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 1308, 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 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

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. Anton Kapustin and Matthew J. Strassler, 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
• W^2,p estimates
• C^{1, α} estimates
• C^{2, 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

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.

Fall 2024

RITs ("Research Interaction Teams") are informal groups designed to foster interaction between faculty, students, and postdocs, and to get students interested in current research. Most of them meet as informal seminars with active student participation (and in many cases, student organization as well).  Course credit is possible for most RITs under the course numbers MATH489, MATH689, and AMSC689.  (Contact the faculty organizer of the particular RIT for more info.)  In addition to the RITs, there are several student seminars which are run by students for students. 

• RIT on Applied Partial Differential Equations
  • Organizers: Jeffrey Kuan and Matei Machedon
  • Meeting Time: 3:00pm - 3:50pm Mondays, MTH 1311.  Organizational meeting Monday, September 9, 2024.
  • Description: We will study mathematical aspects of applied partial differential equations. These might include well-posedness, long-time behavior, attractor dynamics, stability of coherent structures, asymptotic limits, and the relationship between chaos and stochasticity. However the best description is the list of talks given on the website.
• RIT on Geometry and Physics
  • Organizers: S. Jim Gates (Physics), Amin Gholampour (Math), Tristan Hubsch (Howard (Physics) and UMd), Jonathan Rosenberg (Math), Richard Wentworth (Math)
  • Meeting Time: Thursdays at 3:30 PM in MTH1308.  Organizational meeting on Thursday, Aug. 29.  Meetings also available on Zoom.
  • Description: This interdisciplinary RIT will aim to foster interactions between mathematicians and physicists on topics of mutual interest, such as supersymmetry, string theory, topological states of matter, and gauge theory.  Contact one of the organizers for more information.
• RIT on Numerical Continuation Methods
  
  • Organizer: Harry Dankowicz (), Mechanical Engineering and AMSC
  • Topic: Through a collaborative, team-based learning environment,
    participants will develop awareness of and competence in theoretical and computational tools for
    analyzing parameter-dependent sets of nonlinear equations, with emphasis on boundary-value
    problems describing multi-segment periodic trajectories in nonlinear dynamical systems,
    including in problems with delay or uncertainty.
  • Materials and Resources:
    
    1. Allgower & Georg (2003) Introduction to Numerical Continuation Methods, SIAM:
       https://epubs.siam.org/doi/book/10.1137/1.9780898719154
    2. Dankowicz & Schilder (2013) Recipes for Continuation, SIAM:
       https://epubs.siam.org/doi/book/10.1137/1.9781611972573
    3. Govaerts (2000) Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM:
       https://epubs.siam.org/doi/book/10.1137/1.9780898719543
    4. COCO, Toolboxes for Parameter Continuation and Bifurcation Analysis:
       https://sourceforge.net/projects/cocotools/
    5. MATCONT, Numerical Bifurcation Analysis Toolbox in Matlab:
       https://sourceforge.net/projects/matcont/
    6. AUTO, Software for Continuation and Bifurcation Problems in Ordinary Differential
       Equations: https://sourceforge.net/projects/auto-07p/
    7. Ahsan, Dankowicz, Li & Sieber (2022) “Methods of Continuation and Their Implementation in the COCO Software Platform with Application to Delay Differential Equations,” Nonlinear Dynamics 107, pp. 3181-3243.
       https://link.springer.com/article/10.1007/s11071-021-06841-1
  • Meeting Time: 3-4 pm on Tuesdays, starting August 27, 2164 Glenn Martin Hall
  • Participants: Open to undergraduate and graduate students, postdoctoral scholars, and faculty
  • Expectations: Student participants in an RIT are expected to contribute actively to the learning process, presenting content from the list of materials and resources to the whole group in a professional and structured manner. Participants may propose complementary learning materials and are encouraged to explore areas of personal interest. Participants should plan to attend every meeting of the RIT in order to ensure an environment of mutual respect and trust.
    Student participants wishing to receive independent study course credit should identify themselves to the organizer at the first meeting of the RIT or by email.
• RIT on High-Dimensional Statistics
  
  • Organizers: Vince Lyzinski and Eric Slud
  • Meeting Time: Organizational meeting Wed. August 28 at 12pm.  Regular meetings will be Wednesdays 12pm-12:50 in MTH 0201 beginning Sept. 4. 
  • Description: The RIT will be mostly expository, covering interesting material not currently available in our courses. Following Wainwright's book High-Dimensional Statistics, we will mostly emphasize non-asymptotic results related to statistical problems where the observations have dimension larger, and sometimes much larger, than the number of independent data-records. The book covers probability theory of tail-probability and concentration-of-measure  inequalities, some material on metric entropy and empirical-process inequalities and on random matrices and reproducing kernel Hilbert spaces, and extensive statistical applications. Those statistical problems relate to many "big-data" problems arising in modern Data Science. The book's audience is "first-year graduate students": it is very clearly written, and much of it will make sense to students with undergraduate-level probability background, but probably not to students without statistical background at the level of Stat 700. 
• RIT on Quantitative Ecological and Evolutionary Dynamics (QEED)
  
  • Organizers: Joshua Weitz (Biology), Emme Bruns (Biology), Bill Fagan (Biology), Phil Johnson (Biology), Vadim Karatayev (Biology), Evan Economo (Entomology), Stephanie Yarwood (ES&T), Michelle Girvan (Physics), Jim Yorke (Mathematics)
  • Meeting Time: Fridays, 11am-noon, ES&J 2204
  • Description: QEED will support early career scientists in developing and deploying quantitative modeling approaches to understand the impact of ecological and evolutionary dynamics on environmental and human health. By fostering an interdisciplinary research and training community, QEED will enable intellectual exchange, provide opportunities for trainees to present research in progress, expand understanding of research in related areas, improve research skills, and build collaborations.
• RIT on Algebraic Geometry: K3 Surfaces
  
  • Organizer: Dori Bejleri
  • Meeting Time: Organizing meeting Tuesday, September 3 at 11am in 3206.  Most meetings will be Fridays at 2:00.  The schedule will be posted on a discord server; contact Dr. Bejleri for the topic identifier.
    
  • Description: K3 surfaces are perhaps the most interesting examples of smooth 2-dimensional algebraic varieties.  This RIT will study their structure following the excellent book Lectures on K3 Surfaces by Daniel Huybrechts, Cambridge Univ. Press, 2016.  A preliminary version is online here.  RIT credit is available for participating students.
    
• RIT on Applied Harmonic Analysis
  • Organizer: Radu Balan
  • Meeting Time: 1:00pm-1:50pm Mondays, MTH 1310
  • Description: We plan to discuss topics in harmonic analysis and related fields (functional analysis, operator and representation theory) with applications to various fields such as signal processing, machine learning, graph representations, quantum information theory.
• RIT on Mathematical Finance/Financial Mathematics
  
  • Organizer: Dilip Madan, Finance
  • Meeting Time: Mondays, 4-5 PM, VMH2509
  • Course Credit: available for interested students under the number AMSC689, section no. 6505. 
• RIT on Weather, Chaos, and Data Assimilation
  
  • Organizers: Kayo Ide  and Brian Hunt
  • Meeting Time: Mondays 2-3pm; Contact organizers for more info.
  • Description: We study prediction and estimation problems for nonlinear dynamical systems with main applications in (but not limited to) earth system sciences. Emphasis is put on uncertainty quantification and reduction, and a rapidly emerging field for the integration of data assimilation and machine learning/artificial intelligence. 
• RIT on Physical Oceanography
  
  • Organizers: Professor James Carton () and Dr. Luyu Sun ()
  • Meeting Time: Every Monday; 12:00pm-1:00pm.
  • Where: Online (The link will be determined later)
  • Description:  This RIT in physical oceanography aims to bring together researchers, students, and professionals from applied mathematics, atmospheric science, and oceanic science who share an interest in the physical processes governing ocean behavior. Our discussions will span a wide range of topics, including:
    • Ocean Circulation: Explore large-scale circulation patterns such as the Gulf Stream and thermohaline circulation, and understand the impact of mesoscale eddies on heat and nutrient transport.
    • Air-Sea Interactions: Delve into the exchange of momentum, heat, and freshwater between the ocean and atmosphere, and their influence on weather and climate systems.
    • Data Assimilation Techniques: Discuss the application of optimization and statistical methods, such as Kalman filtering, to integrate observational data (e.g., satellite measurements, buoys, and drifters) into ocean models for more accurate predictions.
    • Numerical Simulation and Analysis: Examine numerical techniques used in simulating complex oceanic systems, from finite difference methods to machine learning approaches, and their applications in solving real-world challenges.
    Each meeting will feature a brief presentation on key concepts and recent findings, followed by an open forum for discussion. Participants are encouraged to share their research, insights, and questions in a collaborative environment.
  RIT on Stochastic Optimization
  • Organizers: Michael Fu, Management Science
  • Meeting Time: Not meeting Spring 2024
  • Description: The focus of this RIT will be gradient-based stochastic optimization methodologies and applications, including techniques for stochastic gradient estimation in simulation and other data-driven settings. Other statistical ranking & selection approaches are also considered, as well as Markov decision processes and reinforcement learning. Potential application areas include queueing systems, manufacturing, supply chain management, and financial engineering.
  • Prerequisites: strong background in prob/stats at the advanced undergraduate level; recommended: real analysis and measure theory
• RIT on Optimization and Equilibrium Problems with Applications in Engineering
  
  • Organizer:  Steven A. Gabriel, Dept. of Mechanical Engineering,
  • Meeting Time: Not meeting Spring 2023, but may restart in Fall 2023.
  • Description: We will cover a variety of problems in optimization and equilibrium modeling, a subject that includes convex optimization, game theory, economics, and has a strong connection to integer programming as well.  Applications are in energy, transportation, and other engineering-economic areas. Professor Gabriel will start with an overview of the field and some suggested applications.  Course credit is available for interested students.  Any student who wants credit will need to give a presentation on a topic of his/her choice in optimization/equilibrium modeling.
• RIT on Operations Research and Management Science
  
  • Organizers: Alex Estes, Bruce Golden, Raghu Raghavan, and Zhi-Long Chen
  • Meeting Time: Thursdays at 3:00 PM at VMH 4335
  • Description: The topic of the seminar will be operation research and management science. Topics may include stochastic processes, integer programming, stochastic optimization, and machine learning, and applications of these methods. Potential applications include inventory planning, drone and vehicle routing for package delivery, air traffic management, network design, and organ matching. 
• RIT on Quantum Information Science
  • Organizers: Maria Cameron, Carl Miller, Konstantina Trivisa
  • Meeting time: Mondays, 4pm-5pm, Kirwan Hall 3206.
  • Overview: In this seminar, we are interested in all aspects of research at the intersection between quantum information science and mathematics. In the Spring 2024 semester, we will focus on hybrid talks where speakers will (i) provide an overview of their research or a QIS topic, followed by (ii) working out a mathematical concept related to the QIS topic on the board, along with the other participants.
    Goals for the seminar include:
    • Studying recent research results in quantum information from a mathematical angle;
    • Finding examples (old and new) in which existing tools from mathematics have been adapted for application in quantum information;
    • Studying quantum algorithms for mathematical problems.
  • Course credit: Available for interested students. Contact Daniel Serrano <> for details.
• Informal Geometric Analysis Seminar
  • Organizers: Tamas Darvas and Yanir Rubinstein
  • Meeting Time: Tuesdays 3:30-5:00 PM
  • Location: MTH 2300
• RIT on Deep Learning
  
  • Organizers: Wojtek Czaja,  Turner Pepper, and Gabriel Vilarroubi
  • Meeting Time: Fridays at 1PM in MTH 1310.
• Student Geometry-Topology Seminar
  • Organizer: Jacob Erickson
  • Meeting Time: Fridays at 3:00PM in MTH 1308.
• Student Algebra-Number Theory Seminar
  • Organizer: Jackson Hopper
  • Meeting Time: to be determined.  Contact Jackson if you want to have input on this.
• Joint AMSC, MATH and STAT (JAMS) Student Seminar
  
  • Organizers: tRevati Jadhav (), Brandon Kolstoe (), and the Graduate Student Committee (GSCAMS) ()
  • Meeting Time: Thursdays, 4:30 - 5:30 PM
  • Location: John S. Toll Physics Building 2208
  • Description: The goal of this student seminar is to provide elementary introductions to a diverse array of mathematical topics in the style of the popular "What is ...?" series, prominantly featured in the AMS Notices.
• RIT on Mathematics of Infectious Diseases
  • Orgainizers: Abba Gumel
  • Meeting Time: Thursdays 2:00PM - 4:00PM, MTH 1311