo's conjecture

    Speaker: Pazit Haim-Kislev (Institute for Advanced Study) - https://www.ias.edu/scholars/pazit-haim-kislev

    When: Wed, March 12, 2025 - 3:15pm
    Where: Kirwan Hall 3206
  • The P=W Conjecture

    Speaker: Davesh Maulik (MIT) - https://math.mit.edu/directory/profile.html?pid=177

    When: Wed, March 26, 2025 - 3:15pm
    Where: Kirwan Hall 3206
  • Proper graph colouring, optimization, and paradoxical decompositions

    Speaker: Robert Simon (London School of Economics) - https://www.lse.ac.uk/Mathematics/people/Robert-Simon

    When: Wed, April 9, 2025 - 3:15pm
    Where: Kirwan Hall 3206
  • The Unitary Dual (this colloquium is cancelled)

    Speaker: Jeff Adams (IDA-CCS and UMD) - https://www.math.umd.edu/~jda/

    When: Wed, April 16, 2025 - 3:15pm
    Where: Kirwan Hall 3206