Department of


Seminar Calendar
for algebraic geometry seminar events the next 12 months of Monday, August 22, 2016.

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More information on this calendar program is available.
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 ( ). 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.

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

Hadrian Quan (UIUC Math)

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.

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

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.

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 (, 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.

Tuesday, December 6, 2016

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

To Be Announced

John Calabrese (Rice University)

Thursday, January 19, 2017

3:00 pm in 345 Altgeld Hall,Thursday, January 19, 2017

Organizational meeting

Tuesday, January 24, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, January 24, 2017

On the Behrend function and its motivic version in Donaldson-Thomas theory

Yungfeng Jiang (U Kansas Math)

Abstract: The Behrend function, introduced by K. Behrend, is a fundamental tool in the study of Donaldson-Thomas invariants. In his foundational paper K. Behrend proves that the weighted Euler characteristic of the Donaldson-Thomas moduli space weighted by the Behrend function is the Donaldson-Thomas invariants defined by R. Thomas using virtual fundamental cycles. This makes the Donaldson-Thomas invariants motivic. In this talk I will talk about the basic notion of the Behrend function and apply it to several other interesting geometries. If time permits, I will also talk about the motivic version of the Behrend function and the famous Joyce-Song formula of the Behrend function identities.

Friday, January 27, 2017

3:00 pm in 243 Altgeld Hall,Friday, January 27, 2017

Raindrop. Droptop. Symmetric functions from DAHA.

Josh Wen (UIUC Math)

Abstract: In symmetric function theory, various distinguished bases for the ring of (deformed) symmetric functions come from specifying an inner product on said ring and then performing Gram-Schmidt on the monomial symmetric functions. In the case of Jack polynomials, there is an alternative characterization as eigenfunctions for the Calogero-Sutherland operator. This operator gives a completely integrable system, hinting at some additional algebraic structure, and an investigation of this structure digs up the affine Hecke algebra. Work of Cherednik and Matsuo formalize this in terms of an isomorphism between the affine Knizhnik-Zamolodichikov (KZ) equation and the quantum many body problem. Looking at q-analogues yields a connection between the affine Hecke algebra and Macdonald polynomials by relating the quantum affine KZ equation and the Macdonald eigenvalue problem. All of this can be streamlined by circumventing the KZ equations via Cherednik's double affine Hecke algebra (DAHA). I hope to introduce various characters in this story and give a sense of why having a collection of commuting operators can be a great thing.

Tuesday, January 31, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, January 31, 2017

Kirwan surjectivity for quiver varieties

Tom Nevins (UIUC)

Abstract: Many interesting hyperkahler, or more generally holomorphic symplectic, manifolds are constructed via hyperkahler/holomorphic symplectic reduction. For such a manifold there is a “hyperkahler Kirwan map,” from the equivariant cohomology of the original manifold to the reduced space. It is a long-standing question when this map is surjective (in the Kahler rather than hyperkahler case, this has been known for decades thanks to work of Atiyah-Bott and Kirwan). I’ll describe a resolution of the question (joint work with K. McGerty) for Nakajima quiver varieties: their cohomology is generated by Chern classes of “tautological bundles.” If there is time, I will explain that this is a particular instance of a general story in noncommutative geometry. The talk will not assume prior familiarity with any of the notions above.

Friday, February 3, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 3, 2017

Syzygies and Implicitization of tensor product surfaces

Eliana Duarte (UIUC Math)

Abstract: A tensor product surface is the closure of the image of a map $\lambda:\mathbb{P}^1\times \mathbb{P}^1\to \mathbb{P}^3$. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of $\lambda$ in $\mathbb{P}^{3}$. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map $\lambda$ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of $\lambda$ are closely related to the geometry of the set of points at which $\lambda$ is undefined.

Friday, February 10, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 10, 2017

The KP-CM correspondence

Matej Penciak (UIUC Math)

Abstract: In this talk I will describe how two seemingly unrelated integrable systems have an unexpected connection. I will begin with the classical story first worked out by Airault, McKean, and Moser. I will then describe a more modern interpretation of the relation due to Ben-Zvi and Nevins.

Friday, February 17, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 17, 2017

What is a Topological Quantum Field Theory?

Lutian Zhao (UIUC Math)

Abstract: In this talk we will introduce the physicists' definition of topological quantum field theory, mainly focusing on cohomological quantum field theory introduced by Witten. We will discuss topological twisting and see what topological invariant is actually computed. If time permits, we will see how Gromov-Witten invariants are constructed by physics.

Friday, February 24, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 24, 2017

Quantum cohomology of Grassmannians and Gromov-Witten invariants

Sungwoo Nam (UIUC Math)

Abstract: As a deformation of classical cohomology ring, (small) quantum cohomology ring of Grassmannians has a nice description in terms of quantum Schubert classes and it has (3 point, genus 0) Gromov-Witten invariants as its structure constants. In this talk, we will describe how 'quantum corrections' can be made to obtain quantum Schubert calculus from classical Schubert calculus. After studying its structure, we will see that the Gromov-Witten invariants, which define ring structure of quantum cohomology of Grassmannians, are equal to the classical intersection number of two-step flag varieties. If time permits, we will discuss classical and quantum Littlewood-Richardson rule using triangular puzzles.

Tuesday, February 28, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, February 28, 2017

BPS Counts on K3 surfaces and their products with elliptic curves

Sheldon Katz (UIUC)

Abstract: In this survey talk, I begin by reviewing the string theory-based BPS spectrum computations I wrote about with Klemm and Vafa in the late 1990s. These were presented to the algebraic geometry community as a prediction for Gromov-Witten invariants. But our calculations of the BPS spectrum contained much more information than could be interpreted via algebraic geometry at that time. During the intervening years, Donaldson-Thomas invariants were introduced, used by Pandharipande and Thomas in their 2014 proof of the original KKV conjecture. It has since become apparent that the full meaning of the KKV calculations, and more recent extensions, can be mathematically interpreted via motivic Donaldson-Thomas invariants. With this understanding, we arrive at precise and deep conjectures. I conclude by surveying the more recent work of myself and others in testing and extending these physics-inspired conjectures on motivic BPS invariants.

Tuesday, March 7, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, March 7, 2017

Bernstein-Sato polynomials for maximal minors

Andras Lorincz (Purdue University)

Abstract: Initially introduced for hypersurfaces, Bernstein-Sato polynomials have been recently defined for arbitrary varieties by N. Budur, M. Mustata and M. Saito. Nevertheless, they are notoriously difficult to compute with very few explicit cases known. In this talk, after giving the necessary background, I will discuss some techniques that allow the computation of the Bernstein-Sato polynomial of the ideal of maximal minors of a generic matrix. Time permitting, I will also talk about connections to topological zeta functions and show the monodromy conjecture for this case.Initially introduced for hypersurfaces, Bernstein-Sato polynomials have been recently defined for arbitrary varieties by N. Budur, M. Mustata and M. Saito. Nevertheless, they are notoriously difficult to compute with very few explicit cases known. In this talk, after giving the necessary background, I will discuss some techniques that allow the computation of the Bernstein-Sato polynomial of the ideal of maximal minors of a generic matrix. Time permitting, I will also talk about connections to topological zeta functions and show the monodromy conjecture for this case.

Tuesday, March 14, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, March 14, 2017

Categorical Gromov-Witten Invariants

Junwu Tu (University of Missouri )

Abstract: In this talk, following Costello and Kontsevich, we describe a construction of Gromov-Witten type invariants from cyclic A-infinity categories.

Tuesday, March 28, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, March 28, 2017

A variety with non-finitely generated automorphism group

John Lesieutre (UIC)

Abstract: If X is a projective variety, then Aut(X)/Aut^0(X) is a countable group, but little is known about what groups can occur. I will construct a six-dimensional variety for which this group is not finitely generated, and discuss how the construction can adapted to give an example of a complex variety with infinitely many non-isomorphic real forms.

Tuesday, April 4, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, April 4, 2017

Rational points of generic curves and the section conjecture

Tatsunari Watanabe (Purdue University)

Abstract: The section conjecture comes from Grothendieck's anabelian philosophy where he predicts that if a variety is "anabelian", then its arithmetic fundamental group should control its geometry. In this talk, I will introduce the section conjecture and the generic curve of genus g >=4 with no marked points as an example where the conjecture holds. The primary tool used is called weighted completion of profinite groups developed by R Hain and M Matsumoto. It linearizes a profinite group such as arithmetic mapping class groups and is relatively computable since it is controlled by cohomology groups.

Friday, April 7, 2017

3:00 pm in 243 Altgeld Hall,Friday, April 7, 2017

An introduction to quantum cohomology and the quantum product

Joseph Pruitt (UIUC Math)

Abstract: The quantum cohomology ring of a variety is a q-deformation of the ordinary cohomology ring. In this talk I will define the quantum cohomology ring, discuss attempts to describe the quantum cohomology rings of toric varieties via generators and relations, and I will close with some methods to actually work with the quantum product.

Tuesday, April 11, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, April 11, 2017

Enriched Hodge Structures

Deepam Patel (Purdue University)

Abstract: It is well known the the category of mixed Hodge structures does not give the right answer when studying cycles on possibly open/singular varieties. In this talk, we will discuss how the category of mixed Hodge structures can be `enriched’ to a category appropriate for studying algebraic cycles on infinitesimal thickenings of complex analytic varieties. This is based on joint work with Madhav Nori and Vasudevan Srinivas.

Tuesday, April 18, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, April 18, 2017

Stable quotients and the B-model

Rahul Pandharipande (ETH Zurich)

Abstract: I will give an account of recent progress on stable quotient invariants, especially from the point of view of the B-model and present a geometrical derivation of the holomorphic anomaly equation for local CY cases (joint work with Hyenho Lho).

Friday, April 21, 2017

3:00 pm in 243 Altgeld Hall,Friday, April 21, 2017

Maximal tori in the symplectomorphism groups of Hirzebruch surfaces

Hadrian Quan (UIUC Math)

Abstract: In this talk, I'll discuss some beautiful results of Yael Karshon. After introducing the family of Hirzebruch surfaces, I'll highlight how certain toric actions identify these spaces with trapezoids in the complex plane. Finally, I'll describe the necessary and sufficient conditions she finds to determine when any two such surfaces are symplectomorphic. No knowledge of symplectic manifolds or toric varieties will be assumed.

Friday, May 5, 2017

3:00 pm in 243 Altgeld Hall,Friday, May 5, 2017

Complete intersections in projective space

Jin Hyung To (UIUC Math)

Abstract: We will go over complete intersection projective varieties (projective algebraic sets).