
Date Time 
Location  Speaker 
Title – click for abstract 

08/19 3:00pm 
BLOC 302 
A. Conner Harvard 
Geometry and the complexity of matrix multiplication
Strassen's well known algorithm for multiplying matrices is at core a certain method to multiply 2 by 2 matrices using 7 multiplications. The data describing this method is equivalently an expression to write the structure tensor of the 2 by 2 matrix algebra as a sum of 7 decomposable tensors. Any such decomposition of an n by n matrix algebra yields a Strassen type algorithm, and Strassen showed that one essentially cannot do better than algorithms coming from such decompositions. Bini later showed all of the above remains true when we allow the decomposition to depend on a parameter and take limits.
I discuss a technique for lower bounds for this decomposition problem, border apolarity. Two key ideas to this technique are (i) to not just look at the sequence of decompositions, but the sequence of ideals of the point sets determining the decompositions and (ii) to exploit the symmetry of the tensor of interest to insist that the limiting ideal has an extremely restrictive structure. I discuss its applications to the matrix multiplication tensor and other tensors potentially useful for obtaining upper bounds via Strassen's laser method. 

08/23 4:00pm 
BLOC 302 
Haohua Deng Duke 
Completion of twoparameter period mappings and applications
The problem of completing period mappings with geometric & Hodgetheoretic meaningful boundary data has been longstanding since Griffiths' question in 1970. In this talk I will survey my recent works joint with Colleen Robles which provided an answer for any period mappings coming from a variation of polarized Hodge structures over a quasiprojective surface. I will also talk about some applications including the construction of generalized Neron models and computing generic degree of period maps. 

08/30 4:00pm 
BLOC 302 
Justin Lacini Princeton 
Syzygies of adjoint linear series on projective varieties
Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In recent joint work with Purnaprajna Bangere we give a positive answer to this question.


09/09 3:00pm 
BLOC 628 
Kent Vashaw UCLA 
An introduction to tensor triangular geometry
For even some of the smallest and most wellunderstood finite groups, classifying indecomposable representations over a field of positive characteristic is impossible. Since the development of support varieties in the 1980s, one rougher attempt to understand these categories of representations is to classify indecomposable objects up to a suitable equivalence; formally, this goal amounts to classifying the thick ideals of the category, and a full classification for finite groups was given by Benson—Carlson—Rickard. Tensortriangular geometry, initiated in the early 2000s by Paul Balmer, gives a vast generalization of these techniques, and produces a topological space, the Balmer spectrum, to any tensor triangulated category; these categories have a tensor product which behaves in a similar way to the tensor product of vector spaces, and the Balmer spectrum is analogous to the prime spectrum of a commutative ring, where the tensor product plays the role of multiplication. We will discuss some recent progress in extending the Benson—Carlson—Rickard theorem to representation categories of finitedimensional Hopf algebras, which is joint with Nakano and Yakimov. 

09/13 4:00pm 
BLOC 302 

no seminar (ams regional) 

09/16 3:00pm 
BLOC 302 
Wanlin Li Washington U. St. Louis 
Nonvanishing of Ceresa and GrossKudlaSchoen cycles
The Ceresa cycle and the Gross—Kudla—Schoen modified diagonal cycle are algebraic $1$cycle associated to a smooth algebraic curve with a chosen base point. They are algebraically trivial for a hyperelliptic curve and nontrivial for a very general complex curve of genus $\ge 3$. Given a pointed algebraic curve, there is no general method to determine whether the Ceresa and GKS cycles associated to it are rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem. 

09/23 3:00pm 
BLOC 302 
Daniel Bragg U. Utah 
A Stacky Murphy’s Law for the Stack of Curves
We show that every DeligneMumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich. 

10/07 3:00pm 
BLOC 302 
K. Ganapathy U. Michigan 
TBA 

10/11 4:00pm 
BLOC 302 
JM Landsberg 
How to find hay in a haystack (tensor version)? (colloquium style talk)
This will be a general audience talk.
The "hay in a haystack" problem is to find an explicit object that behaves like a random one
in terms of its complexity.
A tensor version of this problem is to find an explicit tensor in some tensor space that has maximal
border rank (i.e., maximal complexity), in other words the border rank of a random tensor. A stronger version is to find an explicit sequence, of
say (m,m,m) tensors as m goes to infinity, with maximal border rank. The state of the art
for this problem is embarrassing. Arora and Barak refer to lower bounds as
"Complexity theory's Waterloo". I'll report on the little that is known and try to explain why the problem
is so difficult. 

10/25 4:00pm 
BLOC 302 
Keller VandeBogert Notre Dame 
From Total Positivity to Pure Free Resolutions
Schur functors are fundamental objects sitting at the intersection of representation theory, combinatorics, and algebraic geometry. There are many ways to try to generalize such objects, but one perspective is to view the classical JacobiTrudi identity as saying that Schur functors are built "with respect to" the symmetric algebra. This leads to the following question: given an arbitrary algebra A, do there exist Schur functors "with respect to" the algebra A? In this talk, I'll make this question more precise, and discuss an answer to this question which has deep connections to both the equivariant analogues of positivity in combinatorics and BoijSoederberg theory of nonregular rings in commutative algebra. This is joint work with Steven V. Sam.


10/28 3:00pm 
BLOC 302 
J. Buczynski U. Warsaw 
Fujita vanishing, sufficiently ample line bundles, and cactus varieties
For a fixed projective scheme X, we say that a property P(L) (where L is a line bundle on X) is satisfied by sufficiently ample line bundles if there exists a line bundle M on X such that for any ample L the property P(L+M) holds. I will discuss which properties of line bundles are satisfied by the sufficiently ample line bundles  for example, can you figure out before the talk, whether a sufficiently ample line bundle must be very ample? The grandfather of such properties and a basic ingredient used to study this concept is Fujita's vanishing theorem, which is an analogue of Serre's vanishing for sufficiently ample line bundles. At the end of the talk I will define cactus varieties (an analogue of secant varieties) and sketch a proof that cactus varieties to sufficiently ample embeddings of X are (settheoretically) defined by minors of matrices with linear entries. The topic is closely related to conjectures of EisenbudKohStillman (for curves) and SidmanSmith (for any varieties). Based on a joint work with Weronika Buczyńska and Łucja Farnik. 

11/11 3:00pm 
BLOC 302 
Ian Anderson Utah State University 
TBA 

11/18 3:00pm 
BLOC 302 
Therese Wu University of Houston 
TBA 

11/22 4:00pm 
BLOC 302 
M. Forbes UIUC 
TBA
TBA 