Department of

Mathematics


Seminar Calendar
for Algebra, Geometry and Combinatorics events the year of Monday, January 1, 2018.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    December 2017           January 2018          February 2018    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                 1  2       1  2  3  4  5  6                1  2  3
  3  4  5  6  7  8  9    7  8  9 10 11 12 13    4  5  6  7  8  9 10
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 17 18 19 20 21 22 23   21 22 23 24 25 26 27   18 19 20 21 22 23 24
 24 25 26 27 28 29 30   28 29 30 31            25 26 27 28         
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Thursday, February 1, 2018

3:00 pm in 345 Altgeld Hall,Thursday, February 1, 2018

Standard Rothe Tableaux

Neil J. Y. Fan (Sichuan University and UIUC)

Abstract: Edelman and Greene constructed a bijection between the set of standard Young tableaux and the set of balanced Young tableaux of the same shape. Fomin, Greene, Reiner and Shimozono introduced the notion of balanced Rothe tableaux of a permutation w, and established a bijection between the set of balanced Rothe tableaux of w and the set of reduced words of w. In this talk, we introduce the notion of standard Rothe tableaux of w, which are tableaux obtained by labelling the cells of the Rothe diagram of w such that each row and each column is increasing. We show that the number of standard Rothe tableaux of w is smaller than or equal to the number of balanced Rothe tableaux of w, with equality if and only if w avoids the four patterns 2413, 2431, 3142 and 4132. When w is a dominant permutation, i.e., 132-avoiding, the Rothe diagram of w is a Young diagram, so our result generalizes the result of Edelman and Greene.

Thursday, February 8, 2018

3:00 pm in 345 Altgeld Hall,Thursday, February 8, 2018

Polynomial Families from Combinatorial $K$-Theory

Cara Monical (UIUC)

Abstract: Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings. We use this combinatorial model to give $K$-theoretic extensions of Demazure atoms, key polynomials, quasisymmetric Grothendieck functions, and quasikey polynomials. We then connect these new bases to existing $K$-theoretic analogues. This is joint work with O. Pechenik and D. Searles.

Wednesday, February 21, 2018

12:00 pm in 443 Altgeld Hall,Wednesday, February 21, 2018

Bounding Betti numbers of patchworked real hypersurfaces by Hodge numbers

Kristin Shaw (MPI Leipzig)

Abstract: The Smith-Thom inequality bounds the sum of the Betti numbers of a real algebraic variety by the sum of the Betti numbers of its complexification. In this talk I will explain our proof of a conjecture of Itenberg which refines this bound for a particular class of real algebraic projective hypersurfaces in terms of the Hodge numbers of its complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is a powerful combinatorial method for constructing topological types of real algebraic varieties. To prove the bounds conjectured by Itenberg, we develop a real analogue of tropical homology and use spectral sequences to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov. Their homology theory gives the Hodge numbers of a complex projective variety from its tropicalisation. Lurking in the spectral sequences of the proof are the keys to controlling the topology of the real hypersurface produced from a patchwork. This is joint work in preparation with Arthur Renaudineau.

Thursday, March 1, 2018

3:00 pm in 2 Illini Hall,Thursday, March 1, 2018

Subword Complexes, Alternating Sign Matrices, and Prism Tableaux

Anna Weigandt (UIUC)

Abstract: Subword complexes were introduced by A. Knutson and E. Miller to study Schubert polynomials. The pipe dream formula for Schubert polynomials has a natural interpretation as a sum over the facets of a subword complex. In this talk, we will discuss a generalization of subword complexes, defined by alternating sign matrices (ASMs). The geometry of these complexes is governed by the poset structure of ASMs. Prism tableaux were introduced in joint work with A. Yong to study Schubert polynomials. There is a map from prism tableaux of a fixed shape to a generalized subword complex. Restricting to a distinguished set of prism tableaux produces a bijection with the top dimensional facets of this complex.

Thursday, March 8, 2018

3:00 pm in 322 David Kinley Hall,Thursday, March 8, 2018

Ideals of QSym, shuffle-compatibility and exterior peaks

Darij Grinberg (U. Minnesota)

Abstract: In recent work (arXiv:1706.00750), Gessel and Zhuang have introduced the concept of a shuffle-compatible permutation statistic, and shown that various "descent statistics" (e.g., the descent set, the descent number, the major index, the peak set, the peak number) are shuffle-compatible. Every such statistic leads to an ideal of the algebra QSym of quasisymmetric functions. I shall discuss one particular statistic -- the "exterior peak set" -- whose shuffle-compatibility I prove (it was left open by Gessel and Zhuang). I will then proceed to extend the notion of shuffle-compatibility to a stronger notion that distinguishes between left and right shuffles. Proving that the exterior peak set still satisfies this stronger version of shuffle-compatibility leads us through several algebraic structures on QSym: the Hopf antipode, the dendriform structure, and two further operations. (Preprint: http://www.cip.ifi.lmu.de/~grinberg/algebra/gzshuf2.pdf )

Monday, April 2, 2018

4:00 pm in 243 Altgeld Hall,Monday, April 2, 2018

Irving Reiner lectures: Lectures on Quantum Schubert Calculus I

Leonardo C. Mihalcea (Virginia Tech )

Abstract: The quantum cohomology ring of a complex projective manifold X is a deformation of the ordinary cohomology ring of X. It was defined by Kontsevich in the mid 1990’s in relation to physics and enumerative geometry. Its structure constants - the Gromov-Witten invariants - encode numbers such as how many conics pass through 3 general points in the Grassmann manifold of 2-planes in the 4-space. The quantum cohomology ring is best understood when X has many symmetries, or good combinatorial properties, and these lectures will focus to the case when X is a Grassmann manifold or a flag manifold. The subject is quite rich, and intensively studied, with connections to algebraic combinatorics, algebraic and symplectic geometry, representation theory, and integrable systems. My goal is to introduce the audience to some of the basic ideas and techniques in the subject, such as how to calculate effectively in the quantum cohomology rings, and what are the geometric ideas behind the calculations, all illustrated by examples. The lectures are intended for graduate students, in particular I am not assuming prior knowledge of quantum cohomology. I plan to include the following topics. The ‘quantum = classical’ phenomenon of Buch, Kresch and Tamvakis: how a ‘quantum’ calculation can be performed in the ‘classical’ cohomology of an auxiliary space - this leads to formulas based on Knutson and Tao’s puzzles; the technique of curve neighborhoods and the quantum Chevalley formula: what are these, and how they help to get recursive formulas for the equivariant Gromov-Witten invariants; the quantum Schubert Calculus of Grassmannians: a presentation for the quantum ring and polynomial representatives for Schubert classes; quantum K-theory: what is it, what we know, and why is everything so much harder in this case. If time permits, I may briefly mention the connection between quantum cohomology and Toda lattice (B. Kim’s theorem), and the ‘quantum=affine’ phenomenon (D. Peterson’s conjecture, proved by T. Lam and M. Shimozono).

Tuesday, April 3, 2018

4:00 pm in 243 Altgeld Hall,Tuesday, April 3, 2018

Irving Reiner lectures: Lectures on Quantum Schubert Calculus II

Leonardo C. Mihalcea (Virginia Tech )

Abstract: The quantum cohomology ring of a complex projective manifold X is a deformation of the ordinary cohomology ring of X. It was defined by Kontsevich in the mid 1990’s in relation to physics and enumerative geometry. Its structure constants - the Gromov-Witten invariants - encode numbers such as how many conics pass through 3 general points in the Grassmann manifold of 2-planes in the 4-space. The quantum cohomology ring is best understood when X has many symmetries, or good combinatorial properties, and these lectures will focus to the case when X is a Grassmann manifold or a flag manifold. The subject is quite rich, and intensively studied, with connections to algebraic combinatorics, algebraic and symplectic geometry, representation theory, and integrable systems. My goal is to introduce the audience to some of the basic ideas and techniques in the subject, such as how to calculate effectively in the quantum cohomology rings, and what are the geometric ideas behind the calculations, all illustrated by examples. The lectures are intended for graduate students, in particular I am not assuming prior knowledge of quantum cohomology. I plan to include the following topics. The ‘quantum = classical’ phenomenon of Buch, Kresch and Tamvakis: how a ‘quantum’ calculation can be performed in the ‘classical’ cohomology of an auxiliary space - this leads to formulas based on Knutson and Tao’s puzzles; the technique of curve neighborhoods and the quantum Chevalley formula: what are these, and how they help to get recursive formulas for the equivariant Gromov-Witten invariants; the quantum Schubert Calculus of Grassmannians: a presentation for the quantum ring and polynomial representatives for Schubert classes; quantum K-theory: what is it, what we know, and why is everything so much harder in this case. If time permits, I may briefly mention the connection between quantum cohomology and Toda lattice (B. Kim’s theorem), and the ‘quantum=affine’ phenomenon (D. Peterson’s conjecture, proved by T. Lam and M. Shimozono).

Wednesday, April 4, 2018

4:00 pm in 243 Altgeld Hall,Wednesday, April 4, 2018

Irving Reiner lectures: Lectures on Quantum Schubert Calculus III

Leonardo C. Mihalcea (Virginia Tech )

Abstract: The quantum cohomology ring of a complex projective manifold X is a deformation of the ordinary cohomology ring of X. It was defined by Kontsevich in the mid 1990’s in relation to physics and enumerative geometry. Its structure constants - the Gromov-Witten invariants - encode numbers such as how many conics pass through 3 general points in the Grassmann manifold of 2-planes in the 4-space. The quantum cohomology ring is best understood when X has many symmetries, or good combinatorial properties, and these lectures will focus to the case when X is a Grassmann manifold or a flag manifold. The subject is quite rich, and intensively studied, with connections to algebraic combinatorics, algebraic and symplectic geometry, representation theory, and integrable systems. My goal is to introduce the audience to some of the basic ideas and techniques in the subject, such as how to calculate effectively in the quantum cohomology rings, and what are the geometric ideas behind the calculations, all illustrated by examples. The lectures are intended for graduate students, in particular I am not assuming prior knowledge of quantum cohomology. I plan to include the following topics. The ‘quantum = classical’ phenomenon of Buch, Kresch and Tamvakis: how a ‘quantum’ calculation can be performed in the ‘classical’ cohomology of an auxiliary space - this leads to formulas based on Knutson and Tao’s puzzles; the technique of curve neighborhoods and the quantum Chevalley formula: what are these, and how they help to get recursive formulas for the equivariant Gromov-Witten invariants; the quantum Schubert Calculus of Grassmannians: a presentation for the quantum ring and polynomial representatives for Schubert classes; quantum K-theory: what is it, what we know, and why is everything so much harder in this case. If time permits, I may briefly mention the connection between quantum cohomology and Toda lattice (B. Kim’s theorem), and the ‘quantum=affine’ phenomenon (D. Peterson’s conjecture, proved by T. Lam and M. Shimozono).

Thursday, April 26, 2018

3:00 pm in 345 Altgeld Hall,Thursday, April 26, 2018

Hearing the Limiting Shape of a Hypersurface Configuration

Robert Walker (U. Michigan)

Abstract: There is a niche body of work on limiting shapes, i.e., asymptotic Newton polyhedra, of symbolic generic initial systems considered for polynomial rings in characteristic zero (e.g., by my academic sister Sarah Mayes-Tang, and separately by Dumnicki, Szemberg, Szpond, and Tutaj-Gasinska, a quartet of Polish mathematicians). In particular, in one joint paper the latter four authors compute the limiting shape for ideals defining zero-dimensional star configurations in projective space--star configurations turn out to be a steady source of a lot of interesting "ALGECOM" phenomenology. In this talk, we discuss work-in-progress to generalize their computation to the case of ideals defining zero-dimensional configurations in projective space determined by hypersurfaces of a common fixed degree. Along the way, we draw connections to a 2015 investigation (published in Transactions of the AMS in 2017) of select homological and asymptotic properties of hypersurface and matroidal configurations by Geramita, Harbourne, Migliore, and Nagel. I'll aim to close the talk by indicating how we might see--or "hear"--select asymptotic numerical invariants in the limiting shape: Waldschmidt constants, asymptotic Castelnuovo-Mumford regularity, and resurgences for homogeneous polynomial ideals

Thursday, May 3, 2018

3:00 pm in 345 Altgeld Hall,Thursday, May 3, 2018

A closed non-iterative formula for straightening fillings of Young diagrams

Reuven Hodges (Northeastern University )

Abstract: Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process, due to Alfred Young, that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. It has been a long-standing open problem to give a non-iterative, closed formula for this straightening process. In this talk I will give such a formula, as well as a simple combinatorial description of the coefficients that arise. Moreover, an interpretation of these coefficients in terms of paths in a directed graph will be explored. I will end by discussing a surprising application of this formula towards finding multiplicities of irreducible representations in certain plethysms and how this relates to Foulkes' conjecture.

Tuesday, May 29, 2018

2:00 pm in 241 Altgeld Hall,Tuesday, May 29, 2018

Wall-crossing phenomena for Newton-Okounkov bodies

Laura Escobar (UIUC)

Abstract: A Newton-Okounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes. Work of Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. As an application we show how the wall-crossing formula for the tropicalization of Gr(2, n) is an instance of our phenomenon for Newton-Okounkov bodies. Joint work with Megumi Harada.

Thursday, October 11, 2018

3:00 pm in 345 Altgeld Hall,Thursday, October 11, 2018

Schubert polynomials and flow polytopes

Avery St. Dizier   [email] (Cornell University)

Abstract: The flow polytope associated to an acyclic graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. We will discuss a family of subdivisions of certain flow polytopes and an invariant of these different subdivisions. We will then explain how this invariant leads to certain Schubert and Grothendieck polynomials. This connection implies interesting results about the Newton polytopes of these polynomials, some of which are known to hold generally. This is joint work with K. Meszaros. There will be a pre-talk on the same day at 12:30 PM in 441 Altgeld Hall.

Tuesday, October 30, 2018

12:30 pm in 441 Altgeld Hall,Tuesday, October 30, 2018

The coincidental down-degree expectations (CDE) phenomenon

Sam Hopkins   [email] (University of Minnesota)

Abstract: Let P be a finite poset. Consider two probability distributions on P: the uniform distribution; and the distribution where each element occurs with probability proportional to the number of maximal chains passing through it. Reiner, Tenner, and Yong observed that for many posets P of interest to algebraic combinatorialists, the expected Hasse diagram down-degree is the same for both these distributions. They called this the "coincidental down-degree expectations" (CDE) property for posets. I will review the history of the study of CDE posets: its origins in the algebraic geometry of curves, its connections to the tableaux/symmetric function/Schubert calculus world, etc. I will also present the "toggle perspective," which has been useful in establishing that various distributive lattices are CDE. Time permitting I will explain a forthcoming proof of a conjecture of Reiner-Tenner-Yong which says that certain weak order intervals corresponding to vexillary permutations are CDE. The proof involves expanding the "toggle perspective" to the setting of semidistributive lattices (following Barnard and Thomas-Williams).

Thursday, November 29, 2018

3:00 pm in 345 Altgeld Hall,Thursday, November 29, 2018

Orbital Varieties and Conormal Varieties

Rahul Singh   [email] (Northeastern University)

Abstract: I will start by recalling Steinberg's geometric interpretation of the Robinson-Schensted correspondence and use that as a point of introduction to orbital varieties and conormal varieties of Schubert varieties. The main focus of the talk will be the combinatorics and geometry of these varieties when the underlying Schubert variety is cominuscule. In particular, I will present some recent work constructing a resolution of singularities and a system of defining equations. Time permitting, we will also discuss some geometric results in the case where the underlying Schubert variety is a divisor. There will be a pre-talk at 12:30 PM in 441 Altgeld Hall titled "Robinson-Schensted correspondence with a geometric flavor". Abstract: We index the irreducible components of the Steinberg variety in two different ways, once by the symmetric group S_n, and once by pairs of (standard) Young tableaux. This 'naturally' recovers the Robinson-Schensted correspondence. This setup also provides the 'geometric order' on Young tableaux, which remains combinatorially mysterious in its full generality.

Monday, December 17, 2018

11:00 am in 343 Altgeld Hall,Monday, December 17, 2018

Gröbner bases for symmetric determinantal ideals

Laura Escobar   [email] (Washington University in St. Louis)

Abstract: Following techniques from Woo-Yong 2012, we give Gröbner bases for a large class of symmetric determinantal ideals arising from type C Kazhdan-Lusztig varieties. This is joint work with Alex Fink, Jenna Rajchgot and Alex Woo.