Department of

# Mathematics

Seminar Calendar
for algebraic geometry events the next 6 month of Friday, July 1, 2016.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, August 23, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, August 23, 2016

#### Organizational Meeting

Thursday, August 25, 2016

3:00 pm in 441 Altgeld Hall,Thursday, August 25, 2016

#### Organizational meeting

Abstract: Besides scheduling this semesters talks, we'll need to discuss the possibility of grad students giving pretalks at the AG lunch and organizing a summer minicourse program like the do Michigan ( http://www-personal.umich.edu/~takumim/minicourses2016.html ). It'll be lit and stocked with cookies!

Tuesday, August 30, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, August 30, 2016

#### Modules over factorization spaces, and moduli spaces of parabolic G-bundles

###### Emily Cliff (Illinois Math)

Abstract: Beilinson and Drinfeld introduced the notion of factorization algebras, a geometric incarnation of the notion of a vertex algebra. An advantage of working with factorization algebras is that they admit non-linear analogues, called factorization spaces, which can be viewed as both generalizations of and ways to produce examples of factorization algebras from algebraic geometry. The resulting factorization algebras can then be studied via the geometry of the spaces from which they arise. Just as vertex algebras admit interesting categories of representations, so too do factorization algebras and factorization spaces. In this talk we will review the definitions of a factorization algebra and factorization space before introducing the notion of a module over a factorization space. As an example and an application we will construct a moduli space of principal G-bundles with parabolic structures, and discuss how it can be linearized to recover modules over the factorization algebra corresponding to the affine Lie algebra associated to a reductive group G.

Thursday, September 1, 2016

3:00 pm in 441 Altgeld Hall,Thursday, September 1, 2016

#### ADE classifications in Slodowy slices

###### Josh Wen (UIUC Math)

Abstract: Among other things, the ADE Dynkin diagrams classify: finite subgroups of $SL_2(\mathbb{C})$ (McKay correspondence); isolated surface singularities with multiplicity 2 that can be resolved by successive blowups (Kleinian/du Val singularities); and some complex simple Lie algebras (Killing-Cartan/Dynkin classfication). I'll tell you a story about how these three families are tied together in the geometry of the Grothendieck-Springer resolution and Slodowy slices.

Friday, September 2, 2016

4:00 pm in 243 Altgeld Hall,Friday, September 2, 2016

#### Polynomials for symmetric orbit closures on the flag variety

###### Benjamin Wyser   [email] (UIUC Math)

Abstract: The variety of complete flags has many interesting subvarieties. The most famous are the Schubert varieties. In 1982, Lascoux and Sch\"{u}tzenberger defined Schubert polynomials as natural representatives of their cohomology classes. These polynomials have been studied extensively using a wide range of tools from combinatorics, representation theory and algebraic geometry. In the representation theory of real Lie groups, one finds analogues of the Schubert varieties. These are the closures of orbits on the flag variety under the action of a certain symmetric subgroup. I will discuss joint work with Alexander Yong in which we compute analogues of Schubert polynomials in this setting. I will describe the computation and also discuss some combinatorial and geometric properties of our polynomials.

Tuesday, September 13, 2016

2:00 pm in 243 Altgeld Hall,Tuesday, September 13, 2016

#### Sums of Squares on projective varieties

###### Rainier Sinn (Georgia Tech)

Abstract: We will consider the question whether we can write every nonnegative quadratic form on a real projective variety as a sum of squares in its coordinate ring. In this talk, we will focus mostly on irreducible projective varieties and determine the minimal length of sum-of-squares representations on nice varieties. I will mention results from recent joint works with Greg Blekherman, Daniel Plaumann, and Cynthia Vinzant as well as Greg Blekherman and Mauricio Velasco.

Thursday, September 15, 2016

3:00 pm in 441 Altgeld Hall,Thursday, September 15, 2016

#### An introduction to D-modules and crystals

###### Emily Cliff (UIUC Math)

Abstract: We will introduce the notion of a D-module on a variety X, a generalization of the concept of a vector bundle with flat connection. We know that over a smooth manifold, a vector bundle with flat connection is equivalent to a local system, a family of vector spaces over the manifold related to each other by parallel transport along paths in the manifold. It seems hard to translate this picture back to algebraic geometry, for example because we have no good notion of paths in a variety, but luckily Grothendieck introduced the idea of crystals of sheaves: these are sheaves with parallel transport between infinitesimally close points. We will explain this definition, and sketch the proof of the equivalence between crystals and D-modules. We will consider advantages of each approach, and along the way will see naturally occurring (and not-too-scary) examples of stacks and their categories of sheaves.

Tuesday, September 20, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, September 20, 2016

#### Sums of powers of linear and quadratic forms

###### Bruce Reznick (UIUC)

Abstract: Various results on this topic will be discussed from a "first principles" approach, with lots of historical remarks. Particular attention will be paid to writing binary sextic forms as a sum of two (or three) cubes of binary quadratic forms.

Thursday, September 22, 2016

3:00 pm in 441 Altgeld Hall,Thursday, September 22, 2016

#### Intersection Homology and $L^2$ Cohomology

Abstract: Despite its name, singular homology is perhaps not the best topological tool for studying singular spaces. For a non-singular complex projective variety $X$, one has access to a host of classical results: Poincare duality, the de Rham theorem, the Hodge-Dolbeault isomorphism. For a singular variety many of these results no longer hold. One solution is intersection homology, which was developed by Goresky-MacPherson to modify singular homology in order to recover Poincare duality. In this talk we will (with lots of pictures!) motivate and introduce intersection homology and $L^2$ cohomology. Time permitting, we will discuss some open problems concerning what the de Rham theorem might look like for these new invariants.

Tuesday, September 27, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, September 27, 2016

#### The integrality conjecture and the Kac positivity conjecture

###### Ben Davison (EPFL)

Abstract: Without assuming knowledge of any incarnation of Donaldson-Thomas theory, I'll give an introduction to the categorified version of it. I'll also explain what this upgrade of DT theory has to do with proving positivity conjectures, via my favourite example: the Kac positivity conjecture (originally proved by Hausel Letellier and Villegas), stating that polynomials counting absolutely indecomposable representations of quivers over F_q have positive coefficients.

Tuesday, October 4, 2016

3:00 pm in Altgeld Hall,Tuesday, October 4, 2016

#### Categorical Plucker formula and homological projective dual

###### Conan Leung (Chinese University of Hong Kong)

Abstract: A generalised Plucker formula describes changes of intersection numbers of complex Lagrangian under Mukai flop. In a recent joint work with Jiang and Xie, we generalise this to the level of derived category of coherent sheaves.

Friday, October 7, 2016

4:00 pm in 245 Altgeld Hall,Friday, October 7, 2016

#### Topological K-Theory, the Hopf Invariant One Problem, and Real Division Algebras

###### Brian Shin (UIUC Math)

Abstract: The techniques of K-theory were introduced to the world by Grothendieck in the 1957 in the setting of algebraic geometry. These techniques can very naturally be translated into a purely topological setting, resulting in a theory called topological K-theory. In this talk, I will introduce topological K-theory and one of its earliest applications: the Hopf invariant one problem. I will then use these results to answer the question: in which dimensions is it possible to have a division algebra over the real numbers.

Tuesday, October 11, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, October 11, 2016

#### Applications of Numerical Algebraic Geometry: Bertini and Blood Coagulation

###### Francesco Pancaldi (University of Notre Dame)

Thursday, October 13, 2016

3:00 pm in 441 Altgeld Hall,Thursday, October 13, 2016

#### Construction of virtual fundamental class

###### Yun Shi (UIUC Math)

Abstract: Virtual fundamental class is an important construction for modern enumerative geometry. In this talk, I will explain the construction given by Behrend and Fantechi. I will also talk a little about its application to Donaldson-Thomas theory.

Tuesday, October 18, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, October 18, 2016

#### Rational Curves on Large Degree Hypersurfaces in Positive Characteristic

###### Matthew Woolf (UIC)

Abstract: One of the most foundational results about complex hypersurfaces is that there is a dichotomy between hypersurfaces of small degree, which are covered by rational curves, and hypersurfaces of large degree, on which rational curves are rare. Over a field of positive characteristic, however, one can construct smooth hypersurfaces of arbitrarily large degree which are unirational. In this talk, I will discuss joint work with Eric Riedl showing that nonetheless, a general hypersurface of large degree does not have many rational curves.

Thursday, October 20, 2016

3:00 pm in 243 Altgeld Hall,Thursday, October 20, 2016

#### Rees-like Algebras and the Eisenbud-Goto Conjecture

###### Jason McCullough (Rider University)

Abstract: Regularity is a measure of the computational complexity of a homogeneous ideal in a polynomial ring. There are examples in which the regularity growth is doubly exponential in terms of the degrees of the generators, but better bounds were conjectured for "nice" ideals. Together with Irena Peeva, I discovered a construction that overturns some of the conjectured bounds for "nice" ideals - including the Eisenbud-Goto conjecture. Our construction involves two new ideas that we believe will be of independent interest: Rees-like algebras and step-by-step homogenization. I'll explain the constructions and some of their consequences.

Tuesday, October 25, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, October 25, 2016

#### A strange bilinear form on the space of automorphic forms

###### Jonathan Wang (University of Chicago)

Abstract: Let F be a function field and G a reductive group over F. We define a "strange" bilinear form B on the space of K-finite smooth compactly supported functions on G(A)/G(F). For G = SL(2), the definition of B generalizes to the case where F is a number field (and this is expected to be true for any G). The definition of B relies on the constant term operator and the standard intertwining operator. This form is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. To see this, we show the relation between B and S. Schieder's geometric Bernstein asymptotics.

Thursday, October 27, 2016

3:00 pm in 441 Altgeld Hall,Thursday, October 27, 2016

#### The Gopakumar-Vafa Invariants

###### Lutian Zhao (UIUC Math)

Abstract: In 1998, Gopakumar and Vafa argued from M-theory that BPS counts (now known as Gopakumar-Vafa invariants) have the same "generating function" as the Gromov-Witten invariants. In particular, these invariants are integral, and they agree with naive curve counting in many cases. Also, it explains the contribution of multicovering and bubbling phenomena. The basic idea of this counting is to use Lefschetz decomposition on the moduli space of D-Branes to "virtually count" the number of abelian varieties. In this talk, I will discuss why it is a promising counting invariant and give some easy cases of this counting. The serious difficulty of this counting is the definition of moduli of D-Branes, which only have a satisfactory description at g=0. If time permits, I will describe some attempts by Hosono-Saito-Takahashi, Kiem-Li and Maulik-Toda on this theory.

Tuesday, November 1, 2016

2:00 pm in 241 Altgeld Hall,Tuesday, November 1, 2016

#### An adaptation of the Langlands Correspondence in the contexts of Geometry and Quantum Physics

###### Georgios Kydonakis (UIUC )

Abstract: The Langlands Program emerged in the late 1960s as a web of conjectures relating deep questions in number theory, algebraic geometry and the theory of automorphic forms. While in the case of number fields the Langlands Correspondence is still a conjecture (except for special cases), in the case of function fields it is a theorem. A geometric reformulation of the correspondence was suggested in the 1980s, which would make sense over algebraic curves defined over $\mathbb{C}$, i.e. Riemann Surfaces. More recently, a relation between Langlands duality in Mathematics and S-duality in Quantum Field Theory has been established. We will discuss elements of these adaptations through the prism of "analogy" in Mathematics, the way A. Weil described it in his famous 1940 letter from jail.

3:00 pm in 243 Altgeld Hall,Tuesday, November 1, 2016

#### An action of the cactus group on crystals

###### Iva Halacheva (University of Lancaster)

Abstract: Any Lie algebra g which is complex, finite-dimensional, and semisimple has an associated group J(g) built out of its Dynkin diagram, and known as the cactus group. Another type of objects related to g are crystals, each encoding the information of a corresponding g-representation. We describe two realizations of an action of the cactus group on any g-crystal. The first is combinatorial, via the so called Schützenberger involutions. The second is geometric, and comes from the monodromy action for a De Concini-Procesi moduli space, induced by a family of maximal commutative subalgebras in U(g).

Friday, November 4, 2016

3:00 pm in 241 Altgeld Hall,Friday, November 4, 2016

#### KN 1-parameter subgroups for representations of quivers

###### Itziar Ochoa (UIUC Math)

Abstract: Given a projective variety X with an action of a complex reductive group G, the quotient space $X/ G$ may not exist in the category of algebraic varieties. In order to fix this problem, Geometric Invariant Theory gives a construction of a $G$-invariant open subset $X^{ss}$ of X for which the algebraic quotient exists. The Kirwan-Ness (KN) stratification refines $X$ and its unique open stratum coincides with the set $X^{ss}$. When $X$ is a linear representation $G$, the semistable locus $X^{ss}\subset X$ and a KN stratification of $X \backslash X^{ss}$ are associated to a choice of a homomorphism $G\rightarrow \mathbb{G}_m$ . That is, we can write $X=X^{ss}\sqcup \bigsqcup_{\alpha\in \text{KN}}S_\alpha,$ where $S_{\alpha}$ are locally closed smooth pieces and KN indexes the 1-parameter subgroups that determine the stratification. In this situation, Nevins and McGerty give an algorithm to find the KN 1-parameter subgroups. I will ilustrate the algorithm with some examples and at the end I will focus on the case when $X$ is a representation of the cyclic quiver.

Thursday, November 10, 2016

3:00 pm in 243 Altgeld Hall,Thursday, November 10, 2016

#### Blowup algebras of rational normal scrolls

###### Alessio Sammartano (Purdue University)

Abstract: The Rees ring and the special fiber ring of a polynomial ideal $I$, also known as the blowup algebras of $I$, play an important role in commutative algebra and algebraic geometry. A central problem is to describe the defining equations of these algebras. I will discuss the solution of this problem when $I$ is the homogeneous ideal of a rational normal scroll.

Tuesday, November 15, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, November 15, 2016

#### Cartier descent and p-curvature in mixed characteristic.

###### Chris Dodd (UIUC)

Abstract: I'll review the classical notions of the title in positive characteristic, and then explain some recent progress in "lifting" these notions to mixed characteristic; with applications to p-adic differential equations.

Tuesday, November 29, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, November 29, 2016

#### Uniform Asymptotic Growth on Symbolic Powers of Ideals

###### Robert Walker (University of Michigan)

Abstract: Symbolic powers ($I^{(N)}$) in Noetherian commutative rings are mysterious objects from the perspective of an algebraist, while regular powers of ideals ($I^s$) are essentially intuitive. However, many geometers tend to like symbolic powers in the case of a radical ideal in an affine polynomial ring over an algebraically closed field in characteristic zero: the N-th symbolic power consists of polynomial functions "vanishing to order N" on the affine zero locus of that ideal. In this polynomial setting, and much more generally, a challenging problem is determining when, given a family of ideals (e.g., all prime ideals), you have a containment of type $I^{(N)} \subseteq I^s$ for all ideals in the family simultaneously. Following breakthrough results of Ein-Lazarsfeld-Smith (2001) and Hochster-Huneke (2002) for, e.g., coordinate rings of smooth affine varieties, there is a slowly growing body of "uniform linear equivalence" criteria for when, given a suitable family of ideals, these $I^{(N)} \subseteq I^s$ containments hold as long as N is bounded below by a linear function in s, whose slope is a positive integer that only depends on the structure of the variety or the ring you fancy. My thesis (arxiv.org/1510.02993, arxiv.org/1608.02320) contributes new entries to this body of criteria, using Weil divisor theory and toric algebraic geometry. After giving a "Symbolic powers for Geometers" survey, I'll shift to stating key results of my dissertation in a user-ready form, and give a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd from "Ant-Man," I hope this talk will be awesome.

Thursday, December 1, 2016

3:00 pm in 243 Altgeld Hall,Thursday, December 1, 2016

#### Random Toric Surfaces and a Threshold for Smoothness

###### Jay Yang (University of Wisconsin)

Abstract: I will present a construction of a random toric surface inspired by the construction of a random graph. With this construction we show a threshold result for smoothness of the surface. The hope is that this inspires further application of randomness to Algebraic Geometry and Commutative Algebra.

Tuesday, December 6, 2016

3:00 pm in 243 Altgeld Hall,Tuesday, December 6, 2016