1. S.G., “Surface Operators and Knot Homologies,” arXiv:0706.2369 S.G., A.Iqbal, C.Kozcaz, C.Vafa, “Link homologies and the refined topological vertex,” arXiv:0705.1368 S.G., E.Witten, “Gauge theory, ramification, and the geometric Langlands program,” hep- th/0612073 S.G., H.Murakami, “SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial,” math.gt/0608324 Geometric Structures in Gauge Theory Sergei Gukov

2. During the past year I have been mainly working on topological string theory and topological gauge theory which, roughly speaking, describe the supersymmetric sector of the physical string theory/gauge theory.

3. Applications Physical Applications F-terms in string theory compactifications Black Hole physics dynamics of SUSY gauge theory Mathematical Applications Enumerative geometry Homological algebra Low-dimensional topology Representation theory Gauge theory

4. Chern-Simons Theory In Chern-Simons theory Example: G = SU(2) [E.Witten] polynomial in q Wilson loop operator Jones polynomial

5. Question (M.Atiyah): Why integer coefficients?

6. Knot Homologies Khovanov homology: Example : [M.Khovanov] j 5793 1 3 0 1 2 i

7. Physical Interpretation BPS state: membrane ending on the Lagrangian five-brane space of BPS states [S.G., A.Schwarz, C.Vafa] M-theory on M5-brane on Earlier work: [H.Ooguri, C.Vafa] [J.Labastida, M.Marino, C.Vafa] (conifold) Lagrangian

8. This Physical Interpretation Leads to Many New Results and Surprising Predictions new theory, which unifies all the existing knot homologies [N.Dunfield, S.G., J.Rasmussen] generalization to arbitrary groups and representations [S.G., A.Iqbal, C.Kozcaz, C.Vafa] mathematical construction of these is out of reach [S.G., J.Walcher]

9. Codimension 3: Line operators: Codimension 4: Local operators Wilson line‘t Hooft line Codimension 2: Surface operators much studied in AdS/CFT Realization in 4D Gauge Theory Surface operators

10. Transform in an interesting way under Electric-Magnetic duality OPE algebra of line operators becomes non- commutative It turns out that many interesting 4D gauge theories admit (supersymmetric) surface operators, which have a number of nice properties: surface operator S Braid group actions on D-branes [S.G., E.Witten]

11. It turns out that many interesting 4D gauge theories admit (supersymmetric) surface operators, which have a number of nice properties: Mathematical applications –to the so-called ramified case of the Geometric Langlands program –to Knot Homologies Correlation function = vector space vector space

12. Open questions and further directions New types of surface operators and their transformation under S-duality Realization of Chern-Simons invariants in four-dimensional gauge theory Geometric construction of representations using surface operators in gauge theory …