Department of

Mathematics


Seminar Calendar
for Algebraic Geometry events the next 12 months of Sunday, January 1, 2017.

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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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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 ďhyperkahler Kirwan map,Ē 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). Iíll describe a resolution of the question (joint work with K. McGerty) for Nakajima quiver varieties: their cohomology is generated by Chern classes of ďtautological bundles.Ē 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.

Wednesday, February 8, 2017

3:00 pm in 243 Altgeld Hall,Wednesday, February 8, 2017

Stability and wall-crossing in algebraic geometry

Rebecca Tramel (UIUC)

Abstract: I will discuss two notions of stability in algebraic geometry: slope stability of vector bundles on curves, and Bridgeland stability for complexes of sheaves on smooth varieties. I will try and motivate both of these definitions with questions from algebraic geometry and from physics. I will then work through a few detailed examples to show how varying our notion of stability affects the set of stable objects, and how this relates to the geometry of the space we are studying.

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.

Monday, February 27, 2017

4:00 pm in 245 Altgeld Hall,Monday, February 27, 2017

Algebra, Combinatorics, Geometry

Hal Schenck (Department of Mathematics, University of Illinois)

Abstract: I'll give an overview of the spectacular success of algebraic methods in studying problems in discrete geometry and combinatorics. First we'll discuss the face vector (number of vertices, edges, etc.) of a convex polytope and recall Euler's famous formula for polytopes of dimension 3. Then we'll discuss graded rings, focusing on polynomial rings and quotients. Associated to a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P, and gives a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture using this machinery, and also touch on connections between P and toric varieties, which are objects arising in algebraic geometry.

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 `enrichedí 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.

Thursday, April 13, 2017

2:00 pm in 241 Altgeld Hall,Thursday, April 13, 2017

L-values, Bessel moments and Mahler measures

Detchat Samart   [email] (UIUC)

Abstract: We will discuss some formulas and conjectures relating special values of L-functions associated to modular forms to moments of Bessel functions and Mahler measures. Bessel moments arise in the study of Feynman integrals, while Mahler measures have received a lot of attention from mathematicians over the past few decades due to their apparent connection with number theory, algebraic geometry, and algebraic K-theory. Though easy to verify numerically with high precision, most of these formulas turn out to be ridiculously hard to prove, and no machinery working in full generality is currently known. Some available techniques which have been used to tackle these problems will be demonstrated. Time permitting, we will present a meta conjecture of Konstevich and Zagier which gives a general framework of how one could verify these formulas using only elementary calculus.

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.

4:00 pm in 241 Altgeld Hall,Friday, April 21, 2017

Itís hard being positive: symmetric functions and Hilbert schemes

Joshua Wen (UIUC Math)

Abstract: Macdonald polynomials are a remarkable basis of $q,t$-deformed symmetric functions that have a tendency to show up various places in mathematics. One difficult problem in the theory was the Macdonald positivity conjecture, which roughly states that when the Macdonald polynomials are expanded in terms of the Schur function basis, the corresponding coefficients lie in $\mathbb{N}[q,t]$. This conjecture was proved by Haiman by studying the geometry of the Hilbert scheme of points on the plane. Iíll give some motivations and origins to Macdonald theory and the positivity conjecture and highlight how various structures in symmetric function theory are manifested in the algebraic geometry and topology of the Hilbert scheme. Also, if you like equivariant localization computations, then youíre in luck!

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).

Thursday, August 31, 2017

11:00 am in 241 Altgeld Hall,Thursday, August 31, 2017

Polynomial Roth type theorems in Finite Fields

Dong Dong (Illinois Math)

Abstract: Recently, Bourgain and Chang established a nonlinear Roth theorem in finite fields: any set (in a finite field) with not-too-small density contains many nontrivial triplets $x$, $x+y$, $x+y^2$. The key step in Bourgain-Chang's proof is a $1/10$-decay estimate of some bilinear form. We slightly improve the estimate to a $1/8$-decay (and thus a better lower bound for the density is obtained). Our method is also valid for 3-term polynomial progressions $x$, $x+P(y)$, $x+Q(y)$. Besides discrete Fourier analysis, algebraic geometry (theorems of Deligne and Katz) is used. This is a joint work with Xiaochun Li and Will Sawin.

Wednesday, September 6, 2017

4:00 pm in 141 Altgeld Hall,Wednesday, September 6, 2017

Some fun with Hilbert schemes of points on surfaces

Joshua Wen (Illinois Math)

Abstract: The Hilbert scheme of points of $X$, parametrizes zero-dimensional subschemes of $X$. These spaces can be messy in general, but in the case that $X$ is a smooth surface, the Hilbert scheme is smooth as well. Rather than being some esoteric moduli space, the Hilbert scheme in this case is something one can get to know. Iíll introduce the Nakajima-Grojnowski construction of a Heisenberg algebra action that can be used to compute its Borel-Moore homology in a somewhat surprising way. Focusing on the case where $X$ is the plane, Iíll highlight connections with symmetric function theory. The goal is to give an overview of the surprising and useful structures the Hilbert scheme has, and perhaps this semester we can either study those structures in more detail or study larger-picture explanations for why those structures exist in the first place (moduli of sheaves on surfaces, 4d-gauge theory, etc.).

Thursday, September 7, 2017

1:00 pm in Altgeld Hall 243,Thursday, September 7, 2017

Character values, combinatorics, and some p-adic representations of finite groups

Michael Geline (Northern Illinois University)

Abstract: There are many questions which remain open from Brauer's modular representation theory of finite groups, and little agreement exists over the extent to which they ought to ultimately depend on the classification of finite simple groups. In studying a classification-free approach to one such question, involving lattices over the p-adics, I was led to an elementary combinatorial problem which is interesting in its own right and somewhat amenable to analysis by means of character values. I will present this problem, the original conjecture of Brauer which gave rise to it, and my own progress in the area. If it is necessary to mention algebraic geometry, there is something of a long-shot analogy between the character values and the Weil conjectures.

Tuesday, September 12, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, September 12, 2017

To Be Announced

Chris Dodd (UIUC)

Wednesday, September 13, 2017

4:00 pm in Altgeld Hall 141,Wednesday, September 13, 2017

Factorization Algebra and Spaces

Matej Penciak (Illinois Math)

Abstract: In this talk I will introduce the notions of factorization algebras and spaces, and give an idea of where they fit into modern representation theory.

Tuesday, September 19, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, September 19, 2017

Character values, combinatorics, and some p-adic representations of finite groups

Michael Geline (Northern Illinois University)

Abstract: There are many questions which remain open from Brauer's modular representation theory of finite groups, and little agreement exists over the extent to which they ought to ultimately depend on the classification of finite simple groups. In studying a classification-free approach to one such question, involving lattices over the p-adics, I was led to an elementary combinatorial problem which is interesting in its own right and somewhat amenable to analysis by means of character values. I will present this problem, the original conjecture of Brauer which gave rise to it, and my own progress in the area. If it is necessary to mention algebraic geometry, there is something of a long-shot analogy between the character values and the Weil conjectures.

Wednesday, September 20, 2017

4:00 pm in Altgeld Hall 141,Wednesday, September 20, 2017

Higgs bundle and related "physics"

Lutian Zhao (Illinois Math)

Abstract: Higgs bundle was a math term introduced by Nigel Hitchin as a rough analogue of Higgs boson in standard model of physics. It turns out that it is deeply rooted in the N=4 super Yang-Mills world, where Kapustin and Witten realize the geometric Langlands correspondence as a special case of S-duality. In this talk. I'll introduce the history and notion of the Higgs bundle, and try to talk about some of the idea in their "proof". No knowledge of physics is assumed.

Tuesday, September 26, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, September 26, 2017

CANCELLED

Mark Penney (University of Oxford / Max Planck Institute (Bonn))

Wednesday, September 27, 2017

4:00 pm in Altgeld Hall 141,Wednesday, September 27, 2017

Introduction to Higgs Bundles and Spectral Curves

Matej Penciak (Illinois Math)

Abstract: In this talk I will define Higgs bundles, and try to motivate them as "parametrized linear algebra". Along the way I will emphasize the spectral description of Higgs bundles. By the end, the hope is to define the Hitchin integrable system which plays a central role in algebraically integrable systems.

Wednesday, October 4, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 4, 2017

Parabolic Higgs bundles

Georgios Kydonakis (Illinois Math)

Abstract: The Narasimhan and Seshadri theorem, one of the seminal first results in the study of the moduli space of vector bundles over a Riemann surface, relates degree zero, stable vector bundles on a compact Riemann surface $X$ with unitary representations of ${{\pi }_{1}}\left( X \right)$. One direction to generalize this theorem is by allowing punctures in the Riemann surface and the correspondence, which now involves parabolic bundles, was carried out by Mehta and Seshadri. The version for fundamental group representations of the punctured Riemann surface into Lie groups other than $G=\text{U}\left( n \right)$ entails introducing the notion of parabolic Higgs bundles. We will describe these holomorphic objects and see examples of those corresponding to Fuchsian representations of the fundamental group of the punctured Riemann surface.

Tuesday, October 10, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, October 10, 2017

Dynamics, geometry, and the moduli space of Riemann surfaces

Alex Wright (Stanford)

Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

3:00 pm in 243 Altgeld Hall,Tuesday, October 10, 2017

Koszul duality and characters of tilting modules

Pramod Achar (Louisiana State University)

Abstract: This talk is about the "Hecke category," a monoidal category that appears in various incarnations in geometric representation theory. I will explain some of these incarnations and their roles in solving classical problems, such as the celebrated Kazhdan-Lusztig conjectures on Lie algebra representations. These conjectures (proved in 1981) hinge on the fact that the derived category of constructible sheaves on a flag variety is equipped with an obvious monoidal action of the Hecke category on the right. It turns out that there is also a second, "hidden" action of the Hecke category on the left. The symmetry between the "hidden" left action and the "obvious" right action leads to the phenomenon known as Koszul duality. In the last part of the talk, I will discuss new results on Koszul duality with coefficients in a field of positive characteristic, with applications to characters of tilting modules for algebraic groups. This is joint work with S. Makisumi, S. Riche, and G. Williamson.

Wednesday, October 11, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 11, 2017

Hurwitz number

Hao Sun (Illinois Math)

Abstract: I will say something about Hurwitz number in algebraic geometry.

Tuesday, October 17, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, October 17, 2017

A Tate duality theorem for local Galois symbols

Evangelia Gazaki (University of Michigan)

Abstract: Let $K$ be a $p$-adic field and $M$ a finite continuous $Gal(\overline{K}/K)$-module annihilated by a positive integer $n$. Local Tate duality is a perfect duality between the Galois cohomology of $M$ and the Galois cohomology of its dual module, $Hom(M,\mu_n)$. In the special case when $M=A[n]$ is the module of the $n$-torsion points of an abelian variety, Tate has a finer result. In this case the group $H^1(K,A[n])$ has a "significant subgroup", namely there is a map $A(K)/n\rightarrow H^1(K,A[n])$ induced by the Kummer sequence on $A$. Tate showed that under the perfect pairing for $H^1$, the orthogonal complement of $A(K)/n$ is the corresponding part, $A^\star(K)/n$, that comes from the points of the dual abelian variety $A^\star$ of $A$. The goal of this talk will be to present an analogue of this classical result for $H^2$. We will see that the "significant subgroup" in this case is given by the image of a cycle map from zero cycles on abelian varieties to Galois cohomology, while the orthogonal complement under Tate duality is given by an object of integral $p$-adic Hodge theory. We will then discuss how this computation fits with the expectations of the Bloch-Beilinson conjectures for abelian varieties defined over algebraic number fields.

Wednesday, October 18, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 18, 2017

Moment maps in Algebraic and Differential Geometry

Hadrian Quan   [email] (UIUC)

Abstract: In geometry, group actions are both ubiquitous and convenient. In this talk, Iíll survey an interesting circle of ideas relating notions of stability for orbits of an action to the complex geometry of the space being acted on. Time permitting, Iíll mention how some of this story generalizes after passing from finite to infinite dimensional groups.

Tuesday, October 24, 2017

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

To Be Announced

Visu Makam (University of Michigan)

Tuesday, November 7, 2017

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

To Be Announced

Daniele Rosso (Indiana University Northwest)