Informal Mathematical Physics Seminar

Schedule of talks for Fall 2016:

A note to the speakers: this is an informal seminar, meaning that the talks are longer than usual (1:30) and are expected to include a good introduction to the subject as well as a maximally accessible (i.e. minimally general & minimally technical) discussion of the main result. The bulk of the audience is typically formed by beginning graduate students. Blackboard talks are are particularly encouraged.

Abstracts

September 12

Classical Chow varieties parametrizing effective algebraic cycles are defined as closed subsets in a projective space, with a reduced scheme structure. This does not give a good concept of a "family of effective cycles" over an arbitrary base scheme S, which is needed in applications. Some work in this direction has been done by Kollar (in the case when S is semi-normal) and Angeniol (characteristic zero, involving conditions which are rather hard to verify). We propose a new approach based on a concept of Intersection

Bundles introduced by Deligne in the 80s as a part of his program in the study of determinant of cohomology (later implemented by Elkik, Munoz-Garcia and Ducrot). We turn this approach inside out and say that a family of effective cycles is "whatever defines Intersection Bundles on the base". This gives a "Chow functor" which is automatically equipped with the "Quot to Chow morphism" for flat families of subschemes or coherent sheaves.

September 19

Suppose you have a finite group G and you want to study certain related structures (random walks, expander graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some important conjectures. The “standard" method to construct representations of finite classical group is due to Deligne and Lusztig (1976). However, it seems that their approach has relatively little to say about the small representations.

This talk will discuss a joint project with Roger Howe (Yale), where we introduce a language to define, and a new method for systematically construct, the small representations of finite classical groups.

I will demonstrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).

October 3

This will be a colloquium-style talk, followed by a half-hour more technical talk.

First hour:

The development of homological mirror symmetry has led to deep conjectures on the relationship between birational geometry and invariants of algebraic varieties which are more homological in nature. Most notable is the conjecture, due to Bondal and Orlov, that two varieties which differ by a flop have equivalent "derived categories." I will discuss how equivariant geometry sheds new light on this and related conjectures, leading to a proof in many new higher dimensional examples arising as moduli spaces. The key technique is a new theory of ``Theta-stratifications" which are analogous to equivariant Morse stratifications in the setting of algebraic geometry.

Next half-hour:

The Verlinde formula is a celebrated and classic computation of the dimension of the space of sections of certain line bundles on the moduli space of principal G-bundles on a smooth algebraic curve. I will discuss how the same method of stratification from the first part of the talk can be used to prove a version of this formula on the moduli space of Higgs bundles on a curve.

October 10

The moduli spaces of local systems on decorated surfaces enjoy many nice properties. In particular, it was shown by Fock and Goncharov that they form examples of cluster varieties, which means that they are Poisson varieties with a positive atlas of toric charts, and thus admit canonical quantizations. I will describe joint work with A. Shapiro in which we embed the quantized enveloping algebra U_q(sl_n) into the quantum character variety associated to a punctured disk with two marked points on its boundary. The construction is closely related to the (quantized) multiplicative Grothendieck-Springer resolution for SL_n. I will also explain how the R-matrix of U_q(sl_n) arises naturally in this topological setup as a (half) Dehn twist. Time permitting, I will describe some potential applications to the study of positive representations of the split real quantum group U_q(sl_n,R)

October 24

I will explain how to refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and to prove the first non-trivial cases of these conjectures. These results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder-Stevens-Varchenko and yield precise formulations for affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the speaker to relate these conjectures for U_q(sl_2 hat) to evaluations of certain theta hypergeometric integrals defined by Felder-Varchenko. We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral of Spiridonov.

October 31

Bloch-Okounkov studied certain functions on partitions f called shifted symmetric polynomials. They showed that certain q-series arising from these functions (the so-called q-brackets <f>q) are quasimodular forms. We revisit a family of such functions, denoted Qk, and study the p-adic properties of their q-brackets. To do this, we define regularized versions Qk(p) for primes p. We also use Jacobi forms to show that the <Qk(p)>q are quasimodular and find explicit expressions for them in terms of the <Qk>q

November 7

Associated to a pair of algebras quantizing the same graded, Poisson algebra, we have a category of Harish-Chandra bimodules. These have been studied with some detail in the context of universal enveloping algebras, finite W-algebras and hypertoric enveloping algebras, among others. I will introduce this concept in the setting of rational Cherednik algebras, with an emphasis on the relationship between Harish-Chandra bimodules and category O. This relationship is more clearly seen in type A and I will focus on this case. Time permitting, I will say how things change in other types, too.

November 21

In number theory there is a method for extracting Fourier coefficients from an automorphic function, which underlies the notion of Whittaker model. Frenkel, Gaitsgory, and Vilonen carried out an analogous construction in algebraic geometry by defining categories of Whittaker sheaves on variants of Drinfeld's compactification. In this talk I will discuss the degeneration of a Whittaker sheaf as its Fourier coefficients go to zero. This degeneration can be encoded in a perverse sheaf on Drinfeld's compactification using the operation of nearby cycles. I will describe the fibers of this sheaf in terms of the Langlands dual Lie algebra, proving along the way that it is tilting with respect to the stratification by defect. I will also compute its Jordan-Holder series.

November 28

I will present a construction of the q-deformed W-algebra of type gl_r that avoids the free field realization. The upshot is to allow us to construct an action of this algebra on the K-theory of the moduli space of rank r framed sheaves, and finally to interpret the Carlsson-Okounkov Ext operator as a W-algebra intertwiner. I will briefly survey much of the physics that goes into the problem, explicitly present several constructions of W-algebras, and mention possible extensions of the problem to moduli of semistable sheaves on projective surfaces.

December 5

The construction of an effective action for many field theories requires

some form of renormalization. This talk will be on the inductive

position space renormalization procedure developed by Costello for this

purpose. We will introduce the procedure, using the example of a scalar

field theory for simplicity, and show how the procedure was clarified

for closed manifolds and extended to a class of compact manifolds with boundary

December 12

I'll show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, I'll define a q-deformation of this algebra, called cyclotomic DAHA. Namely, I'll give a q-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory. In addition, I'll explain that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, I'll apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of q-deformed quasiinvariants. This is joint work with A. Braverman and M. Finkelberg.

organized by Igor
Krichever and Andrei
Okounkov

Mondays, 5:30, Room 507

To sign up for dinner click hereSchedule of talks for Fall 2016:

Sept 5 | no seminar | Labor Day |

Sept 12 | Vladimir Baranovsky | Chow functor in the projective case |

Sept 19 | Shamgar Gurevich | Small Representations of finite classical groups |

Sept 26 | no seminar | |

Oct 4, 622 Note special day and location |
Dan Halpern-Leistner |
On the methods of equivariant Morse theory
in algebraic geometry |

Oct 10 | Gus Schrader |
Quantum groups from character varieties |

Oct 17 | no seminar | |

Oct 24 | Yi Sun | Affine Macdonald conjectures and special
values of Felder-Varchenko functions |

Oct 31 | Sarah Trebat-Leder | On p-adic Modular Forms and the
Bloch-Okounkov Theorem |

Nov 7 | José Simental | Harish-Chandra bimodules for rational
Cherednik algebras |

Nov 14 | no seminar | |

Nov 21 | Justin Campbell |
Nearby cycles of Whittaker sheaves |

Nov 28 | Andrei Negut |
q-deformed W-algebras, Ext operators and the
AGT-W relations |

Dec 5 2:30-4:00 (note special time) |
Ben Albert |
Heat Kernel Renormalization on Manifolds with
Boundary |

Dec 12 | Pavel Etingof | Cyclotomic Double affine Hecke algebras |

A note to the speakers: this is an informal seminar, meaning that the talks are longer than usual (1:30) and are expected to include a good introduction to the subject as well as a maximally accessible (i.e. minimally general & minimally technical) discussion of the main result. The bulk of the audience is typically formed by beginning graduate students. Blackboard talks are are particularly encouraged.

Abstracts

September 12

Classical Chow varieties parametrizing effective algebraic cycles are defined as closed subsets in a projective space, with a reduced scheme structure. This does not give a good concept of a "family of effective cycles" over an arbitrary base scheme S, which is needed in applications. Some work in this direction has been done by Kollar (in the case when S is semi-normal) and Angeniol (characteristic zero, involving conditions which are rather hard to verify). We propose a new approach based on a concept of Intersection

Bundles introduced by Deligne in the 80s as a part of his program in the study of determinant of cohomology (later implemented by Elkik, Munoz-Garcia and Ducrot). We turn this approach inside out and say that a family of effective cycles is "whatever defines Intersection Bundles on the base". This gives a "Chow functor" which is automatically equipped with the "Quot to Chow morphism" for flat families of subschemes or coherent sheaves.

September 19

Suppose you have a finite group G and you want to study certain related structures (random walks, expander graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these formulas seemed to be lack of knowledge over the low dimensional representations of G. In fact, the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge of them might lead to proofs of some important conjectures. The “standard" method to construct representations of finite classical group is due to Deligne and Lusztig (1976). However, it seems that their approach has relatively little to say about the small representations.

This talk will discuss a joint project with Roger Howe (Yale), where we introduce a language to define, and a new method for systematically construct, the small representations of finite classical groups.

I will demonstrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).

October 3

This will be a colloquium-style talk, followed by a half-hour more technical talk.

First hour:

The development of homological mirror symmetry has led to deep conjectures on the relationship between birational geometry and invariants of algebraic varieties which are more homological in nature. Most notable is the conjecture, due to Bondal and Orlov, that two varieties which differ by a flop have equivalent "derived categories." I will discuss how equivariant geometry sheds new light on this and related conjectures, leading to a proof in many new higher dimensional examples arising as moduli spaces. The key technique is a new theory of ``Theta-stratifications" which are analogous to equivariant Morse stratifications in the setting of algebraic geometry.

Next half-hour:

The Verlinde formula is a celebrated and classic computation of the dimension of the space of sections of certain line bundles on the moduli space of principal G-bundles on a smooth algebraic curve. I will discuss how the same method of stratification from the first part of the talk can be used to prove a version of this formula on the moduli space of Higgs bundles on a curve.

October 10

The moduli spaces of local systems on decorated surfaces enjoy many nice properties. In particular, it was shown by Fock and Goncharov that they form examples of cluster varieties, which means that they are Poisson varieties with a positive atlas of toric charts, and thus admit canonical quantizations. I will describe joint work with A. Shapiro in which we embed the quantized enveloping algebra U_q(sl_n) into the quantum character variety associated to a punctured disk with two marked points on its boundary. The construction is closely related to the (quantized) multiplicative Grothendieck-Springer resolution for SL_n. I will also explain how the R-matrix of U_q(sl_n) arises naturally in this topological setup as a (half) Dehn twist. Time permitting, I will describe some potential applications to the study of positive representations of the split real quantum group U_q(sl_n,R)

October 24

I will explain how to refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and to prove the first non-trivial cases of these conjectures. These results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder-Stevens-Varchenko and yield precise formulations for affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the speaker to relate these conjectures for U_q(sl_2 hat) to evaluations of certain theta hypergeometric integrals defined by Felder-Varchenko. We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral of Spiridonov.

This talk presents joint work with E. Rains and A. Varchenko posted
at arXiv:1610.01917

October 31

Bloch-Okounkov studied certain functions on partitions f called shifted symmetric polynomials. They showed that certain q-series arising from these functions (the so-called q-brackets <f>q) are quasimodular forms. We revisit a family of such functions, denoted Qk, and study the p-adic properties of their q-brackets. To do this, we define regularized versions Qk(p) for primes p. We also use Jacobi forms to show that the <Qk(p)>q are quasimodular and find explicit expressions for them in terms of the <Qk>q

November 7

Associated to a pair of algebras quantizing the same graded, Poisson algebra, we have a category of Harish-Chandra bimodules. These have been studied with some detail in the context of universal enveloping algebras, finite W-algebras and hypertoric enveloping algebras, among others. I will introduce this concept in the setting of rational Cherednik algebras, with an emphasis on the relationship between Harish-Chandra bimodules and category O. This relationship is more clearly seen in type A and I will focus on this case. Time permitting, I will say how things change in other types, too.

November 21

In number theory there is a method for extracting Fourier coefficients from an automorphic function, which underlies the notion of Whittaker model. Frenkel, Gaitsgory, and Vilonen carried out an analogous construction in algebraic geometry by defining categories of Whittaker sheaves on variants of Drinfeld's compactification. In this talk I will discuss the degeneration of a Whittaker sheaf as its Fourier coefficients go to zero. This degeneration can be encoded in a perverse sheaf on Drinfeld's compactification using the operation of nearby cycles. I will describe the fibers of this sheaf in terms of the Langlands dual Lie algebra, proving along the way that it is tilting with respect to the stratification by defect. I will also compute its Jordan-Holder series.

November 28

I will present a construction of the q-deformed W-algebra of type gl_r that avoids the free field realization. The upshot is to allow us to construct an action of this algebra on the K-theory of the moduli space of rank r framed sheaves, and finally to interpret the Carlsson-Okounkov Ext operator as a W-algebra intertwiner. I will briefly survey much of the physics that goes into the problem, explicitly present several constructions of W-algebras, and mention possible extensions of the problem to moduli of semistable sheaves on projective surfaces.

December 5

The construction of an effective action for many field theories requires

some form of renormalization. This talk will be on the inductive

position space renormalization procedure developed by Costello for this

purpose. We will introduce the procedure, using the example of a scalar

field theory for simplicity, and show how the procedure was clarified

for closed manifolds and extended to a class of compact manifolds with boundary

December 12

I'll show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, I'll define a q-deformation of this algebra, called cyclotomic DAHA. Namely, I'll give a q-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory. In addition, I'll explain that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, I'll apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of q-deformed quasiinvariants. This is joint work with A. Braverman and M. Finkelberg.

Seminar arxiv: Spring
2016 Fall
2015 Spring
2015 Fall
2014 Spring
2014 Fall
2013 Spring
2013 Fall
2012 Spring
2012