# Conference on Algebraic Geometry, Representation theory and Mathematical Physics

From April 29 to May 1, 2019 the CMSA will be hosting a Conference on Algebraic Geometry, Representation theory and Mathematical Physics. This workshop is organized by Bong Lian (Brandeis) and Artan Sheshmani (CMSA) . The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.

List of registrants

#### Speakers:

Monday, April 29

 Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Wei Zhang, MIT Title: The arithmetic fundamental lemma for diagonal cycles Abstract: I’ll recall the Gross–Zagier theorem and a high dimensional generalization, the arithmetic Gan-Gross-Prasad conjecture, which relates the height pairing of arithmetic diagonal cycles on certain shimura varieties to the first order derivative of certain L-functions.  The arithmetic fundamental lemma conjecture arises from the relative trace formula approach to this conjecture. I will recall the statement of the arithmetic fundamental lemma and outline a proof. 10:00 – 10:30am Break 10:30 – 11:30am Yuri Tschinkel, NYU Title: Equivariant birational geometry and modular symbols Abstract: We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups (joint with M. Kontsevich and V. Pestun). 11:30 – 1:30pm Lunch 1:30 – 2:30pm Alexander Efimov, Moscow Title: Torsionness for regulators of canonical extensions Abstract: I will sketch a generalization of the results of Iyer and Simpson arXiv:0707.0372 to the general case of a normal-crossings divisor at infinity. 2:30 – 3:00pm Break 3:00 – 4:00pm Amin Gholampour, Maryland Title: Euler Characteristics of punctual quot schemes on threefolds Abstract: Let F be a homological dimension 1 torsion free sheaf on a nonsingular quasi-projective threefold. The first cohomology of the derived dual of F is a 1-dimension sheaf G supported on the singular locus of F. We prove a wall-crossing formula relating the generating series of the Euler characteristics of Quot(F, n) and Quot(G,n), where Quot(-,n) denotes the quot scheme of length n quotients. We will use this relation in studying the Euler characteristics of the moduli spaces of stable torsion free sheaves on nonsingular projective threefolds. This is a joint work with Martijn Kool. 4:00 – 4:30pm Break 4:30 – 5:30pm Maksym Fedorchuck, BC Title:  Stability of one-parameter families of weighted hypersurfaces Abstract:  We define a notion of stability for fibrations over a curve with generic fibers being weighted hypersurfaces (in some weighted projective space) generalizing Kollár’s stability for families of hypersurfaces in a projective space.  The stability depends on a choice of an effective line bundle on the parameter space of weighted hypersurfaces and different choices pick out different birational model of the total space of the fibration. I will describe enumerative geometry that goes into understanding these stability conditions, and, if time permits, examples where this machinery can be used to produce birational models with good properties.  Joint work with Hamid Ahmadinezhad and Igor Krylov.

Tuesday, April 30

 Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Brendan Hassett, Brown Title: Rationality for geometrically rational threefolds Abstract: We consider rationality questions for varieties over non-closed fields that become rational over an algebraic closure, like smooth complete intersections of two quadrics.  (joint with Tschinkel) 10:00 – 10:30am Break 10:30 – 11:30am Dennis Gaitsgory, Harvard Title: The Fundamental Local Equivalence in quantum geometric Langlands Abstract: The Fundamental Local Equivalence is statement that relates the q-twisted  Whittaker category of the affine Grassmannian for the group G and the category of modules over the Langlands dual “big” quantum group. The non-triviaiity of the statement lies is the fact that the relationship between the group and its  dual is combinatorial, so to prove the FLE one needs to express both sides in combinatorial terms. In the talk we will indicate the proof of a related statement for the “small” quantum group. The combinatorial link is provided by the category of factorization modules over a certain factorization algebra, which in itself is a geometric device that concisely encodes the root data. 11:30 – 1:00pm Lunch 1:00- 2:00pm Andrei Negut, MIT Title: AGT relations in geometric representation theory Abstract: I will survey a program that seeks to translate the Alday-Gaiotto-Tachikawa correspondence (between gauge theory on R^4 and conformal field theory) into the language of algebraic geometry. The objects of study become moduli spaces of sheaves on surfaces, and the goal is to connect them with the W-algebra of type gl_n. 2:00 – 2:15pm Break 2:15 – 3:15pm Dan Abramovich, Brown Title: Resolution in characteristic 0 using weighted blowing up Abstract: Given a variety $X$, one wants to blow up the worst singular locus, show that it gets better, and iterate until the singularities are resolved. Examples such as the whitney umbrella show that this iterative process cannot be done by blowing up smooth loci – it goes into a loop. We show that there is a functorial way to resolve varieties using \emph{weighted} blowings up, in the stack-theoretic sense. To an embedded variety $X \subset Y$ one functorially assigns an invariant $(a_1,\ldots,a_k)$, and a center locally of the form $(x_1^{a_1} , \ldots , x_k^{a_k})$, whose stack-theoretic weighted blowing up has strictly smaller invariant under the lexicographic order. This is joint work with Michael Tëmkin (Jerusalem) and Jaroslaw Wlodarczyk (Purdue), a side product of our work on functorial semistable reduction. A similar result was discovered by G. Marzo and M. McQuillan. 3:15 – 3:30pm Break 3:30 – 4:30pm Fedor Bogomolov, NYU Title: On the base of a Lagrangian fibration for a compact hyperkahler manifold. Abstract: In my talk I will discuss our proof with N. Kurnosov that the base of such fibration for complex projective manifold hyperkahler manifold of dimension $4$ is always a projective plane $P^2$. In fact we show that the base of such fibration can not have a singular point of type $E_8$. It was by the theorem of Matsushita and others that only quotient singularities can occur and if the base is smooth then the it is isomorphic to $P^2$. The absence of other singularities apart from $E_8$ has been already known and we show that $E-8$ can not occur either. Our method can be applied to other types of singularities for the study of  Lagrangian fibrations in higher dimensions More recently similar result was obtained by Huybrechts and Xu. 4:30 – 4:45pm Break 4:45 – 5:45pm Dawei Chen, BC Title: Volumes and intersection theory on moduli spaces of Abelian differentials Abstract: Computing volumes of moduli spaces has significance in many fields. For instance, Witten’s conjecture regarding intersection numbers on moduli spaces of Riemann surfaces has a fascinating connection to the Weil-Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. In this talk I will introduce an analogue of Witten’s intersection numbers on moduli spaces of Abelian differentials to compute the Masur-Veech volumes induced by the flat metric associated with Abelian differentials. This is joint work with Moeller, Sauvaget, and Zagier (arXiv:1901.01785).

Wednesday, May 1

 Time Speaker Title/Abstract 8:30 – 9:00am Breakfast 9:00 – 10:00am Pavel Etingof, MIT Title: Short star-products for filtered quantizations Abstract: This is joint work with Eric Rains and Douglas Stryker. 10:00 – 10:30am Break 10:30 – 11:30am Roman Bezrukavnikov, MIT Title: Stability conditions and representation theory Abstract: I will recall the concept of real variation of stabilities (introduced in my work with Anno and Mirkovic)and its relation to modular Lie algebra representations. I will also address a potential generalization of that picture to modular representations of affine Lie algebras related to the classical limit of geometric Langlands duality and its local counterpart. 11:30 – 11:45am Break 11:45 – 12:45pm Qile Chen, BC Title: Counting curves in critical locus via logarithmic compactification Abstract: An R-map consists of a pre-stable map to possibly non-GIT quotient together with sections of certain spin bundles. The moduli of R-maps are in general non-compact. When the target of R-maps is equipped with a super-potential W with compact critical locus, using Kiem-Li cosection localization it has been proved by many authors in various settings that the virtual cycle of R-maps can be represented by the cosection localized virtual cycle which is supported on the proper locus consisting of R-maps in the critical locus of W. Though the moduli of R-maps is equipped with a natural torus action by scaling of the spin bundles, the non-compactness of the R-maps moduli makes such powerful torus action useless. In this talk, I will introduce a logarithmic compactification of the moduli of R-maps using certain modifications of stable logarithmic maps. The logarithmic moduli space carries a canonical virtual cycle from the logarithmic deformation theory. In the presence of a super-potential with compact critical locus, it further carries a reduced virtual cycle. We prove that (1) the reduced virtual cycle of the compactification can be represented by the cosection localized virtual cycle; and (2) the difference of the canonical and reduced virtual cycles is another reduced virtual cycle supported along the logarithmic boundary. As an application, one recovers the Gromov-Witten invariants of the critical locus as the invariants of logarithmic R-maps of its ambient space in an explicit form. The latter can be calculated using the spin torus action. This is a joint work with Felix Janda and Yongbin Ruan. 12:45 – 2:30pm Lunch 2:30 – 3:30pm Si Li, Tsinghua Title: Semi-infinite Hodge structure: from BCOV theory to Seiberg-Witten geometry Abstract: I will explain how the semi-infinite Hodge theory extends Kodaira-Spencer gravity (Bershadsky-Cecotti-Ooguri-Vafa theory of B-twisted closed topological string field theory) into a full solution of Batalin-Vilkovisky master equation. This allows us to formulate quantum B-model via a rigorous BV quantization method and construct integrable hierarchies arising naturally from the background symmetry. In the second part of the talk, I will explain the recent discovery of the connection between K.Saito’s primitive form and 4d N=2 Seiberg-Witten geometry arising from singularity theory. 3:30 – 4:00pm Break 4:00 – 5:00pm Ludmil Katzarkov, Moscow Title: PDE’s non commutative  motives and HMS. Abstract: In this talk we will discuss the theory of central manifolds and the new structures in geometry it produces. Application to Bir.  Geometry will be discussed.