Spring/Summer 2024

Representation Theory and Applications

UNIST Mathematical Sciences Undergraduate/Graduate Course

Confirmed Schedule:
Mondays and Wednesdays, 13:00 - 14:15
February - June 2024

Location:
Natural Sciences Building 108
Department of Mathematical Sciences
Ulsan National Institute of Science and Technology (UNIST)

Fall/Winter 2023

General
Topology

UNIST Mathematical Sciences Undergraduate Course

Confirmed Schedule:
Mondays and Wednesdays, 13:00-14:15
September - December 2023

Location:
Natural Sciences Building 108
Department of Mathematical Sciences
Ulsan National Institute of Science and Technology (UNIST)

Spring/Summer 2023

Mathematical Foundations of Machine Learning

UNIST Mathematical Sciences Undergraduate Course

Confirmed Schedule:
Mondays and Wednesdays, 10:30 - 11:45
February - June 2023

Location:
Natural Sciences Building 108
Department of Mathematical Sciences
Ulsan National Institute of Science and Technology (UNIST)

Fall/Winter 2022

General
Topology

UNIST Mathematical Sciences Undergraduate Course

Confirmed Schedule:
Mondays and Wednesdays, 9:00-10:15
September - December 2022

Location:
Natural Sciences Building 108
Department of Mathematical Sciences
Ulsan National Institute of Science and Technology (UNIST)

Spring/Summer 2022

Mathematical Foundations of Machine Learning

UNIST Mathematical Sciences Undergraduate Course

Confirmed Schedule:
Mondays and Wednesdays, 13:00-14:15
February - June 2022

Location:
Natural Sciences Building 108
Department of Mathematical Sciences
Ulsan National Institute of Science and Technology (UNIST)

Summer/Fall 2021

Algebraic Topology

UNIST Mathematical Sciences Graduate Course

Geometrical and topological complexity is abundant even beyond the realms of mathematics. The aim of this course is to give a glimpse into how geometric structures can be studied by interpreting them algebraically, allowing us to computationally measure their complexity. The course will illustrate connections between algebraic topology and other areas of mathematics as well as mathematical physics. The course will cover a selection of standard topics in algebraic topology ranging from homotopy and fundamental groups to homology and cohomology of arbitrary spaces.

Confirmed Schedule:
Tuesdays and Thursdays, 16:00-17:15
September - December 2021

Location:
Natural Sciences Building 108
Department of Mathematical Sciences
Ulsan National Institute of Science and Technology (UNIST)

Lecture 1 [August 31, 2021]:
1.0 Why Algebraic Topology?
1.1 Fundamental Concepts: Spaces
1.2 Equivalence and Topology
1.3 Topological Spaces
1.4 Topological Equivalence

Lecture 2 [September 2, 2021]:
2.0 Recap Lecture 1
2.1 Topology and Topological Spaces
2.2 Topological Equivalence
2.3 Paths and Connectedness
2.4 Deformation and Retraction
2.5 Homotopy

Lecture 3 [September 7, 2021]:
3.0 Recap Lecture 2
3.1 Homotopy
3.1 Equivalence Relations and Homotopy Classes
3.2 Paths Revisited

Lecture 4 [September 9, 2021]:
4.0 Recap Lecture 3
4.1 Products of Paths
4.2 Homotopy and Products of Paths
4.3 Loops
4.4 Fundamental Group

Lecture 5 [September 14 2021]:
5.0 Recap Lecture 4
5.1 Fundamental Group Revisited
5.2 Properties of the Fundamental Group

Lecture 6 [September 16 2021]:
6.0 Recap Lecture 5
6.1 Properties of the Fundamental Group II
6.2 Group Homomorphisms and the Fundamental Group
6.3 Homotopy and the Fundamental Group

Lecture 7 [September 23 2021]:
7.0 Recap Lecture 6
7.1 Contractible Spaces
7.2 Simply-Connected Spaces
7.3 Fundamental Group of the Circle

Lecture 8 [September 30 2021]:
8.0 Recap Lecture 7
8.1 Path Lifting Property
8.2 Homotopy Lifting Theorem
8.3 Degree
8.4 Degree Maps
8.5 Fundamental Group of other Spaces

Lecture 9 [October 4 2021]:
9.0 Recap Lecture 8
9.1 Homotopy Equivalence
9.2 Deformation Retract
9.3 Fundamental Group of the Punctured Plane
9.4 Fundamental Group of a Cylinder
9.5 Fundamental Group of the Torus
9.6 Seifert-Van Kampen Theorem

Lecture 10 [October 8 2021]:
10.0 Recap Lecture 9
10.1 Geometrically Independent Points
10.2 Hyperplanes
10.3 k-Simplex

Lecture 11 [October 12 2021]:
11.0 Recap Lecture 10
11.1 More about k-Simplices
11.2 Open k-Simplex and Faces
11.3 Simplicial Complex
11.4 Polyhedra and Triangulations

Lecture 12 [October 14 2021]:
12.0 Recap Lecture 11
12.1 Polyhedra and Skeletons
12.2 Simplicial Maps
12.3 Simplicial Approximation

Lecture 13 [October 25 2021]:
13.0 Recap Lecture 12
13.1 Star-Related Simplicial Complexes
13.2 Barycentric Subdivision
13.3 Simplicial Approximation Theorem

Lecture 14 [October 28 2021]:
14.0 Recap Lecture 13
14.1 Preview of Simplicial Homology Groups
14.2 Orientation of Simplicial Complexes
14.3 Simplicial Chain Complex

Lecture 15 [November 2 2021]:
15.0 Recap Lecture 14
15.1 Chain Group

Lecture 16 [November 4 2021]:
16.0 Recap Lecture 15
16.1 Boundary Homomorphism
16.2 Cycles and Cycle Groups

Lecture 17 [November 11 2021]:
17.0 Recap Lecture 16
17.1 Boundaries and Boundary Groups
17.2 Homology Groups

Lecture 18 [November 12 2021]:
18.0 Recap Lecture 17
18.1 Homology Groups of a 2-Simplex
18.2 Homology Groups of a 3-Simplex

Lecture 19 [November 16 2021]:
19.0 Recap Lecture 18
19.1 Homology Groups of a Circle
19.2 Homology Groups of a Cylinder
19.3 Homology Groups of a Torus
19.4 Classification of Homology Groups of 2-Manifolds

Lecture 20 [November 18 2021]:
20.0 Recap Lecture 19
20.1 Homology Groups of a Klein bottle
20.2 Properties of Integral Homology Groups
20.3 Betti numbers and Torsion Coefficients

Lecture 21 [November 23 2021]:
21.0 Recap Lecture 20
21.1 Euler characteristic
21.2 Euler-Poincare Theorem

Lecture 22 [November 30 2021]:
22.0 Recap Lecture 21
22.1 Induced Homomorphisms
22.2 Topological Invariance of Homology Groups

Lecture 23 [December 2 2021]:
23.0 Recap Lecture 22
23.1 Homology Groups as Homotopy Invariants
23.2 Subdivision Chain Map
23.3 Homotopy Invariance

Lecture 24 [December 7 2021]:
24.0 Recap Lecture 23
24.1 Revisiting the Simplicial Chain Complex
24.2 Co-Chains and the Co-Boundary Map
24.3 Simplicial Co-Chain Complex and the Cohomology Group

Lecture 25 [December 9 2021]:
25.0 Recap Lecture 24
25.1 Examples of Cohomology
25.2 0-Dimensional Cohomology
25.3 Poincare Duality

Lecture 26 [December 10 2021]:
26.0 Recap Lecture 25
26.1 Consequences of Poincare Duality
26.2 Kenneth Formula
26.3 Cup Product

Spring/Summer 2018

Brane Dynamics and Supersymmetric Gauge Theories

Tsinghua University YMSC
Graduate Course

This course is an introduction to branes in string theory and how they give rise to supersymmetric gauge theories in various dimensions. We will first cover the classification of different types of branes that arise in string theory and the bound states they form. We will then move on to the brane realizations of supersymmetric gauge theories in 2,3,4,5 and 6 dimensions. While doing so, we will explore how gauge theory phenomena such as dualities arise in terms of branes and how brane dynamics gives a powerful geometrical interpretation of such gauge theory phenomena. This course is intended to be a pedagogical introduction to the topic of branes in string theory with examples taken from modern developments in string theory, including topics like mirror symmetry and the Higgs and Coulomb branch of N=4 theories, branes at Calabi-Yau singularities and recent applications of Hanany-Witten brane setups.

Prerequisite:
Quantum Field Theory, Supersymmetry, Representation Theory

Confirmed Schedule:
Tuesdays and Thursdays, 17:00-18:40
March 6 - June 14, 2018

Location:
Yau Mathematical Sciences Center, Tsinghua University,
Conference Room 1, Jin Chun Yuan West Building

Announcement:
http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=1056

Lecture 1 [March 6, 2018]:
1.0 Why String Theory?
1.1 Strings & Branes
1.2 Spacetime Revisited
1.3 Symmetries & Spacetime
1.4 SO(8), D4

Lecture 2 [March 8, 2018]:
2.0 Recap Lecture 1
2.1 Representation Theory Revisited: SU(n) & SO(2n) Irreps, Characters
2.2 Supersymmetry & Chirality
2.3 10d Massless Supergravity Multiplets
2.4 10d SO(8) vs 11d SO(9)

Lecture 3 [March 13, 2018]:
3.0 Recap Lecture 2
3.1 Representation Theory Revisited: SU(2n+1) Irreps, Characters
3.1 Dimensional Reduction & Branching Rules
3.2 What about d<10?
3.3 Maxwell's Equations Revisited

Lecture 4 [March 15, 2018]:
4.0 Recap Lecture 3
4.1 Rewriting Maxwell's Equations
4.2 Brane Spectroscopy (M, IIA, IIB)

Lecture 5 [March 20, 2018]:
5.0 Recap Lecture 4
5.1 Electromagnetic Duality
5.2 The Vacuum and Moduli Spaces
5.3 Strings on Branes
5.4 Gauss' Law
5.5 Towards Branes ending on Branes

Lecture 6 [March 22, 2018]:
6.0 Recap Lecture 5
6.1 D-Branes and Magnetically Charged Objects
6.2 Vector Multiplets
6.3 Symmetries and Symmetry Breaking under a D-Brane
6.4 M2 ending on M5

Lecture 7 [March 27, 2018]:
7.0 Recap Lecture 6
7.1 M2 on M5 and Electromagnetic Duality

Lecture 8 [March 29, 2018]:
8.0 Recap Lecture 7
8.1 NS5-Branes
8.2 Multiplets in 5d
8.3 NS5- and D5-Branes

Lecture 9 [April 3, 2018]:
9.0 Recap Lecture 8
9.1 Counting Supersymmetries
9.2 Multiplets and Supersymmetry
9.3 The Tensor Multiplet and M-Theory
9.4 Towards Actions

Qingming Festival (Tomb-Sweeping Holiday) [April 5, 2018]

Lecture 10 [April 10, 2018]:
10.0 Recap Lecture 9
10.1 Comments about Basic Interaction Terms
10.2 Gauge Invariance
10.3 Towards the Worldsheet Action
10.4 Preview of new term in the action

Lecture 11 [April 12, 2018]:
11.0 Recap Lecture 10
11.1 Two Branes approaching each other
11.2 The Chern-Simons Term

Lecture 12 [April 17, 2018]:
12.0 Recap Lecture 11
12.1 Electromagnetic Duality on a Brane
12.2 Examples of Branes ending on Branes

Lecture 13 [April 19, 2018]:
13.0 Recap Lecture 12
13.1 D2 on D4
13.2 F1 on D1
13.3 D5 on NS5
13.4 IIB and SL(2,Z)

Lecture 14 [April 24, 2018]:
14.0 Recap Lecture 13
14.1 The Theory on a Dp-Brane
14.2 Tension of a String and Couplings
14.3 S-duality

Lecture 15 [April 26, 2018]:
15.0 Recap Lecture 14
15.1 Dimensional Reduction Revisited
15.2 S1 Compactification: M to IIA
15.3 Momentum Modes and KK Monopoles
15.4 M-Theory on 2-Torus

May 1 Week Holiday [May 1 & 3, 2018]

Lecture 16 [May 8, 2018]:
16.0 Recap Lecture 15
16.1 M-Theory on T2 vs IIA/IIB on S1
16.2 T-duality
16.3 S-duality, T-duality and U-duality

Lecture 17 [May 10, 2018]:
17.0 Recap Lecture 16
17.1 Duality Chains and Brane Configurations
17.2 Revisiting the Worldvolume Theory on a D-Brane
17.3 Dp on Dp+2 and Dp on NS5
17.4 Vector and Hypermultiplets

Lecture 18 [May 15, 2018]:
18.0 Recap Lecture 17
18.1 Chan-Paton Factors
18.2 A NS5, D5 and D3 configuration
18.3 The Field Theory on the D3-brane

Lecture 19 [May 17, 2018]:
19.0 Recap Lecture 18
19.1 Vector and Hypermultiplets in 3d N=4
19.2 Brane Singularities
19.3 The Coulomb Branch and Magnetic Monopoles in 3d N=4

Lecture 20 [May 18, 2018 - extra lecture]:
20.0 Recap Lecture 19
20.1 The Construction of the 3d N=4 Coulomb Branch
20.2 Special Examples of the Coulomb Branch
20.3 A Hypermultiplet Paradox?

Lecture 21 [May 22, 2018]:
21.0 Recap Lecture 20
21.1 Hanany-Witten Transitions
21.2 Linking Numbers for 5-Branes
21.3 Linking Number Properties

Lecture 22 [May 24, 2018]:
22.0 Recap Lecture 21
22.1 Multiple Branes and Hanany-Witten Transitions
22.2 Linking Numbers and Partitions
22.3 Gaiotto-Witten Theories
22.4 Mirror Symmetry and Partitions

Lecture 23 [May 29, 2018]:
23.0 Recap Lecture 22
23.1 Conditions on Partitions and Non-Supersymmetric Theories
23.2 Higgs and Coulomb Branches of 3d N=4 Theories
23.3 Kraft-Procesi Transitions and Hasse Diagrams

Lecture 24 [May 31, 2018]:
24.0 Recap Lecture 23
24.1 Examples of Hasse Diagrams
24.2 4d N=1 Theories and SQCD
24.3 F-and D-Terms and Vacuum Moduli Spaces
24.4 The Moduli Space for SQCD

Lecture 25 [June 1, 2018]:
25.0 Recap Lecture 24
25.1 Different Moduli Spaces for SQCD
25.2 Seiberg Duality for 4d N=1 Theories
25.3 Abelian Orbifolds and Orbifold Theories

Lecture 26 [June 5, 2018]:
26.0 Recap Lecture 25
26.1 4d Orbifold Theories
26.2 Classification of Abelian Orbifolds
26.3 The Theory for k U(N) Instantons and the ADHM Construction
26.4 The Theory for k U(N) Vortices

Lecture 27 [June 7, 2018]:
27.0 Recap Lecture 26
27.1 Plethystics and Hilbert Series
27.2 Algebraic Varieties and Coordinate Rings
27.3 Counting Gauge Invariant Operators

Lecture 28 [June 8, 2018]:
28.0 Recap Lecture 27
28.1 Quiver Gauge Theories and Toric Varieties
28.2 Klebanov-Witten Theory and the Conifold
28.3 Brane Tilings

Fall/Winter 2018

Calculus I

Tsinghua University YMSC
Undergraduate Course

This introductory course covers topics on undergraduate-level Calculus.

Confirmed Schedule:
Mondays and Thursdays, 19:20-21:45
September 2018 - January 2019

Location:
Teaching Building 6, Tsinghua University,
Room 6A018